Wikiversity talk:Main Page

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{{attention}} To request an edit to the [[Wikiversity:Page protection|protected]] Main Page, add {{tl|editprotected}} to your request. Such requests should either be obvious or uncontroversial, or be discussed to show consensus, so please do not make vague requests here. If possible, describe exactly what changes should be made so that any custodian can quickly satisfy the request.&lt;br&gt;
{{attention}} To raise general topics about [[Wikiversity]], make general suggestions about Wikiversity, to ask questions, or to talk about anything else of a general nature, use the [[Wikiversity:Colloquium|Colloquium]].&lt;br&gt;
{{attention}} To discuss the structure, appearance, etc. of the [[Wikiversity:Main Page|Main Page]], go to the [[Wikiversity:Main page learning project]] and the [[Wikiversity talk:Main page learning project|talk page for the main page learning project]].
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'''''If you wish to post something below, go ahead. It's a talk page. But you are more likely to get a response by going to the [[Wikiversity:Colloquium|Colloquium]], which is where the main talking at Wikiversity goes on! See you there.'''''

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== The Wikiversity:Main page learning project ==

The [[Wikiversity:Main page learning project]] was launched after the redesign of the main page in December 2007. The [[Wikiversity:Main page learning project]] has as its goal &quot;the promotion of responsible involvement of the Wikiversity community in an efficient, productive, open and inclusive maintenance of the Wikiversity main page as a flagship of the activity and values of the Wikiversity community&quot;. If you would like to get involved in the design of the main page, this is where to go. 

If you have general comments about the main page, but you don't especially want to get involved in the main page project, then you can also leave comments on the [[Wikiversity_talk:Main page learning project|talk page for the main page learning project]].

:I've suggested that it might be time to retire the &quot;quote of the day&quot; project and remove the quotes from the Main Page. See: [[Wikiversity talk:Main page learning project/QOTD]]. It might also be appropriate to deprecate the inactive [[Wikiversity:Main page learning project]] and archive it. Thoughts? --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 23:37, 29 November 2019 (UTC)

== add new language university ==
Now that Chinese Wikiversity is created, please add a cross-wiki link to it. --[[User:WQL|WQL]] ([[User talk:WQL|discuss]] • [[Special:Contributions/WQL|contribs]]) 12:52, 12 August 2018 (UTC)

:{{Done}} -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:29, 12 August 2018 (UTC)
::What about zulu language [[User:Lucky Shabalala|Lucky Shabalala]] ([[User talk:Lucky Shabalala|discuss]] • [[Special:Contributions/Lucky Shabalala|contribs]]) 05:57, 30 April 2025 (UTC)

== Edit request from 204.234.101.112, 14 February 2019 ==

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[[Special:Contributions/204.234.101.112|204.234.101.112]] ([[User talk:204.234.101.112|discuss]]) 21:17, 14 February 2019 (UTC)

:{{Not done}} Empty request -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 01:11, 15 February 2019 (UTC)

== Georgian (ka) wikiversity ==
PLEASE 

Help me to make Georgian (ka) wikiversity--[[User:ჯეო|ჯეო]] ([[User talk:ჯეო|discuss]] • [[Special:Contributions/ჯეო|contribs]]) 17:23, 1 March 2019 (UTC)

:{{at|ჯეო}} See https://beta.wikiversity.org/wiki/Main_Page. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:00, 1 March 2019 (UTC)

დიდი მადლობა (Didi Madloba-Thank You)!--[[User:ჯეო|ჯეო]] ([[User talk:ჯეო|discuss]] • [[Special:Contributions/ჯეო|contribs]]) 08:44, 2 March 2019 (UTC)

::Please see [[betawikiversity:Category:KA]]. That is the appropriate place to create learning pages in this language. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 14:11, 10 March 2019 (UTC)

== new langueages ==

we should admit crosing of languajes to have a better understanding--[[Special:Contributions/201.208.239.198|201.208.239.198]] ([[User talk:201.208.239.198|discuss]]) 19:34, 25 July 2019 (UTC)

:This is the English Wikiversity. See [[:es:Portada|Wikiversidad]] for Wikiversity in Spanish. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 22:39, 25 July 2019 (UTC)

== How to change an username? ==

How to change an username? --[[User:Josephina Phoebe White|Josephina Phoebe White]] ([[User talk:Josephina Phoebe White|discuss]] • [[Special:Contributions/Josephina Phoebe White|contribs]]) 07:27, 28 August 2019 (UTC)
*{{ping|Josephina Phoebe White}} You can request at [[Special:GlobalRenameRequest]] --[[User:94rain|94rain]] ([[User talk:94rain|discuss]] • [[Special:Contributions/94rain|contribs]]) 07:29, 28 August 2019 (UTC)
Thanks. --[[User:Josephina Phoebe White|Josephina Phoebe White]] ([[User talk:Josephina Phoebe White|discuss]] • [[Special:Contributions/Josephina Phoebe White|contribs]]) 07:45, 28 August 2019 (UTC)

==Religious user names allowed in Wikiversity?==

https://en.m.wikiversity.org/wiki/Wikiversity:Username

Names of religious figures such as &quot;God&quot;, &quot;Jehovah&quot;,&quot;Buddha&quot;,&quot;Jainism&quot;,&quot;Bonadea&quot;,Hinduism or &quot;Allah&quot;, which user names prohibited

Please answer for my question. This Wikiversity user name policy still alive? Religious user names are prohibited? 

:It isn't a policy, but it's a guideline for people who are wanting to register an account are recommended to follow (as per the page, which could be changed with community consensus). I see no reason for this statement to be &quot;dead&quot;. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:15, 2 September 2019 (UTC)
::: Yes: Religious user names are under hedding &quot;Inflammatory usernames&quot;, will be blocked and not allowed.

== LinkedIn ==

I insist that a Wikiversity page should be added on LinkedIn. Wikimedia has its LinkedIn page; Wikipedia, too. But not Wikiversity. I tried to show my Swedish studies but could not choose Wikiversity as the Institution. Why not? Even when it is not a &quot;granting degree&quot; Institution, is is still an Institution, right? When I contacted LinkedIn about this, they sent me the link so that I can create myself the Wikiversity page. But then there is box I must tick: &quot; I confirm I am an approved authority of this Institution to create this page&quot;, which is not the case. But I think there are many Wikiversity experts on here that woud qualify as Wikiversity Linkedin page creators. I can create the page if someone here approves, but I would need some info: # of employees, etc. --[[User:Leonardo T. Cardillo|Leonardo T. Cardillo]] ([[User talk:Leonardo T. Cardillo|discuss]] • [[Special:Contributions/Leonardo T. Cardillo|contribs]]) 23:34, 18 January 2020 (UTC)

:The information would go here [https://www.linkedin.com/company/setup/new/ Wikiversity institution] but it probably should have a bureaucrat or someone from the WMF tick &quot;I verify that I am an authorized representative of this organization and have the right to act on its behalf in the creation and management of this page. The organization and I agree to the additional terms for Pages.&quot; The number of employees (volunteers is not an option but we are unpaid) for our Wikiversity I guess could be the number of active users 201-500. The current logo is File:Wikiversity logo 2017.svg. The website can be https://en.wikiversity.org/wiki/Wikiversity:Main_Page.--[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 00:16, 19 January 2020 (UTC)

{{At|Leonardo T. Cardillo}} Wikiversity is a community. None of us gets to insist that anything happen on behalf of the community unless there is consensus to do so. This requires a discussion in the [[Wikiversity:Colloquium]] and a vote for support or lack thereof. Because this request involves an outside organization, it may also require support from the WMF.

I have some concerns at this point that your passion regarding this issue far exceeds your demonstrated commitment to either Wikiversity or the wider Wikimedia community. It might be better to let this rest for a bit and learn more about how Wikiversity functions before insisting that this be discussed. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 03:29, 19 January 2020 (UTC)

:{{At|Dave Braunschweig}}: I apologize for the use of the word &quot;insist&quot;, I have taken note to not use it anymore here to avoid distractions from the main topic of conversation. Also, I do not like you judge how much my passions should go against my level of contributions. With that being said, and for my personal learning on this environment, can someone please guide me on the very first step I should take to have a Wikiversity page created on LinkedIn? I think you mentioned something like a &quot;poll&quot;, how do I do that? --[[User:Leonardo T. Cardillo|Leonardo T. Cardillo]] ([[User talk:Leonardo T. Cardillo|discuss]] • [[Special:Contributions/Leonardo T. Cardillo|contribs]]) 04:38, 19 January 2020 (UTC)

::{{At|Leonardo T. Cardillo}} I have already guided you on the next step to take. Please read my response carefully. Then slow down and learn more about Wikiversity. We often have people come in with high passions and quick fixes that Wikiversity must make in order to improve. They're typically gone within a month and we're left having to clean up after them. That's not to suggest that this is or isn't a good idea. It is simply to point out that this is a community. You must first learn to work with the community before you try to change it. We look forward to working with you as you figure this out. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 15:31, 19 January 2020 (UTC)

:::{{At|Dave Braunschweig}} Thanks so much for your inputs. I have created this: https://en.wikiversity.org/wiki/Wikiversity:Colloquium#LinkedIn. Please indicate if that is the next step that was intended to be created. Also, please guide on the following ones. Best regards, --[[User:Leonardo T. Cardillo|Leonardo T. Cardillo]] ([[User talk:Leonardo T. Cardillo|discuss]] • [[Special:Contributions/Leonardo T. Cardillo|contribs]]) 16:27, 19 January 2020 (UTC)

== Add New Language ==

Why not bn.wikiversity?  But there is Hindi!  Make it, please.  I am ready to cooperate if needed. [[User:Hirok Raja|Hirok Raja]] ([[User talk:Hirok Raja|discuss]] • [[Special:Contributions/Hirok Raja|contribs]]) 03:07, 1 August 2020 (UTC)

:[[User:Hirok Raja|Hirok Raja]]: please see [[:betawikiversity:|Wikiversity Beta]]. &amp;mdash;Hasley&amp;nbsp;[[user talk:Hasley|&lt;span style=&quot;color: #0645AD; vertical-align: super; font-size: smaller;&quot;&gt;talk&lt;/span&gt;]] 13:04, 1 August 2020 (UTC)

:{{At|Hirok Raja}} Also see [[meta:Wikiversity]]. We are the English Wikiversity. We have no role in setting up new Wikiversity languages. When bn.wikiversity is added, please let us know, and we will add it to our main page. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:59, 1 August 2020 (UTC)

== I'm learning Turkish🤩 ==

Hi(to the person reading this)! I'm learning Turkish and I would like someone(native Turkish speaker) to teach how to pronounce Turkish. I do know some words,alphabets and number☺️ and I'm still learning and I hope someone is willing to help me🥺.

@JinahJady! [[User:JanehJody|JanehJody]] ([[User talk:JanehJody|discuss]] • [[Special:Contributions/JanehJody|contribs]]) 18:14, 4 February 2021 (UTC)

:Hi. Welcome to Wikiversity! Please see our [[Turkish|resources relating to the study of the Turkish language]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:41, 4 February 2021 (UTC)
::Hi,@[[User:JanehJody|JanehJody]] can i help you ::) [[User:MexmetW|MexmetW]] ([[User talk:MexmetW|discuss]] • [[Special:Contributions/MexmetW|contribs]]) 07:47, 28 September 2022 (UTC)
:Hi,@[[User:JanehJody|JanehJody]] I would love to help you to learning turkish :) [[Special:Contributions/85.105.185.109|85.105.185.109]] ([[User talk:85.105.185.109|discuss]]) 07:31, 28 September 2022 (UTC)

== Is it Wikipedia remodeled or a copy of wikipedia? ==

I am confused--[[User:Noukden|Noukden]] ([[User talk:Noukden|discuss]] • [[Special:Contributions/Noukden|contribs]]) 20:45, 24 May 2021 (UTC)

:{{At|Noukden}} None of the above. See [[What is Wikiversity?]] and [[What Wikiversity is not]]. Wikiversity is learning projects. Link to Wikipedia rather than duplicating it and then add hands-on activities so users can learn by doing. See [[IT Fundamentals]] for one approach. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:15, 25 May 2021 (UTC)

== Action in the earliest? ==

I want to know much more of all action that happend in the earliest centuries. [[User:Dilbkhay|Dilbkhay]] ([[User talk:Dilbkhay|discuss]] • [[Special:Contributions/Dilbkhay|contribs]]) 14:57, 21 August 2021 (UTC)

:Depending upon what you mean by &quot;earliest&quot;, have a look at [[Paleanthropology]] or [[Philosophy/Sciences]]. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:07, 20 September 2021 (UTC)

== Biology ==

What are the basic principles of ecology [[User:Aludriyo Dominic|Aludriyo Dominic]] ([[User talk:Aludriyo Dominic|discuss]] • [[Special:Contributions/Aludriyo Dominic|contribs]]) 18:25, 25 January 2022 (UTC)

:{{At|Aludriyo Dominic}} Welcome! See [[Wikipedia:Ecology]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:17, 26 January 2022 (UTC)
:{{ping|Aludriyo Dominic}} I invite you to read [[User:Atcovi/Science/Ecology]] if you're interested in learning about the basics of Ecology. Also check out the wikipedia link above and [[:Category:Ecology|this category]]. Thanks and weclome! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 03:44, 26 January 2022 (UTC)

I will try to study [[User:Aludriyo Dominic|Aludriyo Dominic]] ([[User talk:Aludriyo Dominic|discuss]] • [[Special:Contributions/Aludriyo Dominic|contribs]]) 05:41, 28 January 2022 (UTC)

== Physics ==

Physics Can Be defined as A Pure Science Subject That deals with the Measurement Of Matter In relation to energy. --{{Unsigned|Oyeyemi Abdul-warith|29 January 2022}}

: Welcome to Wikiversity! Here is a landing page that may be helpful: [[Physics]]. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 16:42, 29 January 2022 (UTC)

== Popularize ==

Can someone popularize California or the State of Washington on the Main Page? [[Special:Contributions/2604:3D08:6286:7500:B441:2710:77A4:1304|2604:3D08:6286:7500:B441:2710:77A4:1304]] ([[User talk:2604:3D08:6286:7500:B441:2710:77A4:1304|discuss]]) 03:33, 26 June 2022 (UTC)

:No, sorry, promotion isn't part of the [[Wikiversity:Mission]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 12:06, 26 June 2022 (UTC)

== [[w:Armistice of WWI|Armistice of WWI]], [[w:Paris Peace Conference|Paris Peace Conference]] and Aftermath ==

The best time to feature this on the main page was last week or yesterday;  the second best time is today.

* [[w:Template:First_World_War_treaties]]  (this template should get transcluded or copied to wikiversity, since this doesn't work: {{w:First_World_War_treaties}} although I wish it would)
* [[Wikiversity:Colloquium#Proclaiming_Armistice_of_WWI_Remembrance_and_Veterans_Day_for_11th_Nov]] our course on WWI is woefully inadequate, but this is a good time to start improving it!
[[User:Jaredscribe|Jaredscribe]] ([[User talk:Jaredscribe|discuss]] • [[Special:Contributions/Jaredscribe|contribs]]) 10:22, 12 November 2023 (UTC)

== Can you please add isiZulu plz ==

Because all othere languages her so i can umderstand batter  [[User:Lucky Shabalala|Lucky Shabalala]] ([[User talk:Lucky Shabalala|discuss]] • [[Special:Contributions/Lucky Shabalala|contribs]]) 06:06, 30 April 2025 (UTC)

:Add it how? Add more resources to learn the language? I think that would be fantastic, but it's very labor-intensive and I doubt anyone here has the competence to add that kind of material. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:40, 30 April 2025 (UTC)

== banner ==

says set learning free, propare grammer would be Start learning for free [[User:Ducklan|Ducklan]] ([[User talk:Ducklan|discuss]] • [[Special:Contributions/Ducklan|contribs]]) 20:21, 3 February 2026 (UTC)

:I'm a native American English speaker and this banner is grammatical. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:52, 4 February 2026 (UTC)
::That’s not the problem. I’m wondering if we should more clearly emphasize what Wikiversity is on this banner. Idk maybe it’s fine as it is I would just like it to be clearer[[User:Ducklan|Ducklan]] ([[User talk:Ducklan|discuss]] • [[Special:Contributions/Ducklan|contribs]]) 16:15, 4 February 2026 (UTC)
:::nevermind i just got the banner thought it was supposed to say start learning free, but its actually set learning free(like release learning) [[User:Ducklan|Ducklan]] ([[User talk:Ducklan|discuss]] • [[Special:Contributions/Ducklan|contribs]]) 16:12, 6 February 2026 (UTC)</text>
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== [[Classical guitar pedagogy]] ==

According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC)

:'''Weak delete''' This at least has &lt;em&gt;something&lt;/em&gt; that someone could use, but agreed that it's not particularly useful and not likely to be developed. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)
: '''Move''' to [[w:User:Grégory Leclair/Classical guitar pedagogy]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:18, 23 May 2026 (UTC)
: '''Delete''', the author is no longer here nor at Wikipedia, therefore it's unlikely to be developed for the foreseeable future. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:20, 10 July 2026 (UTC)
:'''Delete''', no significant improvement in nearly two decades.{{diff|Classical guitar pedagogy|2810418|140826}}--[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 19:38, 16 July 2026 (UTC)

== [[Concomitant Strabismus]] ==

Undeveloped with the author not being active on this project in over a decade. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:13, 8 July 2026 (UTC)

:'''Delete''' per above. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:22, 12 July 2026 (UTC)</text>
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      <id>2818433</id>
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      <timestamp>2026-07-16T19:53:54Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <comment>/* Concomitant Strabismus */ Deleted.</comment>
      <origin>2818433</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2355" sha1="5a1qei6qgo34e2zakle8b9ilrpafk8v" xml:space="preserve">{{/header}}
== [[Classical guitar pedagogy]] ==

According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC)

:'''Weak delete''' This at least has &lt;em&gt;something&lt;/em&gt; that someone could use, but agreed that it's not particularly useful and not likely to be developed. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)
: '''Move''' to [[w:User:Grégory Leclair/Classical guitar pedagogy]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:18, 23 May 2026 (UTC)
: '''Delete''', the author is no longer here nor at Wikipedia, therefore it's unlikely to be developed for the foreseeable future. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:20, 10 July 2026 (UTC)
:'''Delete''', no significant improvement in nearly two decades.{{diff|Classical guitar pedagogy|2810418|140826}}--[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 19:38, 16 July 2026 (UTC)

== [[Concomitant Strabismus]] ==
{{archive top|Deleted per consensus. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:53, 16 July 2026 (UTC)}}
Undeveloped with the author not being active on this project in over a decade. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:13, 8 July 2026 (UTC)

:'''Delete''' per above. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:22, 12 July 2026 (UTC)
{{archive bottom}}</text>
      <sha1>5a1qei6qgo34e2zakle8b9ilrpafk8v</sha1>
    </revision>
    <revision>
      <id>2818442</id>
      <parentid>2818433</parentid>
      <timestamp>2026-07-16T23:35:14Z</timestamp>
      <contributor>
        <username>PieWriter</username>
        <id>3039865</id>
      </contributor>
      <comment>/* Classical guitar pedagogy */</comment>
      <origin>2818442</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2552" sha1="e4m6ksoithr9menjp7b9ncy5e4849dt" xml:space="preserve">{{/header}}
== [[Classical guitar pedagogy]] ==
{{archive top|'''Deleted''', per below [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:35, 16 July 2026 (UTC)}}
According to the talk page, the author of this page intended to create this page for Wikipedia. At this moment in time (nearly 20 years later), the page is still riddled with red links and doesn't seem to fit Wikiversity's learning modules. Therefore, I propose that this page should be deleted. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:03, 19 May 2026 (UTC)

:'''Weak delete''' This at least has &lt;em&gt;something&lt;/em&gt; that someone could use, but agreed that it's not particularly useful and not likely to be developed. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:25, 20 May 2026 (UTC)
: '''Move''' to [[w:User:Grégory Leclair/Classical guitar pedagogy]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:18, 23 May 2026 (UTC)
: '''Delete''', the author is no longer here nor at Wikipedia, therefore it's unlikely to be developed for the foreseeable future. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:20, 10 July 2026 (UTC)
:'''Delete''', no significant improvement in nearly two decades.{{diff|Classical guitar pedagogy|2810418|140826}}--[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 19:38, 16 July 2026 (UTC)
{{archive bottom}}

== [[Concomitant Strabismus]] ==
{{archive top|Deleted per consensus. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:53, 16 July 2026 (UTC)}}
Undeveloped with the author not being active on this project in over a decade. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 13:13, 8 July 2026 (UTC)

:'''Delete''' per above. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:22, 12 July 2026 (UTC)
{{archive bottom}}</text>
      <sha1>e4m6ksoithr9menjp7b9ncy5e4849dt</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity talk:Bots</title>
    <ns>5</ns>
    <id>2452</id>
    <revision>
      <id>2818431</id>
      <parentid>2556822</parentid>
      <timestamp>2026-07-16T18:53:02Z</timestamp>
      <contributor>
        <username>Mu301</username>
        <id>3705</id>
      </contributor>
      <minor />
      <comment>-&gt; archive</comment>
      <origin>2818431</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="84" sha1="8ml2gbdycie33ps82tx33k84m29d2j1" xml:space="preserve">Please see [[/Archive 1]], [[/Archive 2]], and [[/Archive 3]] for prior discussions.</text>
      <sha1>8ml2gbdycie33ps82tx33k84m29d2j1</sha1>
    </revision>
  </page>
  <page>
    <title>The necessities in Numerical Methods</title>
    <ns>0</ns>
    <id>119778</id>
    <revision>
      <id>2818447</id>
      <parentid>2818149</parentid>
      <timestamp>2026-07-17T10:39:45Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Non-linear Equations */</comment>
      <origin>2818447</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2914" sha1="k825p7ti5z6l05unr9n124k0o6vo75e" xml:space="preserve">== Calculus == 
=== Numerical Differentiation ===
* Background on Differentiation ([[Media:NM.Diff.1Background.20240625.pdf |pdf]])
* Continuous Function Differentiation ([[Media:NM.Diff.1ContDiff.20241021.pdf |pdf]])
* Discrete Function Differentiation ([[Media:NM.Diff.1Discrete.20241116.pdf |pdf]])
* Forward, Backward, Central Divided Difference
* High Accuracy Differentiation
* Richardson Extrapolation
* Unequal Spaced Data Differentiation
* Numerical Differentiation with Octave
&lt;/br&gt;

=== Non-linear Equations ===
* Bisection Method ([[Media:NM.NLE.1Bisection.20241130.pdf |pdf]])
* Newton-Raphson Method ([[Media:NM.NLE.2Newton.20260713.pdf |pdf]])
* Secant Method
* False-Position Method
&lt;/br&gt;

=== Numerical Integration ===
* Trapezoidal Rule
* Simpson's 1/3 Rule
* Romberg Rule
* Gauss-Quadrature Rule
* Adaptive Quadrature
&lt;/br&gt;

=== Roots of a Nonlinear Equation ===
&lt;/br&gt;

=== Optimization ===
&lt;/br&gt;
&lt;/br&gt;

== Matrix Algebra == 

=== Simultaneous Linear Equations === 
* A system of linear equations ([[Media:SystemLinearEq.20240521.pdf |pdf]])


&lt;/br&gt;

=== Gaussian Elimination === 
&lt;/br&gt;

=== LU Decomposition === 
&lt;/br&gt;

=== Cholesky Decomposition === 
&lt;/br&gt;

=== LDL Decomposition === 
&lt;/br&gt;

=== Gauss-Seidel method === 
&lt;/br&gt;

=== Adequacy of Solutions === 
&lt;/br&gt;

=== Eigenvalue and Singular Value ===
&lt;/br&gt;

=== QRD ===
&lt;/br&gt;

=== SVD ===
&lt;/br&gt;

=== Iterative methods ===
&lt;/br&gt;
&lt;/br&gt;

== Regression ==

=== Linear Regression ===
&lt;/br&gt;

=== Non-linear Regression ===
&lt;/br&gt;

=== Linear Least Squares ===
&lt;/br&gt;
&lt;/br&gt; 


== Interpolation ==

=== Polynomial Interpolation ===
&lt;/br&gt;

=== Linear Splines ===
&lt;/br&gt;

=== Piecewise Interpolation ===
&lt;/br&gt;
&lt;/br&gt; 


== Ordinary Differential Equation ==
&lt;/br&gt;

== Partial Differential Equation ==
&lt;/br&gt;

== FEM (Finite Element Method) ==
&lt;/br&gt;
&lt;/br&gt; 


&lt;/br&gt;


== Using Symbolic Package in Octave ==

* Visit http://octave.sourceforge.net/index.html
* Download symbolic-1.0.9.tar.gz

* In Ubuntu, using the Ubuntu Software Center, I installed GiNac and CLN related software and symbolic package for Octave. But it did not properly installed.

* After extracting files from symbolic-1.0.9.tar.gz, I followed the following steps.
 ./configure 
 ./make 
 ./make INSTALL_PATH=/usr/share/octave/packages/3.2/symbolic-1.0.9 

* While doing this, I got an error message related to mkoctfile. So, I used the following command: sudo apt-get install ocatve3.2-headers. Then I was able to install the symbolic packages in the Ubuntu.

== Read some tutorials about symbolic computation ==

* Symbolic Mathematics in Matlab/GNU Octave (http://faraday.elec.uow.edu.au/subjects/annual/ECTE313/Symbolic_Maths.pdf)

* Symbolic Computations (http://www.math.ohiou.edu/courses/math344/lecture7.pdf)

[[Category:Numerical methods]]

== Using SymPy ( a Python library for symbolic mathematics) == 
&lt;/br&gt;


&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>k825p7ti5z6l05unr9n124k0o6vo75e</sha1>
    </revision>
    <revision>
      <id>2818449</id>
      <parentid>2818447</parentid>
      <timestamp>2026-07-17T10:40:56Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Non-linear Equations */</comment>
      <origin>2818449</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2914" sha1="a9sqnp6o8mwnbj1h1ii3k265r9ss26d" xml:space="preserve">== Calculus == 
=== Numerical Differentiation ===
* Background on Differentiation ([[Media:NM.Diff.1Background.20240625.pdf |pdf]])
* Continuous Function Differentiation ([[Media:NM.Diff.1ContDiff.20241021.pdf |pdf]])
* Discrete Function Differentiation ([[Media:NM.Diff.1Discrete.20241116.pdf |pdf]])
* Forward, Backward, Central Divided Difference
* High Accuracy Differentiation
* Richardson Extrapolation
* Unequal Spaced Data Differentiation
* Numerical Differentiation with Octave
&lt;/br&gt;

=== Non-linear Equations ===
* Bisection Method ([[Media:NM.NLE.1Bisection.20241130.pdf |pdf]])
* Newton-Raphson Method ([[Media:NM.NLE.2Newton.20260714.pdf |pdf]])
* Secant Method
* False-Position Method
&lt;/br&gt;

=== Numerical Integration ===
* Trapezoidal Rule
* Simpson's 1/3 Rule
* Romberg Rule
* Gauss-Quadrature Rule
* Adaptive Quadrature
&lt;/br&gt;

=== Roots of a Nonlinear Equation ===
&lt;/br&gt;

=== Optimization ===
&lt;/br&gt;
&lt;/br&gt;

== Matrix Algebra == 

=== Simultaneous Linear Equations === 
* A system of linear equations ([[Media:SystemLinearEq.20240521.pdf |pdf]])


&lt;/br&gt;

=== Gaussian Elimination === 
&lt;/br&gt;

=== LU Decomposition === 
&lt;/br&gt;

=== Cholesky Decomposition === 
&lt;/br&gt;

=== LDL Decomposition === 
&lt;/br&gt;

=== Gauss-Seidel method === 
&lt;/br&gt;

=== Adequacy of Solutions === 
&lt;/br&gt;

=== Eigenvalue and Singular Value ===
&lt;/br&gt;

=== QRD ===
&lt;/br&gt;

=== SVD ===
&lt;/br&gt;

=== Iterative methods ===
&lt;/br&gt;
&lt;/br&gt;

== Regression ==

=== Linear Regression ===
&lt;/br&gt;

=== Non-linear Regression ===
&lt;/br&gt;

=== Linear Least Squares ===
&lt;/br&gt;
&lt;/br&gt; 


== Interpolation ==

=== Polynomial Interpolation ===
&lt;/br&gt;

=== Linear Splines ===
&lt;/br&gt;

=== Piecewise Interpolation ===
&lt;/br&gt;
&lt;/br&gt; 


== Ordinary Differential Equation ==
&lt;/br&gt;

== Partial Differential Equation ==
&lt;/br&gt;

== FEM (Finite Element Method) ==
&lt;/br&gt;
&lt;/br&gt; 


&lt;/br&gt;


== Using Symbolic Package in Octave ==

* Visit http://octave.sourceforge.net/index.html
* Download symbolic-1.0.9.tar.gz

* In Ubuntu, using the Ubuntu Software Center, I installed GiNac and CLN related software and symbolic package for Octave. But it did not properly installed.

* After extracting files from symbolic-1.0.9.tar.gz, I followed the following steps.
 ./configure 
 ./make 
 ./make INSTALL_PATH=/usr/share/octave/packages/3.2/symbolic-1.0.9 

* While doing this, I got an error message related to mkoctfile. So, I used the following command: sudo apt-get install ocatve3.2-headers. Then I was able to install the symbolic packages in the Ubuntu.

== Read some tutorials about symbolic computation ==

* Symbolic Mathematics in Matlab/GNU Octave (http://faraday.elec.uow.edu.au/subjects/annual/ECTE313/Symbolic_Maths.pdf)

* Symbolic Computations (http://www.math.ohiou.edu/courses/math344/lecture7.pdf)

[[Category:Numerical methods]]

== Using SymPy ( a Python library for symbolic mathematics) == 
&lt;/br&gt;


&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>a9sqnp6o8mwnbj1h1ii3k265r9ss26d</sha1>
    </revision>
  </page>
  <page>
    <title>Understanding Arithmetic Circuits</title>
    <ns>0</ns>
    <id>139384</id>
    <revision>
      <id>2818412</id>
      <parentid>2818342</parentid>
      <timestamp>2026-07-16T14:10:45Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
      <origin>2818412</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3088" sha1="nis3f8map49pjy3s6uwmwegc8hiai4h" xml:space="preserve">
== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.2A.CLA.20260716.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260716.pdf|B]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

== Barrel Shifter ==
* Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]])
&lt;/br&gt;

'''Mux Based Barrel Shifter'''
* Analysis  ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) 
* Implementation 
&lt;/br&gt;

== Multiplier == 
=== Array Multipliers ===
* Analysis  ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]])
&lt;/br&gt;

=== Tree Mulltipliers ===
* Lattice Multiplication  ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]])
* Wallace Tree  ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]])
* Dadda Tree  ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]])
&lt;/br&gt;

=== Booth Multipliers ===
* [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]]
* Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]])
&lt;/br&gt;

== Divider == 
* Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])&lt;/br&gt;

&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>nis3f8map49pjy3s6uwmwegc8hiai4h</sha1>
    </revision>
  </page>
  <page>
    <title>Lua/Background</title>
    <ns>0</ns>
    <id>153308</id>
    <revision>
      <id>2818445</id>
      <parentid>1710927</parentid>
      <timestamp>2026-07-17T04:52:55Z</timestamp>
      <contributor>
        <username>Z. Patterson</username>
        <id>1936818</id>
      </contributor>
      <comment>/* History */ Copy edit</comment>
      <origin>2818445</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1586" sha1="11ulp8i2an81vwcccq3j1b77r7uwluw" xml:space="preserve">{{:{{BASEPAGENAME}}/Sidebar}}
== History ==
Lua was created in 1993 by Roberto Ierusalimschy, Luiz Henrique de Figueiredo, and Waldemar Celes, members of the Computer Graphics Technology Group at PUC-Rio, the Pontifical University of Rio de Janeiro, in Brazil. Versions of Lua prior to version 5.0 were released under a license similar to the BSD license. From version 5.0 onwards, Lua has been licensed under the MIT License.

Some of its closest relatives include Icon for its design and Python for its ease of use by non-programmers. In an article published in Dr. Dobb's Journal, Lua's creators also state that Lisp and Scheme with their single, ubiquitous data structure mechanism (the list) were a major influence on their decision to develop the table as the primary data structure of Lua.

Lua has been used in many commercial applications, such as ''Far Cry'', ''Garry's Mod'', ''Supreme Commander'', ''World of Warcraft'', ''Sonic the Hedgehog'' and Adobe Photoshop Lightroom, as well as non-commercial applications, such as ''Multi Theft Auto'' and ''Angband'' and its variants.

Lua can be embedded in [[W:en:MediaWiki|MediaWiki]], the software behind Wikipedia and Wikiversity, and has been enabled on Wikiversity. Thus the course material can be used directly within the Wikiversity environment.

==Characteristics==
Lua is designed to be [[W:en:Extensible_programming#Extensible_syntax|extensible]], i.e users are able to add new keywords, concepts, and structures to the source language.

Lua is:
* reflective
* imperative 
* procedural

{{subpage navbar}}
{{CourseCat}}</text>
      <sha1>11ulp8i2an81vwcccq3j1b77r7uwluw</sha1>
    </revision>
  </page>
  <page>
    <title>Complex analysis in plain view</title>
    <ns>0</ns>
    <id>171005</id>
    <revision>
      <id>2818419</id>
      <parentid>2818348</parentid>
      <timestamp>2026-07-16T14:31:24Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <origin>2818419</origin>
      <model>wikitext</model>
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      <text bytes="5296" sha1="kg37vdbeorbxqr19viq2o3nnyzhcplu" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260716.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

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    <title>Social Victorians/People/Ormonde</title>
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      <text bytes="13925" sha1="nnf46opysmq5x0nhj3nsv6ecou187bo" xml:space="preserve">== Also Known As ==
*Family name: Butler
*Marquess of Ormonde

== Overview ==
James, Marquess of Ormonde and Elizabeth, Marchioness of Ormonde were descendants of long lines of wealthy and influential people, although their wealth was not at the scale earlier members of this family had enjoyed. Lord Ormonde had close relationships with the Royal Family, particularly [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]].

== Acquaintances, Friends and Enemies ==

== Organizations ==

=== James Edward, Marquess of Ormonde ===

* Royal Yacht Squadron, member; Commodore beginning January 1901

== Timeline ==
'''1871 February 17, Friday''', an Hon. Miss Butler attended the [[Social Victorians/Timeline/1870s#Birmingham Tennis Court Club Ball|&quot;bachelors of the Tennis Court Club&quot; ball in Birmingham]].

'''1876 February 2''', Elizabeth Grosvenor and James Edward, Marquess of Ormonde married, and the wedding was reported on in the ''Morning Post''.&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|date=2024-05-16|title=Elizabeth, Marchioness of Ormonde|url=https://en.wikipedia.org/w/index.php?title=Elizabeth,_Marchioness_of_Ormonde&amp;oldid=1224156453|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Elizabeth,_Marchioness_of_Ormonde.&lt;/ref&gt;

'''1877, early''', the Marquess and Marchioness of Ormonde hosted Prince Arthur, Duke of Connaught and Strathearn at Kilkenny Castle.&lt;ref name=&quot;:6&quot; /&gt;

'''1897 April 10''', letter from Albert Edward, Prince of Wales to James, Marquess of Ormonde, on stationery marked ''R. Y. S. “Britannia.”''

'''1897 June 28, Monday''', according to the ''Morning Post'', the James, Marquis and Elizabeth, Marchioness of Ormonde were invited to the 28 June [[Social Victorians/Diamond Jubilee Garden Party|Queen's Garden Party]], the official end of the Diamond Jubilee celebrations in London.&lt;ref&gt;“The Queen’s Garden Party.” ''Morning Post'' 29 June 1897, Tuesday: 4 [of 12], Cols. 1a–7c [of 7] and 5, Col. 1a–c. ''British Newspaper Archive''  ''&lt;nowiki&gt;https://www.britishnewspaperarchive.co.uk/viewer/BL/0000174/18970629/032/0004&lt;/nowiki&gt;'' and ''&lt;nowiki&gt;https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970629/032/0005&lt;/nowiki&gt;''.&lt;/ref&gt;

'''1897 July 2, Friday''', Elizabeth, Marchioness of Ormonde and her daughters Beatrice and Constance Butler attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House. (Elizabeth, Marchioness of Ormonde is #373 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who attended]]; Beatrice Butler is #45; Constance Butler is #374.)

'''1899 April''', the Marquess and Marchioness of Ormonde hosted George, Duke and Mary, Duchess of York (later King George V and Queen Mary).&lt;ref name=&quot;:6&quot; /&gt;

'''1901 January 22''', James, Marquess of Ormonde succeeded as Commodore of the Royal Yacht Squadron when the Prince of Wales acceded to the throne.&lt;ref name=&quot;:6&quot; /&gt;

'''1901 February 19''', Lady Beatrice Butler and Sir Reginald Pole-Carew married.&lt;ref&gt;{{Cite journal|date=2023-07-12|title=Reginald Pole-Carew (British Army officer)|url=https://en.wikipedia.org/w/index.php?title=Reginald_Pole-Carew_(British_Army_officer)&amp;oldid=1165027568|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Reginald_Pole-Carew_(British_Army_officer).&lt;/ref&gt;

'''1904''', the Marquess and Marchioness of Ormonde hosted King Edward VI and Queen Alexandra at Kilkenny Castle.&lt;ref name=&quot;:6&quot; /&gt;

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==

=== Elizabeth Butler, Marchioness of Ormonde ===
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Elizabeth Butler, Marchioness of Ormonde was dressed as Guinevere,&lt;ref name=&quot;:1&quot;&gt;&quot;Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses.&quot; London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive''  http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.&lt;/ref&gt;{{rp|p. 5, Col. 7a}} and, according to the ''Times'', leading the 21-person procession of Queen Guinevere and the Knights of the Round Table of King Arthur.&lt;ref name=&quot;:0&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt; Especially if she was the leader of the procession, it is surprising that no newspaper accounts exist of her costume, the Lafayette archive does not include photographs of her in costume, and her portrait does not appear in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:42&quot;&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515 (accessed March 2020).&lt;/ref&gt; Another woman, lower ranking in the aristocracy — [[Social Victorians/People/Rodney#Lady Rodney|Corisande, Baroness Rodney]] — was also dressed as Guinevere and her husband [[Social Victorians/People/Rodney#George, Baron Rodney|George, Baron Rodney]] was dressed as King Arthur. The Rodneys do have portraits in the album, and his costume, at least, is described in the London ''Daily News''&lt;nowiki/&gt;'s report about the ball.

Elizabeth, Marchioness of Ormonde's brothers were present at the ball as well. [[Social Victorians/People/Westminster#Lord Arthur Grosvenor|Lord Arthur Hugh Grosvenor]] was dressed as King Arthur in the Queen Guinevere and the Knights of the Round Table of King Arthur procession, and [[Social Victorians/People/Westminster#Lord Gerald Grosvenor|Lord Gerald Grosvenor]] was dressed as Sir Launcelot. Perhaps, then, the ''Times'' is not wrong when it says she led the procession.

On the other hand, in his blog ''Enough of This Tomfoolery!'', woostersauce2014 says that &quot;Lady Ormonde ... ended up being unable to attend due to bereavement so Lord and Lady Rodney went as King Arthur and Queen Guinevere.&quot;&lt;ref&gt;woostersauce2014. &quot;The Devonshire House Ball (1897): Dressing Up on a Grand Scale&quot;. ''Enough of this Tomfoolery!''. 12 September 2017. https://enoughofthistomfoolery.wordpress.com/2017/09/12/the-devonshire-house-ball-1897-dressing-up-on-a-grand-scale/#comments. Retrieved 2025-09-01.&lt;/ref&gt;
[[File:Helen-Mary-Theresa-ne-Vane-Tempest-Stewart-Countess-of-Ilchester-when-Lady-Helen-Stewart-as-the-Archduchess-Marie-Christine-of-Austria.jpg|thumb|alt=Black-and-white photograph of a seated woman richly dressed in an historical costume with a white feather plume in her hair and a fan|Lady Helen Stewart, as the Archduchess Marie Christine of Austria. ©National Portrait Gallery, London.]]
=== Lady Beatrice Butler ===
The ''Times'' lists Elizabeth, Lady Ormonde and Lady Beatrice Butler together as guests of the ball in its report.&lt;ref name=&quot;:0&quot; /&gt;

Lady Beatrice Butler was one of the archduchesses — along with with 3 or 4 other young women — in [[Social Victorians/People/Londonderry#The Entourage of Maria Thérèse|the entourage of the Marchioness of Londonderry]], who led the Austrian procession as Marie Theresa, Empress of the Holy Roman Empire.&lt;ref name=&quot;:0&quot; /&gt; &lt;ref name=&quot;:4&quot; /&gt;{{rp|p. 3, Col. 3a}} These young women were present at the ball as the daughters of Maria Theresa, and the young men dressed as archdukes were present as her sons. Lady Beatrice Butler went as &quot;Archduchess Marie-Karoline in the quadrille of the Austrian Court of Marie Thérèse.&quot;&lt;ref name=&quot;:2&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;{{rp|p. 7, Col. 6b}}

The newspapers report that the archduchesses were all dressed alike, but only one photograph exists of any of these young women in costume — that of [[Social Victorians/People/Londonderry#Helen Mary Theresa Vane-Tempest-Stewart|Helen Mary Theresa Vane-Tempest-Stewart]] (which is shown, right). The newspaper descriptions of and our commentary on her fashion-forward costume are on her page, with her portrait but if, they were indeed dressed identically, then the descriptions apply to all the archduchesses.

=== Lady Constance Butler ===
[[File:ConstanceButler1903CountryLife.tif|thumb|alt=Black-and-white photograph of a serious-looking young woman in a suit, seated, with two small dogs|Constance Butler in 1903]]
Constance Butler was probably dressed as Elaine in the Round Table of King Arthur Procession.

* She is called Elaine in the Round Table procession in the ''London Daily News''.&lt;ref name=&quot;:1&quot; /&gt;{{rp|p. 5, Col. 7a}}
* &quot;Lady Constance Butler and Miss Chaplin each assumed the character of &quot;Elaine,&quot; the former in a white crêpe de chine robe flowing from a band of gold and silver embroidery at the bust, and with long angel sleeves; and the latter in white art silk under chiffon, embroidered in gold, and with a gold girdle.&quot;&lt;ref name=&quot;:4&quot;&gt;“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.&lt;/ref&gt;{{rp|p. 3, Col. 3c}}
*The ''Westminster Gazette'' says, &quot;Lady Constance Butler and Miss Chaplin were pretty as 'Elaine.'&quot;&lt;ref name=&quot;:5&quot;&gt;“The Duchess’s Costume Ball.” ''Westminster Gazette'' 03 July 1897 Saturday: 5 [of 8], Cols. 1a–3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002947/18970703/035/0005.&lt;/ref&gt;{{rp|Col. 1}}
*&quot;Lady Constance Butler and Miss Chaplin were both charming 'Elaines' — the former in a white crepe de chine robe, flowing from a band of gold and silver embroidery at the bust, and with long angel sleeves.&quot;&lt;ref name=&quot;:3&quot;&gt;&quot;The Duchess of Devonshire's Fancy Dress Ball. Special Telegram.&quot; ''Belfast News-Letter'' Saturday 03 July 1897: 5 [of 8], Col. 9c [of 9]–6, Col. 1a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0000038/18970703/015/0005.&lt;/ref&gt;{{rp|p. 5, Col. 9c–6, Col. 1a}}
*The ''Times'' report says she was Lynette in the Queen Guinevere and the Knights of the Round Table procession,&lt;ref name=&quot;:0&quot; /&gt; but it is the only newspaper that does.
The photograph (right) of Lady Constance Butler was the cover of ''Country Life'' 31 January 1903.&lt;ref&gt;Front Cover. ''Country Life''. {{Citation|title=English:  Lady Constance Butler (with two dogs), from a 1903 publication.|url=https://commons.wikimedia.org/wiki/File:ConstanceButler1903CountryLife.tif|date=1903-01-01|accessdate=2023-09-05|first=A. Nielsen|last=photographer}}. https://commons.wikimedia.org/wiki/File:ConstanceButler1903CountryLife.tif.&lt;/ref&gt;

== Notes and Questions ==

#

== Demographics ==
*Nationality: the title is in the Irish peerage.&lt;ref&gt;{{Cite journal|date=2020-09-19|title=Earl of Ormond (Ireland)|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Ormond_(Ireland)&amp;oldid=979289592|journal=Wikipedia|language=en}}&lt;/ref&gt;
=== Residences ===
*Kilkenny Castle, County Kilkenny, Ireland
*32 Upper Brook Street, London, leased from the Duke of Westminster (1881–1921)&lt;ref&gt;{{Cite journal|date=2023-03-08|title=James Butler, 3rd Marquess of Ormonde|url=https://en.wikipedia.org/w/index.php?title=James_Butler,_3rd_Marquess_of_Ormonde&amp;oldid=1143568570|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/James_Butler,_3rd_Marquess_of_Ormonde.&lt;/ref&gt;
== Family ==
*James Edward William Theobald Butler, 3rd Marquess of Ormonde (5 October 1844 – 26 October 1919)&lt;ref&gt;&quot;James Edward William Theobald Butler, 3rd Marquess of Ormonde.&quot; {{Cite web|url=http://www.thepeerage.com/p968.htm#i9679|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;
*Elizabeth Harriet Grosvenor (11 October 1856 – 25 March 1928)&lt;ref&gt;&quot;Elizabeth Harriet Grosvenor.&quot; {{Cite web|url=http://www.thepeerage.com/p968.htm#i9678|title=Person Page|website=www.thepeerage.com|access-date=2020-10-09}}&lt;/ref&gt;
#Beatrice Frances Elizabeth Butler (28 December 1876 – 29 February 1952)
#Constance Mary Butler (26 March 1879 – 20 April 1949)

== Archives and Memoirs ==

# National Library of Ireland. Collection List A 17: Ormond Family Papers. (MSS 2301-2562 and 11,044-11,073). ''XII. iii Correspondence to Ormond family from members of the Royal family''. Ms. 11,057, No. 2: &quot;Letters to James Edward 3rd Marquess of Ormond from Prince of Wales, 1884-1907, The King 1909, Duke of York 1893, German Emperor 1901, Duke of Connaught 1885-1902, Prince Henry of Prussia 1902, Princess Mary Adelaide, Duchess of Teck 1896, Grand Duke of Mecklenburg 1889, Prince Henry of Battenburg, Prince Henry of Prussia 1913.&quot;&lt;ref&gt;Collection List A 17: Ormond Family Papers. (MSS 2301-2562 and 11,044-11,073). National Library of Ireland. April 2024  https://www.nli.ie/sites/default/files/2022-12/a017_ormond.pdf.&lt;/ref&gt; (n.p. [94])
# Butler Family, Marquesses of Ormonde, deeds, family and estate papers, 12th-20th cent, NRA catalogue reference: NRA 11148 Butler; Other reference: See Annual Report 1951-2, 1973-4, 1977; Accessions to Repositories 1978. Email: info@nli.ie. Kildare Street, Dublin, Republic of Ireland 2. ARCHON code: 624. [Use National Archives of Ireland instead.]

== Notes and Questions ==

# James, 3rd Marquess of Ormonde is not mentioned in newspaper reports as having been present at the ball. Did they just miss him? Did he not go? Where was he at this time?
# Elizabeth (Harriet), Marchioness of Ormonde was a Grosvenor belonging to the family of Hugh Lupus Grosvenor, [[Social Victorians/People/Westminster|1st Duke of Westminster]].

== Footnotes ==
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      <text bytes="67562" sha1="9o37fdx9zvtkj952ac5wb6toxit8scb" xml:space="preserve">==Time Line==
[[Social Victorians/Timeline/1840s|1840s]]  [[Social Victorians/Timeline/1850s |1850s]] [[Social Victorians/Timeline/1860s | 1860s]] 1870s [[Social Victorians/Timeline/1880s | 1880s]] [[Social Victorians/Timeline/1890s | 1890s]] [[Social Victorians/Timeline/1900s|1900s]] [[Social Victorians/Timeline/1910s|1910s]] [[Social Victorians/Timeline/1920s-30s|1920s-30s]]

==1870==
&quot;Until 1870 all of the money women earned belonged to their husbands, and until 1882 their property did too, even after a divorce or separation.&quot;&lt;ref name=&quot;:4&quot; /&gt; (698 of 1203)

In 1870 Parliament debated and defeated the first bill for women's suffrage, but allowed &quot;women who owned property ... to stand for election to school boards.&quot;&lt;ref name=&quot;:4&quot; /&gt; (698–699 of 1203)

&quot;The bulk of Irish farmers did not own their land, and instead leased it from landlords, the majority of whom lived in England. In 1870, only 3 percent of agricultural holdings were occupied by owners.&quot;&lt;ref name=&quot;:4&quot; /&gt; (742 of 1203)

Dante Gabriel Rossetti and Arthur Sullivan were at the same dinner party in 1870?

Another dinner party had as guests Charles Dickens, Dante Gabriel Rossetti, John Tenniel and George Du Maurier.

January
February
March
April
May
June
July
August
September
October
November
December

==1871==
Although Queen Victoria had opened Parliament for the first time in February 1866, when people saw her for the first time in years as her open carriage made its way, she was unpopular because it seemed she was not working. Gladstone was Prime Minister.&lt;blockquote&gt;Between 1871 and 1874, eighty-five Republican Clubs were founded in Britain, protesting, among other things, the &quot;expensiveness and uselessness of the monarchy&quot; and Bertie's &quot;immoral example.&quot;&lt;ref name=&quot;:4&quot;&gt;Baird, Julia. ''Victoria the Queen, an Intimate Biography of the Woman Who Ruled an Empire''. Random House, 2016. Apple Books: https://books.apple.com/us/book/victoria-the-queen/id953835024.&lt;/ref&gt; (617 of 1203)&lt;/blockquote&gt;&quot;The 1871 Royal Commission on the Contagious Diseases Acts ... declared there was no comparison to be made between prostitutes and their clients: 'With the one sex the offence is committed as a matter of gain, with the other it is an irregular indulgence of a natural impulse.'&quot;&lt;ref name=&quot;:4&quot; /&gt; (704 of 1203)

=== January ===
Germany is united under King William I of Prussia. Julia Baird says, &quot;At the same time, Italy captured and annexed the Papal States, which had been under the direct rule of the Pope since the 700s and had lost their protector in Napoleon III.&quot;&lt;ref name=&quot;:4&quot; /&gt; (646 of 1203)

==== 4 January 1871, Wednesday ====
&lt;blockquote&gt;INVITATION BALL.
&lt;p&gt;On Wednesday evening last Major Goodman and the Officers of the 5th Dragoon Guards gave an invitation ball, which was held in the Drapers’ Hall (kindly placed at their disposal by the Drapers’ Company). The following ladies and gentlemen were amongst those who received invitations The Marquis and Marchioness of Hertford; the Earl and Countess of Aylesford; Lady A. N. Finch, Lord Guernsey, and the Hon. Mr. Finch; Lord and Lady Leigh and Miss Leigh; Lord and Lady Henley and Miss Henley, Miss Elwes, Lord and Lady Wrottealey, Lord and Lady Manners; C. N. Newdegate, Esq., M.P.; Captain, Mrs., and Miss Adams; E. Petre, Esq., and Lady Gwendoline Petre; J. Beech, Esq., Mrs. and Miss Beech, and Mr. Beech, jun.; Mr. and Mrs. Turner; Mr. and Mrs. Fetherstone Dilke, Mrs. and the Misses Fetherstone, Mr. Fetherstone, and Mr. Beaumont Fetherstone; Mr. and Mrs. P. A. Muntz; Captain and Mrs. Boultbee, of Knowle; Mr. C. M. Caldecott, Mrs. Caldecott, and the Misses Caldecott; the Rev. A. Fanshawe and Mrs. Fanshawe; Captain and Mrs. Battine; the Rev. S. C. Spencer Smith; the Rev. R. H. Baynes, M.A., vicar of St. Michael’s; the Rev. H. T. Harris, (Christ Church); General and Mr. Richmond Jones; Colonel F. Chaplin, and the Officers of the 4th Dragoon Guards, stationed at Northampton; Captain Thornelow, and the Officers of the Royal Artillery, at Weedon; the officers of the 4th Royal Regiment at Weedon; Mr. and Mrs. E. Wood; Mr. and Mrs. Herbert Wood; the Colonel and officers of the First Warwickshire Militia; Mrs. and Miss Alston, and Mr. Alston, jun., of Elmdon; Mr. and Mrs. F. Paget; Mr. and Mrs. Gulson; Captain Thomson; Captain and Mrs. Raleigh King; Mrs. Phillipson; Lord and Lady Mountgarret; the Honourable Miss Butler; Mr. and Mrs. Courtenay Lord; the Hon. Mrs. Twistleton; Mr. and the Misses Conant; Captain and Mrs. J. Marsland; Major and Mrs. Edlman; Mr. and Mrs. Astley; Mr. T. Lant, Mr. R. Lant and Mr. J. Lant, Mrs. and Miss Lant; Mr. W. T. Cavendish; Mr. and Mrs. A. Rotherham; the Marquis of Ormonde, of the first Life Guards; the Earl of Calludon, of the First Life Guards; Mrs. and the Misses Hobson; Mr P. Hobson, and Mrs. Hobson; Mr. and Mrs. Soames; Mr. and Mrs. Adderley, Sir John Rae Reid; Capt. and Mrs. Townshend, of Caldecote Hall; Lieut.-Colonel Swinfen and the Officers of the 5th Dragoon Guards stationed at Leeds; Capt. Marsden and the Officers of the 5th Dragoon Guards stationed at Birmingham; Colonel, Mrs., and Miss Bourne; Mr. and Mrs. Wyley Lord; Captain and Mrs. Thursby; Mr. and Mrs Morrice; Lieut.-Colonel Wirgman; Mr. and Mrs. J. Rotherham; [[Social Victorians/People/Abercorn|Lady Caroline Howard]]; Mr. and Mrs. Rotherham; Mr and Mrs John Sankey and the Misses Sankey; Mrs. and the Misses Murphy; Mr. Bibby (4th Hussars), Captain Gist (7th Hussars), Mr. Gregg (8th Hussars), Mr. Hamilton (7th Dragoon Guards), Colonel Rattray, Mr and Mrs. R. Boyd, &amp;c, &amp;c.&lt;/p&gt;
&lt;p&gt;The string band of the 5th Dragoon Guards, under the direction of Mr. Sidney Jones, performed the following selection of music:— Quadrille, Barbe Bleue; Valse, Marian; Galop, Bonderbryllup; Lancers, Knight of St. Patrick; Valse, Hydropaten; Galop, Flick and Flock; Quadrille, Princess of Trebizonde; Valse, the Belle of the Ball; Galop, the Fox Hunters; Valse, the Dragoon Guards; Lancers, the Gaiety; Valse, the Beautiful Danube; Valse, Wiener Kinder; Quadrille, the Fest; Galop, the Village Rose; Valse, the Geraldine; Lancers, Merry Tunes; Galop, Barbe Bleue; Valse, Various; Galop, Glorioso.&lt;ref&gt;&quot;Invitation Ball.&quot; ''Coventry Standard'' 6 January 1871, Friday: 4 [of 4], Col. 5b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000683/18710106/100/0004. Same print title, n.p.&lt;/ref&gt;&lt;/p&gt;&lt;/blockquote&gt;

=== February ===

==== Birmingham Tennis Court Club Ball ====
1871 February 17, Friday, the &quot;bachelors of the Tennis Court Club&quot; hosted a ball in Birmingham:&lt;blockquote&gt;LEAMINGTON.&lt;p&gt;

B&lt;small&gt;ACHELORS'&lt;/small&gt; B&lt;small&gt;ALL&lt;/small&gt;.&lt;p&gt;— Last night the bachelors of the Tennis Court Club gave a grand ball at the Royal Assembly Rooms, Regent Street. The ball was one of the most brilliant of the season, nearly four hundred of the ''élite'' of the town and neighbourhood having accepted the invitation of the bachelors. The ballroom was specially fitted up for the occasion, and a splendid supper was served in the adjoining rooms, where refreshments were also provided. Coote and Tiney's band was specially engaged for the occasion, and played a selection of the newest and most popular dance music. Amongst the distinguished guests present were — The High Sheriff and Mrs. J. T. Arkwright, Lady Arbuthnott, Lord and Lady Conyers, [[Social Victorians/People/Abercorn|Lady Caroline Howard]], Viscount and Viscountess Mountgarret and the Hon. Miss Butler, Sir John and Lady Blois, Sir Thomas Biddulph, the Hon. Miss Somerville, Sir William and Lady Fairfax, the Hon. Charles L. Butler, Rev. Sir John Rae, General and Mrs. Richmond Jones, Major Eldman, Major and Mrs. James Ashton, Major and Mrs. Boothby, Colonel Ruttie, Colonel Duberly, Colonel and Mrs. Machen, Colonel Rattray, Capt. and Mrs. Kennedy, Capt. W. J. Hall, Capt. Hodge, Capt. and Mrs. Morgan, Capt. and Mrs. Pearse, Capt. Roberts, Capt. Story, Mr. and Mrs. Featherstone Dilke (Maxstoke Castle) and Miss Dixie, Mr. C. M., Miss, and Miss M. A. Caldecott (Holbrooke Grange), Mr. and Mrs. J. Dugdale (Wroxhall Abbey), Mr. E. Greaves, M.P., Mr. and Mrs. C. L. Adderley (Hams Hall), and Capt. and Mrs. Hatherall. Several of the officers from the dragoons and artillery at Coventry and Birmingham were also present. The bachelors who gave the ball were twenty-eight in number.&lt;ref&gt;&quot;Leamington.&quot; &quot;District News.&quot; ''Birmingham Morning News'' 18 February 1871, Saturday: 7 [of 8, print and digital], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005826/18710218/114/0007. Print and digital title are the same.&lt;/ref&gt;&lt;/p&gt;&lt;/blockquote&gt;Another description of this same event, the Bachelors' Ball at the Leamington Spa:&lt;blockquote&gt;The bachelors’ ball at Leamington Spa, which took place on the 17th inst., was a greater success than ever. It was held as usual in the Assembly Rooms, which, by the bye, might be better adapted to such purposes. Theyare not so bad as far as the ball room goes, but to reach the supper room you have to make a pilgrimage up one of the steepest and most uncomfortable staircases ever seen; still, however difficult the journey, a safe arrival will repay one. The room was very prettily decorated, and most sumptuous fare provided. The following is a list of the bachelors who gave the ball: Mr Neville Bagot, Mr Ramsay Clarke, Mr Erasmus Galton, Mr C. H. Gregg (8th Hussars), Mr Ralph C. Gregg, Mr William Gillett, Mr Thomlinson Grant, Col. Hammond, R.A., Capt. Hull, Mr Wm. Harrison, Mr Pulsford Hobson, Mr Sydney Hobson, Mr F. C. Lister Kay, Viscount St. Lawrence, M.P., Capt. Maxwell Lyte (7th Dragoon Guards), Mr Richard Lant, Mr John Lant, Mr Oswald Milne, Mr W. W. Moore, Mr Thomas Norman, Mr Hamilton Osborne, Capt. John Paynter, Capt. Pullin, Mr George Rennie, Mr Alex. G. Stuart, Mr J. H. Sanders, Mr Edmund Vyner, Captain Vandeleur; and nothing that they could do was wanting to make it a most complete success. The frequenters of the subscription balls could scarcely recognise the rendezvous of their fortnightly meetings. A porch had been erected over the entrance in the parade, and the corridors all round the dancing room carpeted with crimson and prettily decorated. Banks of flowers had been arranged in every available corner of the ball room, and a number of mirrors hung against the wall reflected the gay scene. Coote and Tinney’s band played a charming selection, and dancing was kept up with much spirit to a late hour. The company was a large one, the toilettes exceedingly pretty. Among those present were Lord and Lady Conyers, Sir William and Lady Fairfax, [[Social Victorians/People/Abercorn|Lady Caroline Howard]], Viscount and Viscountess Mount-Garrett, [[Social Victorians/People/Ormonde|Hon. Miss Butler]], Sir John Rae Reid, Hon. Mary Somerville, &amp;c.&lt;ref&gt;&quot;Fashionable Entertainments.&quot; ''The Queen'' 25 February 1871, Saturday: 19 [of 24], Col. 3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18710225/121/0019. Print title: The Queen, ''The Lady's Newspaper'', p. 133.&lt;/ref&gt;&lt;/blockquote&gt;The ''Warwick and Warwickshire Advertiser'' has a more detailed account, especially of the people invited and attending (or not):&lt;blockquote&gt;THE BACHELORS' BALL.

This fashionable ''réunion'' of the ''élite'' of the town and neighbourhood took place the Assembly Rooms last evening The large room was beautifully decorated by Mr. Abotta, of Lower Bedford-street, who had the entire management of the preparations. Coote and Tinney's band occupied the orchestra, and played an admirable selection of first-class dance music. Mr. Wheal, of the Lower-parade, supplied the supper. The following gentlemen constituted the committee of management:— Mr. Neville Bagot, Mr. Ramsay Clarke, Mr. Erasmus Gallon, Mr. C. H. Gregg (8th Hussars), Mr. Ralph C. Gregg, Mr. W. Gillett, Mr. Thomlinson Grant, Colonel Hammond, R.A., Captain Hull, Mr. Wm. Harrison, Mr. Pulsford Hob- [Col. 5c–6a] son [Hobson], Mr. Sydney Hobson. Mr. F. C. Lister Kay. Viscount St. Lawrence, M.P., Captain Maxwell Lyte (7th Dragoon Guards), Mr. R. Lant, Mr. J. Lant, Mr. Oswald Milne, Mr. W. W. Moore, Mr. Thos. Norman, Mr. Hamilton Osborne, Captain John Paynter, Captain Pullin, Mr. George Rennie, Mr. Alexander G. Stuart, Mr. J. H. Sanders, Mr. Edmund Vyuer, and Captain Vandeleur. 

The following is a list of the company, alphabetically arranged:— Mr. Mrs. and Miss Andrew, Moseley Lodge; Major Ashton and Mr. James, 28, Lansdowne-place; Miss Ellen Andrew, Moseley Lodge; Mr. and Mrs. J. T. Arkwright, Hatton House, Hatton; Mr. and Mrs. Frank Ashton, Beech-croft, Kenilworth-road; Mr. and Mrs. Adderley, Hams HalI, Warwick; Mr. and Miss Alston, Elmdon Hall, Solihull; Mr. W. and Mrs. T. Alston, Elmdon Hall, Solihull; Mrs. and Miss Ackers, ''chez'' Mountgarrett [?], 34, Lansdowne-place; Lady Arbuthnott, Shenton Hall, Nuneston; Mr. Augustus Arkwright, Hatton House; Miss Adams, 3, Warwick-place; Mr. J. Angerstein, ''chez'' Paynter Denby Villa; Mr. Astley, Hamilton-place; Captain Arthur, George Hotel, Rugby; Sir Theophilus Biddulph, Birdingbury Hall; Mrs. and the Misses (3) Bunowes, 29, Dale-street; Captain and Mrs. Battine, Eathorpe Hall; Captain and Mrs. Charles Blundell, Dun Edin Villa; Mr. George and Miss Brodie, Rowington Vicarage; Sir John and Lady Blois, 31, Clarendon-square; Mr. and Mrs. Barlow, 15, South-parade; Honourable Charles Lennox Butler, Coton House, Rugby; Mr. and Mrs. Boultbee, Springfield, Knowle; Mr. William Blundell, Dun Edin Villa; Miss K. Browne, ''chez'' Beaver Roberts, Thorn Bank; Mr., Mrs., and Miss Beech, Brandon Lodge, Coventry; Mrs. Bame, Clarendon Hotel; Major and Mrs. Boothby, Glencairn; Mr. and Mrs. Rochfort Boyd, ''chez'' Viscountess Mountgarrett; Miss Florence Booth, Huntley Lodge; Major Butter, ''chez'' Majoribanks; Mr. and Mrs. Bowyer, 1, Clarence-crescent; Mr. Philip Bame, Denby Villa; Captain R. Bedford, Knowle Lodge, Lichfield; Mr. T. Beech, jun., Brandon Hall; Miss Boothly [sic], Glencairn; Mr. Mrs, and Miss Brown Clayton, 35, Clarendon Square; Mr.. Mrs., and Miss Chambers, Enstwood [?] Lodge; Captain C. B. Cave, 9th Lancers, Kenilworth; Miss Carles, Leam-terrace; Lord and Lady Conyers, Wellesbourne; Mr. Mrs., and Miss M. A. Caldecott, Holbrook Grange, Rugby; Mr. and Mrs. Aprice Colis, Clarendon-square; Mrs. and Fitzroy Campbell, Wellesbourne; Miss Mary Browne Clayton, Clarendon-square; Captain Stapleton Colton, Kelstone, Southampton; Dr. Collins, 6, Euston-place; Mr. Chamberlayne, Stoney Thorpe, Southam; Mr S. Corbet, Jephson Villa; Mr. J. and Mr. T. Crampton, ''chez'' Knightley, Kineton; Captain and Mrs. Chichester, R.H.A. Coventry Barracks; Mr. M. Campbell, 45, Clarendon-square; Mr. and Mrs. Duppa, 11, Upper-parade; Miss Dixie, Maxstoke Castle; Mr. Beauchamp Downall, 3, Sherbourne-place; Colonel, Mrs. and Miss Duberley, 19, Clarendon-square; Mr. S. Kevill Davies, Darlaston [?] Hall, Coventry; Mr. Paunesfort Duncombe, ''chez'' Viscountess Mountgarrett; Mr. and Mrs. Dugdale, Wroxhall Abbey; Miss Davies, ''chez'' Unett, Castle Froma [?]; Major and Mrs. Edeman, Bentinck House; Miss Edith Featherston, High-street, Warwick; Sir Wm. and Lady Fairfax, 20, Lansdowne-crescent; Captain Minabull [?] Forde, ''chez'' Unett, Castle Froma; Mrs. Fane, Newbold-terrace; Captain W. Featherstone, Warwick; Mr. Beaumont Featherston, Warwick; Mr. and Mrs. G. Greenway. Binswood Cottage; Mr. and Mrs. Newberry George, Grosvenor House; Major, Mr., and Miss Gresley [?], Meriden Lodge; Mr., Mrs., and Miss Grice, Sherbourne; Mrs. and Miss T. Grant, Clarendon-square; Mrs. Georges, Oakfields; Mr. and Mrs. Graham, Oaklands, near Birmingham; Miss Grant, Oakfield, London; Miss Gumson [?], Clarendon-square; Mr. W. Grant, 6th Regiment, ''chez'' Tomlinson Grant, Clarendon-square; Mr. Watson Gooch, Sherboume-place; Miss Grace Granville, ''chez'' Rolfe, Harvey Villa; Captain Georges, Oakfields; Mr. Edward Greaves, M.P., Avonside; Mr. and Mrs. Hunt. Kenilworth-road; Mr. Yates Hunt, Acton Villa; Captain and Mrs. Hatheral, Radford [?] House; Miss Hoey, St. Helen’s; Miss Hope, Milverton Lodge; Mr. and Mrs. Cinton [sic] Henshaw, Lansdowne Villa; Miss Hughes, Newbold-terrace; Mrs. Clement Hoey, St. Helens; Mrs. and the Misses Hobson, Beauchamp-square; Mr. J. T. Hartley, Long Castle, Shiffnal; Mr. T. Harter, The Cedars; Captain Hobson, (3rd Buffs), Avon Lodge; Mr. John Hetherington, Edstone, Henley; Mr. H. Heathfield, Newbold Comyn; [[Social Victorians/People/Abercorn|Lady Caroline Howard]], Waterloo-place; Captain Hodge, ''chez'' Hobson, Beauchamp House; Miss Hurst, ''chez'' Hobson, Beauchamp House; Capt. W. J. Hall, Junior United Service Club; Miss Alice Hartley, Tony Castle, Salop; Mr. and Miss Hodgson, Clopton, Stratford; Mr. Charles Hartley, Tony Castle, Salop; Mr. Edwin Hobson, Beauchamp House-square; Miss Holbech, ''chez'' Hacket, Binswood; Mr. and Mrs. Jeaffresen, Lansdowne-place; General and Mrs. Jones, Clarendon-square; Mr. Cove, Mrs. and Miss Jones, Loxley Hall, Warwick; Mr. Washington Jackson, ''chez'' Harter, The Cedars; Mr. James Jameson, Church-street; the Misses Johnstone, ''chez'' Pigott, Nowbold-terrace; Mrs. King Harman, Ashley Lodge; Miss Lizzie Holliday, Ashley Lodge; Miss Hetherington, Edston Hall; Mr. A. Hillyard, Southam; Mr. Edgar Hibbert, Whitley Abbey; Major Hogge, 16th Regiment, Rugby; Mr. and Mrs. Kay, Lansdowne-place; Mr. Raleigh King, Lillington; Captain and Mrs. Kennedy, 5th Dragoon Guards, Lillington; Rev. Mr. and Mrs. Knightly, Combrooke, Kineton; Mr. Kershaw, United Hotel, Charles-street, St. James; Mr. J. Maxwell Lyte, Magdalen College, Oxford; Misa C. Lyon, Bankfield; Miss Lowes, Clarendon-square; Miss S. Lowndes, Rugby; Mrs. Lockwood, St. Helen's; Mr. Webb Lindsay, Birmingham; Mr. and Mrs. Lucy, Charlecote Hall; Miss Catharine Lyon, Bankfield; Mr. R. Lancaster, Bilton Grange; Mr. T. H. Lowe, Oxford; the Misses Ley (2) Clarendon-square; Viscount and Viscountess Mountgarrett, Lansdowne-place; Hon. Miss Butler, Lansdowne-place; Mr. and Mrs. Majoribanks, Newbold Firs; Mr. and Mrs. W. H. Milne, Beauchamp-square; Mr and Mrs. Male, Euston-place; Mr. Herbert Molyneux, Tennis Court Club; Capt. and Mrs. Morgan Wellington-street; Mr. H. M. McCalmont, Grosvenor-place, London; Mr. and Miss Moore, Knightcott House, Milverton; Mr. J. M. Middleton, Clarendon-square; Mr. and Mrs. Marsland, Huntley Lodge; Mr. A. Myers, Coldstream Guards, ''chez'' Machen, Lillington Lodge; Mr. J. Middleton, Walton-place; Mr. McLeon, Binswood; Mr. MacGregor, Clarendon-square; Miss Miller, Kenilworth House; Miss Majendie, Newbold-terrace; Mr. and Mrs. Tertius Molliet, Lansdowne-circus; Colonel and Mrs. Machen, Lillington; Miss Newbie, Beechcroft; Mr. and Mrs. Philip Pewman, Warwick-road; Miss Newton, ''chez'' Unett, Castle Froma; Captain Norton, 3rd Dragoon Guards, Beauchamp-square; Dr. and Mrs. O'Callaghan, Clarendon-square; head officers of the 2nd and 5th Dragoon Guards, Leeds, Barracks; ditto, detachment of the 5th Dragoon Guards, Birmingham Barracks; ditto, ditto, Coventry Barracks; Miss Osborne, Clarendon-square; Mr. and Mrs. Osborne, Clarendon-square; Mr. and Mrs. Oldham, Castle Froma; Miss Ommancy, Warwick-place; Miss Emily Owen, and Miss Owen, Coleshill House; Mr. F. Osborne, Clarendon-square; Mr. and Mrs. Billingsley Parrey, Newbold Terrace; Mr. and Mrs. Palmer, Clarendon-square; Mr. Mrs. and Miss Paynter, Denby Villa; Mr. Mrs. and Miss Pigott, Newbold-terrace; Captain and Mrs. Pearce, ''chez'' Marjorbanks [sic], Miss and Miss L. Pritchard, Upper-parade; Miss Pixell, South-bank; Mrs. and the Misses Pullin, Waterloo-place; Miss Louisa Passy, Beauchamp-walk; Miss and Miss Ada Pennington, Thickthom, Kenilworth; Mr. Mrs. and Miss Perry, Bitham House, Avon Dassett; Mr. H. K Pullin, Junior, St. James Club; Miss Penny, Warwick-place; Miss Phillips, Clarendon-square; Mr. Pennington, Thickthorn; Miss Henrietta Passy, Beauchamp-walk; General and Mrs. Potter, Holly-walk; Mr. Mrs. and Miss Beaver Roberts, Thorn-bank; Mr. Stewart Roberts, Thorn-bank; Mr. and Mrs. Roundell, Fulham Villa; Colonel and Mrs. Ruthe, Clarence-terrace; Miss Raymond, Douglas House; Mr. Rowley Robertson, South Lodge; Mr. and Mrs. Russell, Newbold-terrace; Miss Ryland, Barford Hall; Sir John Rae Reid, Rugby; Mr. and Mrs. Worley Roberts, Oakley House; Mr. Percy Robertson and Mr. D. Robertson, Newbold-terrace; Miss Neville Rolfe, Dale-street; Colonel Clerk Rattray, Lansdowne-place; Mr. Maurice Raymond, Douglass House; Captain Roberts, Binswood; Mr. Andrew Robertson, Banbury; Miss Read, Clarendon-square; Mr. R. M. Russell, Leek Wootton; Mr. A. P. Roberts, Brazenose [?] College, Oxford; Mr., Mrs., and Miss Scholes, Zelam Lodge; Mon. Mary Somerville, Riber House: Miss Stuart, Clarendon-square; Mr. E. Sanders, Omskirk, Lancashire; Miss Palgrave Simpson, Princes Park, Liverpool; Miss Smythe, Solihull Rectory; Mr. J. F. Starkey. Stratford; Mr. and Mrs. George Stratton, Husband’s Bosworth, Rugby; Mr. Hamilton Stuart, Clarendon-square; Miss Sinclair, Dalestreet; Mr. Sedgwick, Warwick-place; Mr. Spencer Smith, Clarendon-square; Captain Starry; Miss Stallard, Warneford Villa; Mr. W. Stancombe, Magdalen College, Oxford; Miss Seymour, Warwick-road; Miss Sankey, Beauchamp-walk; Mr. Spooner, 11th Regiment, Clarendon-square; Mr. Strongitharm, Norton House; Mr. and Mrs. Molyneux Seal, Milton House; Mr. W. Sinclair. Dale-street; Mr. J. Smith, Dale-street; Mr., Mrs., and Miss Turner, Milverton Lodge; Miss Ellen Turner, ditto; Miss Tomkinson, Dale-street; Miss Thompson, Binswood; Miss Tuite, Warwick-place; Miss Temple, Newbold-terrace; Mr. Dudley Tarleton, Leam-terrace; Mis Tucker, Dale-street; Mr. and Mrs. G. Unett, Castle Froma; Mr. Gwinett, ditto; Mr. and Miss Unett, Portland-street; Mr. and Mrs. White, Beauchamp-walk; Miss Wheler, Bertie-terrace; Miss E. and Miss C. Wise, Shrublands; Mr. and Miss Wollaston, Shenton Hall, Nuneaton; Mr. E. G. Wheler, Bertie-terrace; Mr. and Miss West, Alscot Park, Stratford; Mr. and Mrs. Woodmass, Mosely Lodge; Miss Wardrope, Waterloo-place; Miss Wetherall, Woodcote; Miss Lilly and Miss Alice Wise, Cubbington Grange; Mrs. and Miss Wright, Lansdowne-crescent; Mr. H. White, Ashfield House; Miss Wakefield, Castle Froma, Mr. Herbert Wood, Newbold Revel; Mr. Young, Whitnash Rectory. 

Invitations were also sent to the following but declined for family and other reasons:— Lord and Lady Leigh and Miss Leighs (2); Mrs. General Hall, the Misses Collinson, Mr., Mrs. and Miss Hobson, Avon Lodge; Lieut-Colonel and Mrs. Fiennes; Captain and Mrs. Gregg; Captain and Mrs. Vaughton; Mrs. Frederick Gubbins, Mr. J. P. and Mrs. Gubbins; Mr. Stuart; Miss Maconehy; the Misses Staunton; Miss Galton; Mr. Raleigh King; Mr. Edward Wheler; Miss Miller; Mr. Jennings; Dr. and Mrs. Jephson; Dr. and Mrs. Thomson; Mr. and Mrs. Philpot; Mr. H. and the Misses Baker; Mr. R. Read; Sir Robert and Lady Hamilton; Mr. H. C. and Mr. G. Wise; Colonel and Miss Daniel; Mr. and Mrs. Bigland; Mr. and Mrs. Robertson; Miss Stevenson; Mrs. Osborne; Miss Harter; Mr. and Mrs. Lister Kay; Mr. and Mrs. Henry Chance; Mr., Mrs. and the Misses Bradshaw; Lady Eardly; Miss Stevenson; Captain Turquand; Major Paynter; Captain Tomkinson; Lady Hampson; Mr. Bame; Mr. Augustus Wise; Mr. and Mrs. John Mordaunt; Lady Willoughby de Broke; Mr. Caldecott; Mr. Hamilton and Miss Story; Miss Mabel Hurst; Mr. and Mrs. Bolton King; Miss Kate Fetherston; Sir Charles Mordaunt, Lord and Lady Willoughby de Broke; Miss Rigby; Mrs. and Miss Wise, Woodcote; Mr. E. Wheler, Mr., Mrs. and Miss Pennington, Westfield; Mr., Mrs. and Miss Mackenzie; Miss Wilkins; Major Lee, Mr. and Mrs. Mark Hammond, Miss P. Hughes; Lord and Lady James Murray; Mr. and Mrs. Barker, Mr. and Mrs. James West, Miss Hackett, Mr. and Miss Walker, Mrs. Harman King, Mr. Clement Hoey, Mr. Herbert Wood, Mr. Thomas Lant, the Earl of Howth, Mr. Robertson, Mr. Bookeley and Mr. E. Steward.&lt;ref&gt;&quot;The Bachelors' Ball.&quot; ''Warwick and Warwickshire Advertiser'' 18 February 1871, Saturday: 2 [of 6], Cols. 5c–6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001670/18710218/051/0002. Print title: ''Warwick and Warwickshire Advertiser and Leamington Gazette'', n.p.&lt;/ref&gt; &lt;/blockquote&gt;

=== March ===

=== April ===

==== 18 April 1871 ====
&lt;blockquote&gt;Karl Marx “was commissioned by the General Council of the International to write a pamphlet about the Paris [377–378] Commune.&quot;&lt;ref name=&quot;:3&quot;&gt;Smee, Sebastian. ''Paris in Ruins: Love, War, and the Birth of Impressionism''. W. W. Norton, 2024.&lt;/ref&gt;{{rp|377–378 of 667}}&lt;/blockquote&gt;

===May===

==== 9 May 1871, Tuesday, Queen's Drawing-Room ====
&lt;blockquote&gt;THE QUEEN'S DRAWING-ROOM.

The Queen held a Drawing-room at Buckingham Palace on Tuesday afternoon. The Priuce of Wales, Prince Arthur, Prince Leopold, and Princess Beatrice were present. Her Majesty, accompanied by the Prince of Wales and the other members of the royal family, entered the Throne Room shortly after three o'clock. The Queen wore a black moire antique dress with a train, long white tulle veil with a coronet of diamonds. Her Majesty also wore a necklace of diamonds and amethysts, the Riband and Star of the Order of the Garter, the Orders of Victoria and Albert and Louise of Prussia, and the Saxe Coburg and Gotha Family Order. Princess Beatrice wore a dress of white tulle over a rich white silk petticoat looped up with lilies of the valley and apple blossom; ornaments — pearls and diamonds.

The presentations to Her Majesty were about 280 in number, and included the following:— Mrs Atlay, by the Countess Grey; Miss Backhouse, by her mother, Mrs Backhouse; Miss Charlesworth, by her aunt, Frances Lady Hawke; Miss Backhouse Fox, by her aunt, Mrs Backhouse; [[Social Victorians/People/Abercorn|Lady Caroline Howard]], by her mother, [[Social Victorians/People/Abercorn|the Hon. Mrs Howard]]; the Hon. Gwendoline Fitz-Alan Howard, by the Duchess of Sutherland; [[Social Victorians/People/Abercorn|Lady Alice Howard]], by her mother, Hon. Mrs Howard; [[Social Victorians/People/Abercorn|Lady Louisa Howard]], by her mother, Hon. Mrs Howard; Miss Howard (of Corby), by the Hon. Mrs Philip Stourton; Miss Agnes Howard (of Corby), by the Hon. Mrs Philip Stourton; Sir Henry Ingilby, Bart., by Earl Russell; Mrs Frank Lascelles, by Lady Edward Cavendish; Mrs Gerald Liddell, marriage, by the Countess of Normanby.&lt;ref&gt;&quot;Court and Official News.&quot; ''Yorkshire Post and Leeds Intelligencer'' 11 May 1871, Thursday: 3 [of 4], Col. 4c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000686/18710511/074/0003. Same print title and p.n.&lt;/ref&gt;&lt;/blockquote&gt;

==== 24 May 1871, Wednesday: Derby Day ====
Baron Rothschild's Favonius won. The Prince of Wales attended.

==== 25 May 1871, Thursday, Dinner Party Hosted by Mr. and Mrs. Charltons ====
&lt;blockquote&gt;Mr. and Mrs. Charlton, of Hesleyside, entertained at dinner, on Thursday evening, at 47, Princesgate — his Excellency the Spanish Minister, Count de Beaufort Spontin, Lord and Lady Houghton and the Hon. Miss Milnes, Lord and Lady Acton, the Hon. Lady Williamson, [[Social Victorians/People/Abercorn|Lady Caroline Howard]], Mrs. and Miss Milner Gibson, Viscount Burke, Lord Beaumont, Lord Campbell, the Master of Herries, Major Fife, &amp;c.&lt;ref&gt;&quot;Fashionable World.&quot; ''Morning Post'' 27 May 1871, Saturday: 5 [of 8], Col. 6c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18710527/019/0005. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;June
July
August
September
===October===
'''October 1871'''&lt;blockquote&gt;At Londesborough Lodge near Scarborough, where Lady Londesborough gave a royal house party in October 1871, not only [ 41/42 ] were the bathrooms few but the drains seeped into the drinking water. Several guests, including the Prince [of Wales] and his groom and Lord Chesterfield, contracted typhoid fever. When Chesterfield and the groom died, the doctors abandoned hope for the Prince.&lt;ref name=&quot;:1&quot;&gt;Leslie, Anita. ''The Marlborough House Set''. New York: Doubleday, 1973. Print.&lt;/ref&gt;{{rp|41–42}}&lt;/blockquote&gt;

The Prince of Wales recovered on 14 December 1871.

November
December

==1872==
January
February
March
April
===May===
'''29 May 1872, Wednesday''': Derby Day

June
July
===August===
'''August 1872''': The &quot;dance on the cruiser Ariadne&quot; probably occurred in August 1872:&lt;blockquote&gt;When his [the Prince of Wales'] brother, the Duke of Edinburgh, married the attractive Grand Duchess Marie, daughter of Tsar Alexander II of Russia, her family made a fuss because she was not granted precedence above the Princess of Wales. Albert Edward soothed ruffled feelings by inviting the Tsarevitch and his wife Marie Feodorovna (who was Alexandra's sister) to stay for two months and be entertained at Cowes. ...&lt;p&gt;&lt;/p&gt;

... At the dance on the cruiser Ariadne which the Prince gave in honour of the Tsarevitch and his Grand Duchess,&quot; Lord Randolph Churchill met the 19-year-old &quot;Miss Jennie Jerome of New York.&quot;&lt;ref name=&quot;:1&quot; /&gt;{{rp|42–43}}&lt;/blockquote&gt;

September
October
November
December

==1873==

=== January ===

==== 13 January 1873, Monday ====

==== Ball at the Chief Secretary's Lodge ====
On Tuesday, 14 January 1873, the Dublin Evening Telegraph reported that the Marquis of Hartington's ball had taken place the evening before.&lt;blockquote&gt;The Marquis of Hartington gave a ball last evening at the Chief Secretary's Lodge, to their Excellencies the Lord Lieutenant and the Countess Spencer, who were accompanied by the Dowager Countess Spencer, the Ladies Sarah and Victoria Spencer and the Hon Robert Spencer, Lord and Lady Charles Bruce, and Major Stirling, A D C.&lt;p&gt;

The following had the honour of receiving invitations to meet their Excellencies — The Duke of Leinster, the Marquis and Marchioness of Kildare, the Ladies Fitzgerald, the Marquis and Marchioness of Drogheda, the Earl and Countess of Listowel, Lord and Lady Edward Cavendish, the Earl of Charleville, the Lord Chancellor and Lady O'Hagan, Viscount, Viscountess, the Hon Misses, and Hon Henry Monck; the Archbishop of Dublin, the Hon Mrs and the Misses Trench; Lord Talbot de Malahide and the Hon Francis Talbot, Lord and Lady Sandhurst and Captain Bang, A D C; Lady Cloncurry, Hon Emily and Hon Mary Lawless, Viscount, Viscountess, Hon Georgiana, and Hon Beatrice [de?] Vesci; Lord and Lady Kilmaize [?], Hon Gertrude [?] Browze, Lord and Lady Ventry, Hon Norah Westenra, Lord and Lady Athlumney, Lord, Lady, and Hon D Plunket, M P; Viscountess and the Hon. Miss Netterivlle, Capt the Hon Mrs Vesey, Captain and Lady Julia Follett, Sir Arthur and Lady Olive Guiness and the Ladies White, the Hon H W L Corry, Lord and Lady and the Hon Miss O'Neill, Viscount Hawarden, the Hon Florence Maude, the Hon. Clementina Maude, the Hon Jenico and Mrs Preston, the Hon Henry Leeson, Colonel and the Hon Mrs Caulfield, Mr and the Hon Mrs Robert Hobart, Captain, Lady Mary and Miss Lindsay; Mr Ion [?] Trent Hamilton, M P; Mr Bagwell; the Hon Mrs and the Misses Bagwell, and Mr Bagwell; Colonel the Hon L and Mrs Curzon Smyth, Mr, Lady Margaret, and the Misses Stronge [?]; Mr and the Hon Mrs O'Hagan, Hon Charles Bourke, Hon Mrs Alfred and Lady Kathleen Bury, [[Social Victorians/People/Abercorn|Hon Mrs, Lady Alice, and Lady Louisa Howard]]; Captain, the Hon Mrs, and Miss Donaldson; Dr and Miss Bans, Mrs Grattan Bellew, Sir Edward and Miss Borough, Mr Arthur Cane, Sir Dominic, Lady, and Miss Corrigan; Mr Corrigan, Mr and Mrs Gustavus Cornwall and Miss Cornwall, Mr D'Arcy, M P, and Mrs D'Arcy; Mr Baron Dowse [?], and Mrs and Miss Dowse, Mr Baron Deasy and Mrs Deasy, Dr, Mrs, and Miss de Ricci; Dr and Miss Hatchell, Sir George and Lady Hudson, Mr, Mrs, and the Misses Huband; Mr Arthur Huband, Miss Caroline Huband, Mr and Mrs Arthur Hume, Dr Hughes, Mr Henry Jephsen and Miss Jephsen, Mr Kearney and the Misses Kearney, Captain Kearney, A D C; Captain Lascelles, A D C; Mr, Mrs, and Miss Kirwan; Mr Justice Lawson and Mrs Lawson, Mr and Mrs W Le Fanu, Mr, Mrs, and Miss Lentaigne; Sir George L'Estrange and the Misses L'Estrange, the Lord and Lady Mayoress, and the Misses Mackey; the Lord Chief Justice Monahan, Mrs and Miss Monahan; Sir J, Lady, and Miss Power; Mr John Talbot Power, M P; Col, Mrs, and Miss Radcliffe; the Master of the Rolls, Mrs and Miss Sullivan; Capt and Mrs Moorsom, A D C; General Sir Thomas and Lady Steel, Captain and Mrs Brownrigg, A D C, Mr Granville Milner, Capt, Mrs and Miss Talbot, Colonel, Mrs, and the Misses White; Sir John Stewart Wood, Lady and the Misses Wood; Mrs and the Misses Williams, Mr Justice Fitzgerald and the Hon Mrs Fitzgerald, Mr Fitzgerald, Mr Justice Barry and Mrs Barry, Mr Sergeant Sherlock, M P, Mrs and Miss Sherlock; Mr Sheriock, the Right Hon W H Conan, M P, and Mrs Cogan; Mr Justice Keogh and Mrs Keogh, Mr Keogh, Capt Keogh, R N; Lord Chief Baron and Miss Pigott, Dr, Mrs, and Miss Nugent; General Wardlaw, Colonel M'Kerlie, Mr Sergeant and Mrs and Miss Armstrong; Col, Mrs, and the Misses Maude; Col, Mrs, and Miss Hillier; Mr Heron, M P; Mr and Mrs Watters, Col and Mrs Wynyard, Dr and the Misses Kennedy, the Attorney General and Mrs Palles, the Solicitor General and Mrs Law, Col, Mrs, and Miss Lake; Lady and the Misses Butler, Mr Butler, Col and Mrs Colthurst Vesey, and Miss Walton; Mr, Lady Fanny and Miss Lambert; Mr E C Guinness, Mr and Mrs MMorer O'Ferrall, Mr and Mrs Leonard Morrogh, Sir Bernard and Lady Burke, Mr G and Mrs G Brooke and Miss Brooke, Mr and Mrs Roe, Mr Vance, M P, Mrs and Miss Vance; Col and Mrs Primrose, Lieut Col Ferdall [?], Col and Mrs Goodlake and Miss Alexander, Mr Alison, Mr, Mrs, and Miss Barton, Mr Justice Flanagan, Mrs and Miss Flanagan, Mer J. N. Lentaigne, Mr Johnson, Captain Harrison, Mr, Mrs, and the Misses Maturin; Mr Justice Morris and Mrs Morris, Mr and Mrs Mazlere [?] Brady, Major, Mrs, and Miss Wilkinson; Mr, Mrs, and Miss Donnelly; Mr and Mrs Cruise, Mrs Power, Mr Braon Fitzgerald and Mrs Fitzgerald, Mr Henry Yates Thompson, Mr Courtenay Boyle, Colonel Forster, Mr, Mrs, and Miss Taylor, Mr Bland and Mrs Godfrey Bland, Mr and Miss Dillon, Mr and Mrs Wallace, Mr M'Kenna, Mr Cullinane, Mr Armstrong, Mr C E [?] Dobbin, Mr J A Blake, Major and Mrs Papillon, Capt and Mrs Keane, Mr E Pretty, Mr, Mrs John L O Ferrall and Miss O'Ferrall, Mrs and Miss Walsh, Mr and Mrs R Howard Brook, Mrs and Miss Brook, Mrs and the Misses Blake, Mr and Mrs J Warren, Sir John Gray, M P, Lady, and Miss Gray; Colonel and Mrs Frank Chaplin, Mr, Mrs, and Miss Hemphill; Sir R, Lady and Miss Kane, Mrs and Miss Courtenay, Mr Arthur Courtenay, Mr G Courtenay, Mr E Hardtop, A D C; Mr Bellew, Dr and Mrs Nedley, Dr and Mrs Newell, Mr and Mrs Freeman, Mr and Mrs Geale, Captain Hutten, A D C; Mr and Mrs Adair and Miss Wadsworth, Captain and Mrs J M Benthall, Sir R, Lady, and the Misses M'Causlend [?]; Mr, Mrs, and the Misses Newell Barron; Mr Hawkins, Colonel Goodlake and the Officers of the Coldstream Guards; Captain Spain, R N, and the Officers (4) of her Majesty's ship Vanguard; Colonel Radcliffe and Officers (4), Royal Artillery; Colonel Spade and Officers (4) 1st King's Dragoon Guards; Colonel Ainslie and Officers (4), 1st Royal Dragoons; Colonel Thompson and Officers (4), 14th Hussars; Colonel Ross and Officers (4), 4th Battalion Rifle Brigade; Colonel Hawkins and Officers (4), Royal Engineers; Colonel Gloster and Officers (4), 97th Regiment; Lieutenant-Colonel Maunsell and Officers (4), 13th Regiment.&lt;ref&gt;&quot;Fashionable.&quot; ''Dublin Evening Telegraph'' 14 January 1873, Tuesday: 4 [of 4], Col. 7a–b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002093/18730114/044/0004. Print title ''The Evening Telegraph'', n.p.&lt;/ref&gt; &lt;/blockquote&gt;

==== 29 January 1873, Wednesday ====

==== Drawingroom at Dublin Castle ====
The women listed in the 2nd paragraph, about the members of the Household who were present, were listed as accompanying their husband or father, not as working members of the Household.&lt;blockquote&gt;DRWNINGROOM [sic] AT DUBLIN CASTLE.

His Excellency the Lord Lieutenant and the Countess Spencer held the first Drawingroom for the season at Dublin Castle on Wednesday evening.

Shortly after nine o’clock their Excellencies entered the Throne Room, attended by the following members of the Household:— The under Secretary — Thomas H. Burke, Esq. The Private Secretary — Henry Y. Thompson, Esq; Miss Thompson. The State Steward — Colonel the Hon. Luke White. Comptroller — Lieutenant-Colonel Caulfield; Hon. Mrs. Caulfield. Gentleman Usher — Major the Hon. E. Boyle; Hon. Mrs E. Boyle. Chamberlain — Hon. H. Leeson. Master of the Horse — Lieutenant-Colonel Forster. The Gentleman in Waiting — Lieutenant-Colonel J. M'Donnell and Hon. Mrs. M‘Donnell. The Gentlemen at Large — Lowery Balfour, Esq, Captain Donaldson, and Hon. Mr. Donaldson. Aides-de-Camp — Major Sterling, Lieutenant the Hon. V. Lyttelton, Captain Lascelles, Captain Bridges, Capt. F. Seymour, Captain Kearney, Captain Chaplain, V. C; Lieutenant Hartopp, Lieutenant Wynne Finch, Lieutenant A. Egerton, Captain Hutton, Captain Wood.

The Physician in Ordinary — Thomas Nedley, Esq, M.D. The Surgeon in Ordinary — George Hatchell, Esq., M.D., and Miss Hatohell. The Surgeon to the Household — James S. Hughes, Esq. MD. Her Excellency’s Pages of Honour — Hon. J. Somerville, and Mr. Charles White.

There was very large company present among them being the Lord Mayor and the Lady Mayoress; [sic]

The Lord Chancellor and Lady O’Hagan. The Lord Chief Justice, and Mrs. Whithside. The Lord Chief Baron and Mrs. Pigot, the Attorney-General and Mrs. Palles, the Solicitor-General and Mrs. Law. Major-General Sir Thomas Steele, K.C.B., and Lady Steele (presented.) Captain Brownrigg, A.D.C., and Mrs. Studholm Brownrigg. Colonel Primrose, C.S.I., Deputy Adjutant-General. Colonel the Hon. Leicester Smith, C.B., Deputy Quartermaster-General, and the Hon. Mrs. Leicester Smith. Mr. Porter, Surgeon in Ordinary to the Qneen in Ireland, and Mrs. Porter. Marquis and Marchioness of Kildare, Lady Alice Fitzgerald, and Lady Eva Fitzgerald. Marquis of Headfort, Lady Adelaide Taylour, Lady Florence Taylour. Marquis of Drogheda and Marchioness of Drogheda. Earl and Countess of Shannon, Earl of Kenmare, Countess of Charlemont, Anna Countess of Kingston, Dowager Countess Spencer and Lady Victoria Spencer, Viscount and the Viscountess Monck, and the Hon. Frances Monck, Viscountess Gormanstnwn, Viscountess Netterville, Lord Talbot de Malahide and Hon. Frances Talbot, Lord and Lady Lisgar, Lord Crofton, Lord and Lady Plunket, Lady Sandhurst, Lady Athlumney, Lady Hastings, Lady Cloncurry, Lady Colthurst, Lady Louisa Tenison and Lieutenant-Colonel Tenison, Lady Barbara Chetwynd Stapylton, [[Social Victorians/People/Abercorn|Lady Louisa Howard]], [[Social Victorians/People/Abercorn|Lady Caroline Howard]], Lady Julia Follett and Captain Follett, Lady Georgina Croker, Lady Catherine Bury, Lady Steward Wood, ['''Col. 3c–4a'''] Miss Stewart Wood, and Miss Elvyn Stewart Wood, The Right Hon. J. D. Fitzgerald and the Hon. Mrs. Fitzgerald, the Right Hon. Mr. Justice Morris, the Right Hon. Mr. Justice Barry, and Mrs. Barry, the Right Hon. Baron Dowse, Mrs. Dowse, and Miss Dowse, Judge Woulfe Flanagan, Mrs. and Miss Woulfe Flanagan.

The Provost of Trinity College and Mrs. Lloyd, the Moderator of the General Assembly.

Colonfel Frederick Maude, V.C., C.B., Deputy Inspector General of Auxiliary Forces; Mrs. Frederick Maude, and Miss Ada Cecil Maude (presented). Colonel Lake, C. B. Commissioner of Police, and Miss Lake.

LADIES’ DRESSES.

Her Excellency the Countess Spencer — Train and corsage of rich Lyons peon velvet, lined poult de foie, trimmed bouillones of tulle illusion to match, nœuds of satin and plumes of peacock, and ostrich feathers, same shade; corsage, Raphael, trimmed band of peon velvet, beautifully embroidered in self colours, plumes of ostrich and peon to correspond; petticoat of richest satin antique, with jupes of tulle, beautifully trimmed three broad plisses, with plumes of peacock's tail, headed with shells of velvet all to match train in colour; at sides and backs stoles and broad sashes of peon velvet, beautifully embroidered in self colour; across body of dress was band of velvet, worn like sash; studded with the most magnificent brilliants. Headdress a tiara of diamonds and peon plume; ornaments, diamonds.

The Lady Mayoress, Mansion House — Train and corsage of richest black satin raye, lined blue glace, and trimmed plisses of blue poult de soie; corsage, trimmed a draperie of tulle, with fall of very fine Irish point lace; petticoat of rich blue poult de joie, with volants of Irish point lace, and tulle plaitings, headed blue satin. Head-dress, coart plume, Irish point lace; ornaments, diamonds.

Hon. Mrs. Caulfield, Dublin Castle — Train and corsage of the richest black gros de Suez, lined black taffeta, tastefully trimmed; bouillones of tulle and silver wheat; corsage, trimmed a draperie of tulle, silver wheat, and silver bullion fringe, with a fall fine Brussels point; petticoat of rich black glace under jupe of chantilly; trimmed tablier tulle and satin shells, tunic to correspond, looped black velvet bows, and bouquets of silver wheat. Headdress, court plume, point lappets and diamonds; ornaments, diamonds.

Mrs. Whiteside, Mountjoy-square — Train and corsage of rich pink satin antique, lined with white Florence, beautifully trimmed with bias and nœuds of satin, and a volant of very fine Brussels point; corsage, trimmed draperie of tulle and satin, with fall point lace; petticoat of white satin antique, with jupe of Alencon tulle, tulle plaitings edged with folds of pink satin, and volant fine Brussels point. Headdress.

Lady Butler, Ballintemple, county Carlow — Train and corsage of richest white satin, trimmed bouillones, and pouffs of white tulle de chene, festooned with bouquets of pink laburnum, set rosettes of white tulle de chene; petticoat of white Bruxelles net, trimmed with roulleax of white satin, and bouilloned the waist en pompadour. Headdress, court plume, lappets, and feathers; ornaments, diamonds and pearls.

Miss Wynn, Wynstay, Roebuck — Train and corsage of rouleaux satin, trimmed with pouffs and bouillones of white tulle de chene, and edged with richest blonde lace; petticoat lavender glace, trimmed with rings and frillings of tulle de chene and rich flounce of blonde lace. Headdress, Court plume and lappets ; ornaments, tiara of diamonds.

The Countess of Shannon, Castlemartyr, county Cork — Train of richest white satin, lined marceline, &amp;c., trimmed with white tulle, studded with pearls, and volantes of real Brussels lace; jupe of richest white satin, with tunic of finest real Brussels lace, looped up with chatelaine of pink roses; corsage, a la gracque trimmed en suite. Headdress, plumes of feathers with lappets ; ornaments, diamonds.

Mrs. Murphy, Mount Loftus — Train and corsage of rich mauve gros grain, lined with white satin, and trimmed with Carrickmacross lace and bias folds of silk; petticoat of mauve glace, with mauve tulle, jupe, trimmed en tablier with Carrickmacross lace, and flounce and buillons of tulle. Headdress — Lappets, feathers, and tiara of diamonds. Ornaments, pearls and diamonds.

Mrs. Maxwell, Cruiserath, Clonsilla — Train and corsage of rich ruby velvet, lined with rich white silk, and trimmed with Brussels lace, centre of train trimmed with bows of moire ribbon, the train looped at the side with an echarpe of wide ribbon; corsage to correspond; of rich gros de Suez silk, trimmed with white Brussels lace, flounces headed with ruche of green tulle illusion, studded with green flowers, front trimmed en tablier. Headdeess [sic] — Court plume, Brussels lace lappets, and diadem of diamonds. 

Miss Pigot, 15, Merrion-square, East — Train with pouffe of magnificent black silk, lined with white marcelline, beautifully trimmed with broad bias of lavender satin, ruching of lavender net and Spanish blonde; sash of lavender satin, fastening side under pouffe; corsage, Louis Quinze; petticoat of white poult de soie, with overskirt of white Brussels net, trimmed en tablier, with platings of lavender net and satin, fastening at side, with nœuds of lavender satin. Coiffure — Court plume and tulle veil. Ornaments — Diamonds.

Miss Jackson, Ahanesk, Midleton, Co. Cork — Presentation train, with pouffe and sash of richest white faye silk, lined with marcelline, tastefully trimmed with fluffed plaiting of white silk and satin; corsage, Pompadour style, trimmed with white satin and tulle; jupon of white poult de soie, with overskirt of white tulle, trimmed with alternate plaitings of tulle and white satin. Coiffure — Court plume and tulle veil. Ornaments — Diamonds and pearls.

Mrs. Safford, 97th Regiment — Train and corsage of rich maize satin, lined, richly trimmed with tulle ruche, true-lover’s knots, and nœuds de velour noir, from agrafe; corsage, garnier richment de danlette ancienne; jupe, tulle, maize ruche, richly trimmed to match train. Headdress — Ostrich feather and tulle lappets. Ornaments — Diamonds and pearls.

Miss Mackey—Train and corsage of the richest maize poult de soi, lined with Florence silk, and elegantly trimmed with bouffants of tulle, illusion, and guirlands of cherita leaves; corsage trimmed to correspond; jupe of white tarlatane buillonee and wreaths of cherita leaves. Coiffure — Maize feather and long tulle veil. Ornaments — Silver.&lt;ref&gt;&quot;Drawingroom at Dublin Castle.&quot; ''Cork Constitution'' 31 January 1873, Friday: 3 [of 4], Col. 3c–4b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001646/18730131/056/0003. Print title: ''The Constitution; Or Cork Advertiser'', n.p.&lt;/ref&gt;&lt;/blockquote&gt;February
March
April

===May===
'''28 May 1873, Wednesday''': Derby Day

=== June ===

==== 19 June 1873, Thursday, Polo Match Between Officers of the Royal Horse Guards and Officers of the 9th Lancers ====
&lt;blockquote&gt;THE POLO CLUB.

Although the weather was dull and gloomy yesterday, there was a large company at the club grounds to witness the match between the officers of the Royal Horse Guards (Blue) and the officers of the 9th Lancers. A number of carriages surrounded the enclosure, and many ladies were present, among whom were the Marchioness of Waterford, Viscountess Middelton, Lady Philippa Stanhope, the Countess of Mayo, the Hon. Miss Brodrick, Lady Little, [[Social Victorians/People/Abercorn|Lady Louisa Howard]], [[Social Victorians/People/Abercorn|Lady Caroline Howard]], Lady Harriet Duncombe, Miss Duncoinbe and Miss E. Duncombe, the Hon. Mrs. O'Grady and Miss O'Grady, Lady Knollys and Miss Knollys, the Dowager Lady Craven, Lady Grey de Wilton, Lady Fanny Fitzwigram, Lady Petre, Lady M. Egerton, Misses E. and G. Egerton, the Countess of Gleichen, Lady C. Brineman, Lady Campbell, Lady Emily Ormsby Gore, the Countess of Coventry, Lady Maria Ponsonby, and Lady Henry Somerset. Just before 4 o'clock the competitors took up their stations at the goals, the Hon. H. Boscawen and Sir Beach Cunard being the judges. The Guards, having choice of stations, elected to play from the Pavilion goal, although there was a strong wind blowing against them. Play was called for the first &quot;bully,&quot; and when the ball was tossed into the centre of the ground the advanced guard of both sides missed their blows; and, this brought the others close up, and after some spirited hitting the Guards got the ball nearly to the bottom goal, where it was knocked out of bounds three or four times. Each time it was returned into play some severe rallies ensued, and the scientific hitting and stopping of the Marquis of Worcester, the Hon. C. W. Fitzwilliam, and Lord Kilmarnock met with loud applause, while the play of the whole of the Lancers was so determined and vigorous that the Guards could not break through their defence, but in a good ''mêlée'' [sic] close to the goal the ball was hit just outside the bottom posts. They then had a rest, and the ponies were attended to and carefully watered, and when the ball was hit off the Lancers, playing well together, drove the ball nearly to the top goal, but just missed getting it through the post. The rain now came down and made the turf heavy and slippery, and the play was rather wild, many well-intended hits being lost by the little &quot;tits&quot; slipping when turning sharply at their best speed. Both sides were doing their utmost to obtain the honours; but, although the ball was sent to all parts of the enclosure, and rally after rally came off, each goal being assaulted in its turn, no goal was made. The Guards now got the ball to the bottom end of the ground, and the Marquis of Worcester made a fine drive for victory; the ball, however, did not quite reach the goal, but his Lordship was well backed up by the Hon. C. Fitzwilliam, who, in the midst of a rattling ''mélée'' [sic] close on the posts, cleverly &quot;pushed&quot; the ball through the goal, and scored the first to the Guards, after playing lh. 20min., being the longest time that as [sic] occurred this season. After a rest and a change of ponies the second &quot;bully&quot; was commenced, but, after a short time, during which some fine play was exhibited by both sides, &quot;time&quot; was called by the judges, and the Guards won the game by one goal. Appended will be found the sides:
{| class=&quot;wikitable&quot;
|+
!The Royal Horse Guards
!The Lancers
|-
|Marquis of Worcester,
|Capt. Grissell.
|-
|Lord C. Somerset.
|Lord W. Beresford.
|-
|Hon. C. W. Fitzwilliam.
|Mr. Moore.
|-
|Mr. Egerton.
|Capt. Polaret.
|-
|Lord Kilmarnock.
|Hon. E. Willoughby.
|}
Sides were then chosen by Viscount amentia and Mr. C. de Murrietta, and after some exciting play a goal was got by each.
{| class=&quot;wikitable&quot;
|+Sides
|Lord Valentia.
|Mr. C. de Murietta
|-
|Capt. Middelton.
|Marquis of Queensberry.
|-
|Hon. H. C. Needham.
|Sir Beach Cunard.
|-
|Mr. Green.
|Sir W. Gordon Cumming.
|-
|Hon. R. Neville-Nugent.
|Hon. C. W. Fitzwilliam.
|-
|Mr. A. de Murietta.
|Lord Aberdour.
|-
|
|Mr. Powell.
|}
&lt;ref&gt;&quot;The Polo Club.&quot; ''Hour'' 20 June 1873, Friday: 7 [of 8], Col. 6a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002814/18730620/078/0007. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;

July
August
September

=== October ===

==== 18 October 1873, Saturday, Orange Order Events at Govan ====
This festival seems to have included some speeches and the laying of a foundation stone for an Orange Hall. The speeches were extremely anti-Catholic and bigoted.&lt;blockquote&gt;ORANGE FESTIVAL AT GOVAN.

The third annual festival of the Govan Orangemen and their friends was held in the Govan Hall on Friday night — Br. H. A. Long [?] in the chair. After a service of tea and sake,

The C&lt;small&gt;HAIRMAN&lt;/small&gt; delivered an address, in which he stated, after a few preliminary remarks, that Orangeism had to be looked at from two points of view — one political and the other religious. The political one looked at the Pope and grasped the sword, while the other looked at Christ and opened its arms. One of them was for offence — that was fighting against Popery in all its varied forms, while the other was for the adoption and union of the great system of thrice-blessed Christianity. He congratulated them on living in comparatively happy days, and seeing the complete destruction of the Court of Rome and the Pope's temporal power. Not many years ago, he said, diplomatists came from all parts of the world to the Quirinal or the Vatican, but all that had now passed away, and not left a shadow behind. The chairmen then reviewed at some length the events of Italian history since 1846, and the great contrast in the treatment of priests in Rome at that time and at the present day. It must have been a bitter pill, he went on to say, for the Vatican to swallow when they heard the shouts of triumph of 25,000 Romans rejoicing that they had got free from priestly influence. Mr. Long next referred to the late visit of Victor Emmanuel to the Emperors of Austria and Germany, which he is garded as a pledge of defence against the French nation's interference in Italian affairs. The chairman referred to the immense treasures stored in the Vatican, amounting to eight hundred millions of sovereigns, and to the cramping of the power of the priesthood in Germany by Bismarck[.] The Rev. C. A. M'Kenzie, after apologising for not having any text, gave an interesting sketch of the connection of the North of Ireland with the Western Highlands of Scotland, from the middle of the sixth century, when St. Columba crossed over with his twelve followers, till the perversion of the early Culdee Church by the wife of Malcolm Canmore and her son King David. Popery, he asserted, was an invasion of comparatively recent origin, and the Roman Catholics had no right to the ancient abbeys, to which they seemed inclined to lay claim. In conclusion, he urged upon them, as good Orange-men and followers of the famous King William, of glorious memory, who inscribed on his banner &quot;the liberties of England and the Protestant religion,&quot; never to forget that noble man; and to beware of Puseyism, which was only Popery in disguise. The meeting was afterwards addressed by Mr. Martin, and the proceedings were enlivened with songs by a number of the brethren and their lady friends. After the soiree an assembly took place, and dowering was kept up till an early hour.— ''Glasgow News''.

N&lt;small&gt;EW&lt;/small&gt; O&lt;small&gt;RANGE&lt;/small&gt; H&lt;small&gt;ALL&lt;/small&gt;. — The foundation stone of Staffordstown [?] Orange Hall has been laid by Lady Louisa O'Neill, in presence of Lady O'Neill, [[Social Victorians/People/Abercorn|Lady Caroline Howard]], the Hon. Edward O'Neill, and a large assemblage of Orangemen. After the ceremony, the entire party adjourned to a field adjoining, where a platform had been erected. The lodges present were — Staffordstown L.O.L., 504 [?]; Ballydonnall L.O.L., 306 [?]; Tailorstown True Blues, 544; Grange L.OL., 701; Duneane [?] L.O L., 719; Grange L.O.L., 919; Cranfield L.O.L , 705 [?]; Fenton Invincibles, L.O.L., 1104; and the Fenton Invincibles (juveniles), L.O.L., 1104. Amongst those present on the platform were — Lady O'Neil, the Hon. Edward O'Neill, M.P.; the Hon. Louisa O'Neill, Lady Caroline Howard, William J. Gwynne, Esq.; Richard Lilburn, Esq.; J. J. Carson, Esq., Mrs. Carson, and Miss Carson; Rev. J. B. Greer, Rector of Grange; Rev. J. H. Wright, bector [sic] of Portglenone; Rev. A. Gault, Vicar of Antrim; Rev. William Denham, Presbyterian minister, Duncane; Wm. J. Scully, Esq.; Messrs. John Fulton, John M Kelvey, John Nimmons. W.D.M.; Wm. M'Cullough, Hugh Nicholl, Joshua Hume, James Brooks, Charles Richardson, Robert Chesney, Robert Barton, Wm. Allen, Alexander M'Fadden, Hugh Logan. D. S Beekerstaff, Glenavy District; George French, James M'Manus, John Hume Richardson, Wm. J. Senly. Mr. Gwynne was called to the chair, and the meeting having been opened with prayer, appropriate addresses were afterwards delivered by the chairman, the Hon. Edward O'Neill, the Rev. Mr. Wright, Mr. Lilburn, and the Rev. Mr. Greer. The chairman having made a few concluding remarks, the meeting separated after having given three hearty lowly cheers for Lady O'Neill and party.&lt;ref&gt;&quot;Orange Festival at Govan.&quot; ''Belfast Weekly Telegraph'' 18 October 1873, Saturday: 8 [of 8], Col. 3b–c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003434/18731018/044/0005. Same print title and p.&lt;/ref&gt;&lt;/blockquote&gt;November
December

==1874==
January
February
March
April
===May===

==== 1874 May, Early ====
&lt;blockquote&gt;As monarchists’ hopes flared, the Catholic Church, too, enjoyed a conspicuous revival. The National Assembly approved a design for a new basilica for Paris. Intended as an act of collective atonement, Sacré-Coeur was to perch atop Montmartre, immediately above where Nadar’s balloons had been launched and where the radicals’ insurrection had broken out. Excavations began in early May 1874 ....

But the focus of the penance the basilica was intended to embody gradually shifted from the moral decline of French society in general to the despicable excesses of the Commune. In 1872 Archbishop Darboy’s successor claimed to have had a vision as he climbed the Butte Montmartre. The clouds dispersed, and he realized that it was there, “where the martyrs” were (he meant the murdered generals Lecomte and Clément-Thomas), that a new church should be built. And when the Assembly voted to proceed with the construction, legislators specified that its purpose was to “expiate the crimes of the Commune.”&lt;ref name=&quot;:3&quot; /&gt; (464 of 667)&lt;/blockquote&gt;

===June===
'''3 June 1874, Wednesday''': Derby Day

June
July
August
September

=== October ===
November
===December===
'''8 December 1874, Tuesday''': &quot;CHATSWORTH, Tuesday, December 8th, 1874. — We are come to the last slide of the Chatsworth magic lantern: the Duke of Cambridge and his equerry, a funny little man called Tyrwhitt, of no particular age, in a grey wig; Lord Carlingford and Ly. Waldegrave, the Spencers, Mr. Leveson, Cavendish.&quot;&lt;ref&gt;{{Cite web|url=http://ladylucycavendish.blogspot.com/2010/12/08dec1874-chatsworth-magic-lantern.html|title=Lady Lucy Cavendish: 08Dec1874, The Chatsworth Magic Lantern|last=H|first=Denise|date=2010-12-04|website=Lady Lucy Cavendish|access-date=2025-06-18}}&lt;/ref&gt;

==1875==
Disraeli's progressive legislation for labor rights:&lt;blockquote&gt;In 1875, he passed a series of enlightened acts protecting labor rights, arguing they were as important as property rights. Two of the laws ensured that workers would have the same recourse as employers when contracts were breached, and made peaceful picketing legal, protecting unions from charges of conspiracy.&lt;ref name=&quot;:4&quot; /&gt; (578 of 1203)&lt;/blockquote&gt;After women who owned property were allowed by Parliament to stand for local school-board elections in 1870, &quot;Elizabeth Garrett Anderson, the first woman to qualify as a doctor in Britain — in 1865 — stood and was elected to her local board five years later.&quot;&lt;ref name=&quot;:4&quot; /&gt; (199 of 1203)

The relationship between Swinburne and Lord Houghton:&lt;blockquote&gt;...not all Lord Houghton's children appreciated the catholicity of &quot;Papa's&quot; taste in friends: &quot;Swinburne (in a very excited state) came in in the evening,&quot; wrote Florence Milnes to her brother in 1875: &quot;He is madder than ever, to my astonishment he flopped down on one knee in front of me, &amp; announced that my hair had grown darker. This was rather embarrassing, and he is also so deaf now, which does not make it easier to talk to him.&quot;&lt;ref name=&quot;:2&quot;&gt;Pope-Hennessy Lord Crewe.&lt;/ref&gt;{{rp|5}}&lt;/blockquote&gt;

January
February
March
April
===May===
'''26 May 1875, Wednesday''': Derby Day. The Prince and Princess of Wales attended, as did a number of others of the royal family, including Princess Louise and Lorne.

June
July
===August===
'''August through October 1875''' Richard Monckton Milnes (Lord Houghton) and son Robert Milnes toured the U.S. and Canada:&lt;blockquote&gt;They set off in the steamer s.s Sarmatian from Liverpool in August 1875, stopping at Ireland to pick up the usual load of emigrants bound for the U.S.A. The most interesting among the passengers was 'Mr. Butler, author of Erewhon, who is very amusing and clever though infidel,' but, although he played whist with Samuel Butler, the young man was far more interested in the Eustace Smiths (parents of his friend W. H. Smith), and in a Canadian family named Macpherson, the youngest of whose two daughters, the dark-eyed Isobel, caught his fancy: he saw them afterwards in Toronto, and when they parted she gave him two larger than carte-de-visite photographs of herself, he gave her a smaller one of himself together with the inevitable volume of his father's verse.&quot;&lt;ref name=&quot;:2&quot; /&gt;{{rp|10}}&lt;/blockquote&gt;September
October
November
December

==1876==
Disraeli pushed through the Cruelty to Animals Act in order to please Queen Victoria. This act &quot;forced researchers to demonstrate that any experiments with animals involving pain were absolutely necessary, and ensured they would be anesthetized if so.&quot;&lt;ref name=&quot;:4&quot; /&gt; (679 of 1203)

January
February
March
April
===May===
'''11 May 1876''': In the midst of the Aylesford scandal, the Prince of Wales returned from a journey to Egypt and India, etc.:&lt;blockquote&gt;However harassed and exhausted, the Prince and Princess of Wales would put up a good show. Within an hour of their arrival home they set forth to attend a gala performance at Covent Garden Opera House. It was a brave decision to face the public and allow an immediate opportunity for demonstration. The Prince and Princess were rewarded when the audience rose to its feet to give them a standing ovation before the start of every act, as well as at the end, of Verdi's Ballo in Maschera.&lt;ref name=&quot;:1&quot; /&gt;{{rp|63}}&lt;/blockquote&gt;

'''27 May 1877''': Lily Langtry:&lt;blockquote&gt;Her big moment on May 27, 1877, when Sir Allen Young, the arctic explorer, invited her to late supper in his house, where it had been arranged that the Prince of Wales should meet her after the opera. The result was all that could have been expected. Mrs. Langtry became the Prince's first openly recognised mistress.&lt;ref name=&quot;:1&quot; /&gt;{{rp|69}}&lt;/blockquote&gt;'''31 May 1877, Wednesday''': Derby Day. The Prince and Princess of Wales did not attend, as he was ill.

June
July
August
September
October
November
December

==1877==
&quot;In 1877, unemployment was 4.7 percent; by 1879, it had risen to 11.4 percent.&quot;&lt;ref name=&quot;:4&quot; /&gt; (690 of 1203)

January
February
March
April
===May===
'''30 May 1877, Wednesday''': Derby Day.

June
July
August
September
October
November
===December===
'''15 December 1877'''&lt;blockquote&gt;On Dec. 15, 1877, the Queen honoured Lord Beaconsfield, the Premier, with a visit at Hughenden Manor. Her Majesty, accompanied by Princess Beatrice and attended by General Ponsonby and the Marchioness of Ely, left Windsor at 12.40 and proceeded by special train to High Wycombe, which was reached at 1.15. The Premier received the Queen at the station. A lofty triumphal arch spanned the entrance to the station-yard, and beneath this the royal party drove into the gaily decorated little town. The reception along the route was of the heartiest, and the drive of two miles to Hughenden was one long triumph. Lord Beaconsfield, who had preceded the party, welcomed the Queen at his own door. Lunch was served, and her Majesty remained about two hours. Before leaving she planted a memorial tree.&lt;ref&gt;&quot;The Queen's Glorious Reign.&quot; ''Illustrated London News'' (London, England), Saturday, May 27, 1899; pp. 757–765?; Issue 3136. Queen's Glorious Reign [Supplement]: 762?&lt;/ref&gt;&lt;/blockquote&gt;

==1878==
January
February
March
April
May
===June===
'''5 June 1878, Wednesday''': Derby Day.

July
August
September
October
===November===
'''8 November 1878''': from the journal of George, Duke of Cambridge:&lt;blockquote&gt;''November'' 8. — Gave farewell diner to the Lornes; Louise and Lorne, Augusta, Mary and Francis, Arthur, Leopold, Gleichens, J. Macdonald and self, and played at Nap afterwards. It was a good and nice little dinner.&quot;&lt;ref&gt;Sheppard, Edgar, Ed. ''George, Duke of Cambridge: A Memoir of His Private Life, Based on the Journals and Correspondence of His Royal Highness''. Vol. 2, 1871–1904. New York: Longmans, Green, 1906. http://books.google.com/books?id=dFoMAAAAYAAJ.&lt;/ref&gt;&lt;/blockquote&gt;December

==1879==
===January===
'''12 January 1879'''&lt;blockquote&gt;On 12 January 1879 Robert Milnes came of age, an event celebrated at Fryston by a tenants' ball.&lt;ref name=&quot;:2&quot; /&gt;{{rp|18}}&lt;/blockquote&gt;

'''28 January 1879''': Brett &quot;Harte kicked off his tour at the Crystal Palace in Sydenham on January 28, 1879.&quot;&lt;ref&gt;Nissen, Alex. ''Brett Harte: Prince and Pauper''. Jackson, MS: University Press of Mississippi, 2000.&lt;/ref&gt;{{rp|174}}

February
March

===April===
'''Early April 1879''' or so, probably, Bret Harte got &quot;an invitation to dine the same evening with Arthur Sullivan and the Prince of Wales&quot; as a dinner in Birmingham where Harte met T. Edgar Pemberton.&lt;ref&gt;Scharnhorst, Gary. ''Bret Harte: Opening the American Literary West''. Norman, OK: Univ. of Oklahoma Press, 2000.&lt;/ref&gt;{{rp|152}}
===May===
'''28 May 1879, Wednesday''': Derby Day; the Prince and Princess of Wales attended.

===June===
'''June 1879''', Robert Milnes became engaged to &quot;Sibyl Marcia, a daughter of a North-country baronet, Sir Frederick Graham of Netherby.&quot;&lt;ref name=&quot;:2&quot; /&gt;{{rp|18}} Parties must have followed.

July
August
September
October
November
===December===
'''28 December 1879''': The Tay Bridge Disaster: The Tay Bridge collapsed with a train on it. The weather was very bad, with gale-force winds and rain.

The ''Times'' reported that the average high temperature for the week ending December 31, 1879, was 53° F. and the low was 20° F.

In his column &quot;What the World Says&quot; in the 21 January 1880 World, Edmund Yates writes the following:&lt;blockquote&gt;How am I to describe better the magnificence of the Earl and Countess of Rosslyn’s ball at Euston Lodge last month, than by calling attention to the fact that M. Carlo, the eminent Knightsbridge coiffeur, arrived early in the day to crimp and powder the lacqueys? My informant adds, however, that the curled darlings were rather the worse for the festivities towards night. Was it not enough to turn their heads in every sense of the word?&lt;ref name=&quot;:0&quot;&gt;Edmund Yates, &quot;What the World Says,&quot; ''The World: A Journal for Men and Women''.&lt;/ref&gt;{{rp|21 Jan. 1880, p. 8, col. b.}}&lt;/blockquote&gt;

'''31 December 1879''': Edmund Yates, editor of The World: A Journal for Men and Women, in his column &quot;What the World Says,&quot; describes a private viewing at the Grosvenor Gallery:&lt;blockquote&gt;The private view at the Grosvenor on the last day of the year gave people something to do on a desperately wet afternoon. The artistic dresses were perhaps in greater force than ever; indeed the faces and the hair and the attitudes pursued me to my bed, and gave me many a nightmare. I suppose the plain woman of all time has had the ambition to be looked at: centuries of failure have at last been crowned with a real success. Besides the Cimabue Browns there was an interesting menagerie of real lions, artistic, literary, and clerical. The artists were numerous, and their host and hostess seemed to enjoy themselves very thoroughly.

Frequenters of the picture private views have a new sensation this winter. Last season they mobbed beauty: now hideously-attired unkempt dowdiness provokes the stare. The prize for the new style seems generally awarded to a rhubarb coloured flannel Ulster and a cart-wheel beaver hat, which pervaded both the private views last week. [2 private views last week, one at the Grosvenor]&lt;ref name=&quot;:0&quot; /&gt;{{rp|7 Jan. 1880, p. 9}}&lt;/blockquote&gt;

The official premiere of ''The Pirates of Penzance'' occurred in New York City on 31 December 1879 at the Fifth Avenue Theatre, to establish international copyright. Gilbert and Sullivan were there with the cast. The performance was a social event: attending were Mrs. Vanderbilt and Mrs. Astor.

==Works Cited==
{{reflist}}</text>
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  </page>
  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2818417</id>
      <parentid>2818346</parentid>
      <timestamp>2026-07-16T14:26:50Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2818417</origin>
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      <text bytes="9878" sha1="o7nth01ref6762g621wh9x1n6p9crz0" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

=== Handling Repetition ===
* Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]])
* Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]])

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260716.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
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  </page>
  <page>
    <title>Coordinates Last: Vector Analysis Done Fast</title>
    <ns>0</ns>
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    <revision>
      <id>2818403</id>
      <parentid>2768536</parentid>
      <timestamp>2026-07-16T12:22:26Z</timestamp>
      <contributor>
        <username>Gavin R Putland</username>
        <id>2838145</id>
      </contributor>
      <comment>Restructuring as learning resource.</comment>
      <origin>2818403</origin>
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      <text bytes="344303" sha1="9v5g8sjkrtrux8zloz23s8ww8tm7g9m" xml:space="preserve">{{Author|Gavin R Putland}}{{tertiary}}{{mathematics}}{{physics}}{{engineering}}{{testing}}

== Preface ==

This learning resource (which I call a &quot;paper&quot;, although it's a long one) is an attempt to reduce vector analysis from a second-year undergraduate subject to a ''first''-year undergraduate subject. Its strategy is to delay the use of coordinate systems until their use is required by upcoming topics&amp;mdash;and, behold, assisted by previous topics. It's about ''vector'' analysis as distinct from tensor analysis: it does not deal with dyadics or higher-order tensors, except by way of occasional hints; but, along its unusual path, it ''does'' treat some topics that one might not expect in a &quot;first&quot; course.

[[w:Sheldon Axler|Sheldon Axler]], in his essay &quot;Down with determinants!&quot; ([[#axler-95|1995]]) and his ensuing book ''Linear Algebra Done Right'' (4th Ed., [[#axler-23-|2023–]]), does not eliminate determinants, but introduces them as late as possible, and then exploits them for what he calls their &quot;main reasonable use in undergraduate mathematics&quot;, namely the change-of-variables formula for multiple integrals.&lt;ref&gt;[[#axler-95|Axler, 1995]], &amp;sect;9. The relegation of determinants was anticipated by C.G.&amp;#8239;Broyden ([[#broyden-75|1975]]). But Broyden's approach is less radical: he does not deal with abstract vector spaces or abstract linear transformations, and his eventual definition of the determinant, unlike Axler's, is traditional&amp;mdash;not a product of the preceding narrative.&lt;/ref&gt; Here I treat coordinates in vector analysis somewhat as Axler treats determinants in linear algebra: I introduce coordinate systems as late as possible, and then exploit them in unconventionally ''rigorous'' derivations of vector-analytic identities from (e.g.) vector-algebraic identities. But I contrast with Axler in at least two ways. First, I have no intention of expanding this &quot;paper&quot; into a book. Brevity is of the essence. Second, while one may well avoid determinants in ''numerical''&amp;#8202; linear algebra,&lt;ref&gt;[[#axler-95|Axler, 1995]], &amp;sect;1. But it is Broyden ([[#broyden-75|1975]]), not Axler, who discusses numerical methods at length.&lt;/ref&gt; one can hardly avoid coordinates in ''numerical'' vector analysis! So I cannot offer a coordinate-free path into computation. But I can prepare for computation by expressing the operators of vector analysis in general coordinates and orthogonal coordinates: indeed, readers who stay with me to the end will get a more general treatment of coordinates than is offered by a typical ''book''-length introduction to vector analysis. [''Continued&amp;#8239;&amp;hellip;'']

{{cot|&amp;hellip; Extended content (show or hide)}}

In the meantime, however, coordinates don't get in the way. Familiar coordinates may be mentioned in passing for purposes of illustration; but, until &quot;Cartesian coordinates&quot; are announced under their own heading, I work from ''conceptual'' definitions rather than coordinate-based definitions. This, I submit, keeps the exposition direct and accessible, and facilitates treating related concepts in parallel&amp;mdash;saving time and words, and highlighting similarities and differences.

Something else that doesn't get in the way is an exaggerated pretense of rigor. In the branch of pure mathematics known as ''analysis'', there is a thing called a ''limit'', whereby for every positive ''&amp;epsiv;''&amp;#8201; there exists a positive ''&amp;delta;'' such that if some increment is less than ''&amp;delta;'', some error is less than ''&amp;epsiv;''. In the branch of applied mathematics known as ''[[w:continuum mechanics|continuum mechanics]]'', there is a thing called reality, whereby if the increment is less than some positive ''&amp;delta;'', the assumption of a continuum becomes ridiculous, so that the error cannot be made less than an ''arbitrary &amp;epsiv;''. Yet vector &quot;analysis&quot; (or a superset thereof) is typically studied with the intention of applying it to some form of &quot;continuum&quot; mechanics&amp;mdash;such as the modeling of elasticity, plasticity, fluid flow, or (widening the net) electrodynamics of ordinary matter&amp;mdash;conveniently forgetting that, on a sufficiently small scale, matter is lumpy. (Even if we claim that &quot;particles&quot; of matter are wave functions and therefore continuous, these wave functions are still lumpy on a scale not normally contemplated by continuum mechanics.) One might therefore submit that to express the principles of vector analysis in the language of limits is to strain at a gnat and swallow a camel. Here I avoid that camel by referring to '''elements''' of length or area or volume, each of which is ''small'' enough to allow some quantity or quantities to be considered uniform within it, but, for the same reason, ''large'' enough to allow such local averaging of the said quantity or quantities as is necessary to tune out the lumpiness. We shall see bigger camels, where well-known authors define or misdefine a vector ''operator'' and then derive identities by treating it like an ordinary vector ''quantity''. These I also avoid.

A rough and ready premise is more rigorous than an absurd or meaningless one. Discarding the machinery of limits causes a small lapse in rigor where limits are applicable, but avoids a big lapse where they are not. Maintaining the distinction between operators and quantities cannot cause a loss of rigor, but avoids one wherever the alleged &quot;algebraic&quot; properties of operators get confusing. The resulting standard of rigor is economical but consistent.

This paper is a new arrangement of old knowledge. It does not pretend to offer any new mathematical results, and in that sense does not pretend to be [[original research]]. But, pursuant to its goals as a [[learning resource]], it ''does'' contain independent derivations and independent scholarship.

Much of that scholarship builds on the earlier scholarship of Professor Chen-To Tai, {{serif|FIEEE}}, who died in 2004, and who first came to my attention in 2018 through his invited paper &quot;On the presentation of Maxwell's theory&quot; [''Proc.&amp;#8239;{{serif|IEEE}}'', '''60'''(8):&amp;#8239;936–45, 1972]. In nearly every place where I mention him here, even if I do not accept his conclusion, I am entirely indebted to his works for drawing my attention to the issue raised. In particular, it was through Tai that I became aware of Gibbs's original definitions of the divergence and curl and their suitability for expression in indicial notation ([[#tai-95|Tai, 1995]], pp.&amp;#8239;17,&amp;#8239;21). And although he might not have been pleased, it was through Tai that I first knew with certainty that, if we allow for the variability of the basis vectors, the del-dot and del-cross notations are valid in general coordinates (''ibid.'', pp.&amp;#8239;64–5). Accordingly, this paper is dedicated to him.

{{right|&amp;mdash;&amp;#8201;[[w:User:Gavin R Putland|Gavin R.&amp;#8201;Putland]].}}

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== Overview (for instructors) ==

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The gradient, the curl, the divergence, and the Laplacian are initially defined, without coordinates, as closed-surface integrals per unit volume&amp;mdash;the definition of the Laplacian being indifferent to whether the operand is a scalar field or a vector field. Four integral theorems&amp;mdash;including the divergence theorem&amp;mdash;follow almost immediately, provided that the initial definitions are unambiguous. Their unambiguity, together with some examples of their usefulness, is established as follows, at a level suitable for beginners:
* The gradient is related to an acceleration through an equation of motion;
* The divergence is related to two time-derivatives of density (the partial derivative and the material derivative) through two forms of an equation of continuity;
* The component of the curl in a general direction is expressed as a divergence (now known to be unambiguous);
* The same is done for the general component of the gradient, yielding not only a second proof of unambiguity of the gradient, but also the relation between the gradient and the directional derivative; this together with the original definition of the Laplacian shows that the Laplacian of a ''scalar'' field is the divergence of the gradient and therefore unambiguous. The unambiguity of the Laplacian of a ''vector'' field then follows from a component argument (as for the curl) or a linearity argument.
The derivation of the relation between the gradient and the directional derivative yields a coordinate-free definition of the dot-del operator for a scalar right-hand operand. But, as the directional derivative is also defined for a non-scalar operand, the same relation offers a method of generalizing the dot-del operator, so that the definition of the Laplacian of a general field can be rewritten with that operator. The advection operator&amp;mdash;derived without coordinates, for both scalar and vector properties&amp;mdash;is likewise rewritten.

Meanwhile comparison between the definitions of the various operators leads to coordinate-free definitions of the del-cross, del-dot, and del-squared operators. These together with the dot-del operator allow the four integral theorems to be condensed into a single generalized volume-integral theorem.

If the volume of integration is reduced to a thin curved slab of uniform thickness, with an edge-face perpendicular to the broad faces, the four integral theorems are reduced to their two-dimensional forms, each of which relates an integral over a surface segment to an integral around its enclosing curve, provided that the original ''closed''-surface integral has no contribution from the broad faces of the slab. This proviso can be satisfied by construction in two of the four cases, yielding two general theorems, one of which is the Kelvin&amp;ndash;Stokes theorem. By applying these two theorems to a segment of a closed surface, and expanding the segment to cover the entire surface, it is shown that the gradient is irrotational and the curl is solenoidal.

The next part of the exposition is more conventional, but still coordinate-free. The gradient theorem is derived from the relation between the gradient and the directional derivative. An irrotational field is shown to have a scalar potential. The 1/''r''&amp;#8202; scalar field is shown to be the field whose negative gradient is the inverse-square vector field, whose divergence is a delta function, which is therefore also the negative Laplacian of the 1/''r''&amp;#8202; scalar field. These results enable the construction of a field with a given divergence or a given Laplacian. The wave equation is derived from small-amplitude sound waves in a non-viscous fluid, and shown to be satisfied by a spherical-wave field with a 1/''r''&amp;#8202; amplitude, whose D'Alembertian is a delta function, enabling the construction of a wave function with a given D'Alembertian. But further progress, including the construction of a field with a given ''curl'', seems to require the invocation of a coordinate system.

With the aid of identities already found, expressions are easily obtained for the gradient, curl, divergence, Laplacian, and advection operators in Cartesian coordinates&amp;mdash;with indicial notation and implicit summation, for brevity. While the resulting expressions for the curl and divergence may look unfamiliar, they match the initial definitions given by J.&amp;#8239;Willard Gibbs. The Cartesian expressions are found convenient for deriving further identities: a comprehensive collection (including a multivariate chain rule) is derived, leading to the construction of a field with a given curl in a star-shaped region and, as a by-product, a demonstration that the curl of the velocity field of a rigid body is twice the angular velocity. The curl-of-the-curl identity leads to a second definition of the Laplacian of a vector, the Helmholtz decomposition, and the prediction of electromagnetic waves.

The time-honored method of deriving vector-analytic identities&amp;mdash;treating the divergence and curl as &quot;formal products&quot; with the del operator, varying one field at a time, and adding the results&amp;mdash;is found to be less than rigorous, sometimes less than clear, and hard to justify in view of the ease with which the same thing can be done with Cartesian coordinates, indicial notation, and implicit summation.

The introduction of ''general'' coordinates proceeds through (non-normalized) natural and dual basis vectors, reciprocity, the Kronecker delta, covariance of the natural basis, contravariance of the dual basis, contravariant and covariant components, local bases, contravariance of coordinates, covariance of derivatives w.r.t. coordinates, the Jacobian, and handedness. Reciprocity leads to the dot-product of two vector fields and, via the permutation symbol, to the cross-products of the basis vectors, the definition of one basis in terms of the other, the cross-product of two vector fields, and reciprocity of the covariant and contravariant Jacobians. Thus the stage is set for expressing operators in general coordinates.

The multivariate chain rule leads to expressions for the directional derivative (in terms of the contravariant basis), hence the gradient (del) and advection operators. The identity for the curl of the product of a scalar and a vector leads to an expression for the curl in terms of covariant components. Expressions for the curl and divergence ''operators'' are obtained from the original volume-based definitions, and are found to agree with del-cross and del-dot respectively, with del expressed in the same general coordinates. The volume-based definition of the divergence leads, by a simpler path, to an expression in terms of contravariant components, which in turn yields an expression for the Laplacian.

Affine coordinates are briefly described before proceeding to orthogonal coordinates. In the latter, the Jacobian is simplified and we can choose an orthonormal basis, which is its own reciprocal, so that vectors can be specified in components w.r.t. a single basis. By expressing the old basis vectors and components in terms of the new ones, we can re-express dot-products, cross-products, and differential operators in terms of orthogonal coordinates with an orthonormal basis.

In an appendix, Huygens' principle is mathematized by deriving Green's identities and thence Kirchhoff's integral theorem (''without''&amp;#8202; assuming sinusoidal time-dependence), and then interpreting Kirchhoff's integrand as a distribution of secondary sources.

Some technicalities are relegated to the &quot;Notes&quot;, which are followed by the &quot;Citations&quot;, the &quot;References&quot; cited, and finally&amp;mdash;to compensate for the absence of a &quot;History&quot; section&amp;mdash;some suggested &quot;Further reading&quot;.

=== To-do list ===

Although this resource should be usable already, some improvements are envisaged, namely:
* More illustrations;
* A note on the metric tensor and its determinant.

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== Introduction ==

=== Scalars, vectors, tensors, and coordinates ===

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Elementary calculus concerns differentiation and integration with respect to a ''real'' variable. Vector analysis, or &quot;vector calculus&quot;, concerns what we might call differentiation and integration w.r.t. a ''vector'' variable&amp;mdash;usually the position vector. The function &quot;differentiated&quot; or &quot;integrated&quot; w.r.t. that vector may also be a vector.{{efn|Some authors treat &quot;vector analysis&quot; and &quot;vector calculus&quot; as synonymous. Others, apparently influenced by the difference between elementary &quot;calculus&quot; and real &quot;analysis&quot;, would say that &quot;vector analysis&quot; is more general, more theoretical, and more rigorous than &quot;vector calculus&quot;. That distinction might have surprised the inventors of &quot;vector analysis&quot;, as it was originally called; their motives were specific and practical, and their methods were ad-hoc.}}

But what exactly is a '''vector'''? Mathematicians define a &quot;vector&quot; as a member of a ''[[w:vector space|vector space]]'', which is a [[w:set (mathematics)|set]] whose members satisfy certain basic rules of algebra (called the ''vector-space axioms'') in relation to another set called a ''[[w:field (mathematics)|field]]'' (e.g., the real numbers), which has its own basic rules of algebra (the ''field axioms''), and whose members are called &quot;scalars&quot;. Physicists are more fussy. They typically want a &quot;vector&quot; to be not only a member of a vector space, but also a '''first-order tensor'''&amp;#8239;: a &quot;tensor&quot;, meaning that it exists independently of any coordinate system with which it might be specified; and &quot;first-order&quot; (or &quot;first-degree&quot;, or &quot;first-rank&quot;), meaning that it is specified by a ''one''-dimensional array of numbers. Similarly, a 2nd-order tensor is specified by a 2-dimensional array (a matrix), and a 3rd-order by a 3-dimensional array, and so on. Hence they want a &quot;scalar&quot;, which is specified by a single number (a zero-dimensional array), to be a ''zero-order tensor''. In &quot;vector analysis&quot;, we are greatly interested in applications to physical situations, and accordingly take the physicists' view on what constitutes a vector or a scalar.

So, for our purposes, defining a quantity by three components in (say) a Cartesian coordinate system is not enough to make it a vector, and defining a quantity as a real function of a list of coordinates is not enough to make it a scalar, because we still need to show that the quantity has an independent existence. One method of doing this (''not''&amp;#8202; the method we shall use here!) is to show that the coordinate representation behaves appropriately when the coordinate system is changed. Independent existence of a ''quantity'' means that its coordinate representation changes so as to compensate for the change in the coordinate system.&lt;ref&gt;E.g., Feynman ([[#feynman-63|1963]], vol.&amp;#8239;1, &amp;sect;&amp;#8202;11-5), having defined velocity from displacement in Cartesian coordinates, shows that velocity is a vector by showing that its coordinate representation contra-rotates (like that of displacement) if the coordinate system rotates.&lt;/ref&gt; But independent existence of an ''operator'' means that its expression in one coordinate system (with the operand[s] and the result ''in that system'') gives the same result as the corresponding expression in another coordinate system.&lt;ref&gt;E.g., Feynman ([[#feynman-63|1963]], vol.&amp;#8239;1, &amp;sect;&amp;#8202;11-7), having defined the magnitude and dot-product in Cartesian coordinates, proves that they are scalar functions by showing that the corresponding expressions in rotated (&quot;primed&quot;) coordinates give the same values as the original expressions (in &quot;unprimed&quot; coordinates). And Tai ([[#tai-95|1995]], pp.&amp;#8239;66–7), having found an expression for the &quot;gradient&quot; operator in a general coordinate system (the &quot;unprimed&quot; system), proves the &quot;invariance&quot; of the operator (its vector character in this case) by showing that the corresponding expression in any other general coordinate system (the &quot;primed&quot; system) has the same effect.&lt;/ref&gt;

Here we shall circumvent these complications by the most obvious route: by initially ''defining things without coordinates''. If, having defined something without coordinates, we then need to represent it ''with'' coordinates, we can choose the coordinate system for convenience rather than generality.

For example, without using coordinates, we can define displacements in three-dimensional space by their '''magnitudes''' and '''directions''' and show that they satisfy the vector-space rules, so that they are vectors in the mathematicians' sense, and therefore (because we have defined them without coordinates) in the physicists' sense. Then, by the same rules, we can show that the derivatives of these vectors w.r.t. time are vectors, and that products of these vectors with a scalar (such as mass) are vectors, with the result that not only displacement but also velocity, acceleration, momentum, and force are vectors. Having thus established that these things exist independently of any coordinate system, we can choose convenient coordinates.

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=== Prerequisites ===

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I assume that the reader is familiar with the algebra and geometry of vectors in 3D space, including the dot-product, the cross-product, and the scalar triple product, their geometric meanings, their expressions in Cartesian coordinates, and the identity
:{{big|{{math|'''a''' &amp;times; ('''b''' &amp;times; '''c''')  {{=}}  '''a&amp;sdot;&amp;#8202;c b''' &amp;minus; '''a&amp;sdot;&amp;#8202;b c''' ,}}}}
which we call the &quot;expansion&quot; of the vector triple product.&lt;ref&gt;There are many proofs and interpretations of this identity. My own effort, for what it's worth, is &quot;Trigonometric proof of vector triple product expansion&quot;, ''Mathematics Stack Exchange'', [https://math.stackexchange.com/a/4839213/307861 t.co/NM2v4DJJGo], 2024. The classic is [[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;sect;&amp;#8239;26–7.&lt;/ref&gt; I further assume that the reader can generalize the concept of a derivative, so as to differentiate a vector with respect to a scalar, e.g.
:&lt;math&gt;\mathbf{r}'(t) = \frac{d\mathbf{r}}{dt}
  =\, \lim_{h\to 0} \frac{\mathbf{r}(t+h) - \mathbf{r}(t)}{h} \,,&lt;/math&gt;
or so as to differentiate a function of several independent variables &quot;partially&quot; w.r.t. one of them while the others are held constant, e.g.
:&lt;math&gt;\tfrac{\part}{\part y} \psi\big(x,y,z\big)
  =\, \lim_{h\to 0} \frac{\psi(x,y{+}h~\!,z) - \psi(x,y,z)}{h} \,.&lt;/math&gt;

But, in view of the limited applicability of limits (see the [[#Preface|Preface]]), I also expect the reader to be tolerant of an argument like this: In a short time{{mvar| dt}}, let the vectors {{math|'''r'''}} and{{math| '''p'''}} change by {{math|''d'''''r'''}} and{{math| ''d'''''p'''}} respectively. Then
:&lt;math&gt;\begin{align}
  \tfrac{d}{dt}\big(\mathbf{r}\!\times\!\mathbf{p}\big)
  &amp;= \frac{(\mathbf{r}+d\mathbf{r})\times(\mathbf{p}+d\mathbf{p})
            \,-\, \mathbf{r}\times\mathbf{p}}{dt}\\[1ex]
  &amp;= \frac{\mathbf{r}\!\times\!d\mathbf{p}+d\mathbf{r}\!\times\!\mathbf{p}}{dt}
       ~~\quad [\mathsf{neglecting}~d\mathbf{r}\!\times\!d\mathbf{p}]\\[1ex]
  &amp;=\, \mathbf{r}\times\!\tfrac{d\mathbf{p}}{dt}
       + \tfrac{d\mathbf{r}}{dt}\!\times\mathbf{p}\\[1ex]
  &amp;=\, \mathbf{r}\times\mathbf{\dot{p}}\,+\,\mathbf{\dot{r}}\times\mathbf{p}\,,
\end{align}&lt;/math&gt;
where, as always, the orders of the cross-products matter.{{efn|If {{math|'''r'''}} is the position of a particle and {{math|'''p'''}} is its momentum, the last term vanishes. If the force is toward the origin, the previous term also vanishes, and we are left with ''conservation of angular momentum'' about the origin.}} Differentiation of a ''dot''-product behaves similarly, except that the orders ''don't'' matter; and if&amp;#8239;{{math| '''p'''&amp;#8201;{{=}}&amp;#8201;''m'''''v'''}}, where {{mvar|m}} is a scalar and {{math|'''v'''}} is a vector, then
:&lt;math&gt;~~\mathbf{\dot{p}} = m\mathbf{\dot{v}} + \dot{m}\mathbf{v} \,.&lt;/math&gt;

Or an argument like this:  If&lt;math&gt;~z\!=\!f(x,y)&lt;/math&gt;, then
:&lt;math&gt;\begin{align}
  \frac{\part^2 z}{\part x\,\part y}
  &amp;= \tfrac{\part}{\part x}\,\tfrac{\part}{\part y} f\big(x,y\big)\\
  &amp;= \frac{\part}{\part x}\,\frac{f(x,y{+}dy)-f(x,y)}{dy}\\[1ex]
  &amp;= \frac{\,\frac{f(x{+}dx~\!,\,y{+}dy)\,-\,f(x{+}dx~\!,\,y)}{dy}
           - \frac{f(x,\,y{+}dy)\,-\,f(x,y)}{dy}\,} {dx}\\[2ex]
  &amp;= \frac{\,\frac{f(x{+}dx~\!,\,y{+}dy)\,-\,f(x,\,y{+}dy)}{dx}
           - \frac{f(x{+}dx~\!,\,y)\,-\,f(x,y)}{dx}\,} {dy}\\[1ex]
  &amp;= \frac{\part}{\part y}\,\frac{f(x{+}dx~\!,~\!y)-f(x,y)}{dx}\\[1ex]
  &amp;= \tfrac{\part}{\part y}\,\tfrac{\part}{\part x} f\big(x,y\big)
   = \frac{\part^2 z}{\part y\,\part x} \,;
\end{align}&lt;/math&gt;
that is, we can switch the order of differentiation in a &quot;mixed&quot; partial derivative. If{{mvar| &amp;part;&lt;sub&gt;x&lt;/sub&gt;}} is an abbreviation for {{mvar|{{sfrac|&amp;part;|&amp;part;x}}&amp;#8202;}}, etc., this rule can be written in '''operational''' terms as
:{{big|{{mvar|&amp;part;&lt;sub&gt;x &lt;/sub&gt;&amp;part;&lt;sub&gt;y&lt;/sub&gt; {{=}} &amp;part;&lt;sub&gt;y &lt;/sub&gt;&amp;part;&lt;sub&gt;x &lt;/sub&gt;.}}}}
More generally, if {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}} is an abbreviation for {{mvar|{{sfrac|&amp;part;|&amp;part;x&lt;sub&gt;i&lt;/sub&gt;}}}}&amp;#8202; where&amp;#8202; {{math|''i''&amp;#8239;&amp;#8714;&amp;#8202;&amp;lcub;1,&amp;#8202;2,&amp;hellip;&amp;rcub;,}}&amp;#8202; the rule becomes
:{{big|{{mvar|&amp;part;&lt;sub&gt;i &lt;/sub&gt;&amp;part;&lt;sub&gt;j&lt;/sub&gt; {{=}} &amp;part;&lt;sub&gt;j &lt;/sub&gt;&amp;part;&lt;sub&gt;i &lt;/sub&gt;.}}}}

The above generalizations of differentiation, however, do not go beyond differentiation w.r.t. ''real'' variables, some of which are scalars, and some of which are coordinates. It is now time to consider various kinds of &quot;differentiation&quot; w.r.t. the position vector.

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== Closed-surface integrals per unit volume ==

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The term ''field'', mentioned above in the context of algebraic axioms, has another meaning, which will be its usual meaning from now on: if {{math|'''r'''}} is the position vector, a '''scalar field''' is a scalar-valued function of{{math| '''r''',}} and a '''vector field''' is a vector-valued function of{{math| '''r'''}}; both may also depend on time. These are the functions of which we want &quot;derivatives&quot; w.r.t. the vector{{math| '''r'''}}.

In this section I introduce four such derivatives&amp;mdash;the ''gradient'', the ''curl'', the ''divergence'', and the ''Laplacian''&amp;#8202;&amp;mdash;in a way that will seem unremarkable to those readers who aren't already familiar with them, but idiosyncratic to those who are. The gradient is commonly introduced in connection with a curve and its endpoints, the curl in connection with a surface segment and its enclosing curve, the divergence in connection with a volume and its enclosing surface, and the Laplacian as a composite of two of the above, initially applicable only to a scalar field. Here I introduce all four in connection with a volume and its enclosing surface, and I introduce the Laplacian as a concept in its own right, equally applicable to a scalar ''or vector''&amp;#8202; field; only later do I express the Laplacian in terms of other &quot;derivatives&quot;. My initial definitions of the gradient, the curl, and the Laplacian, although not novel, are usually thought to be more advanced than the common ones&amp;mdash;in spite of being conceptually simpler, and in spite of being obvious variations on the same theme.

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=== Instant integral theorems (with a caveat) ===

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Let {{mvar|V}} be a volume (3D region) enclosed by a surface {{mvar|S}} (a mathematical surface, ''not'' generally a physical barrier). Let &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; be the unit normal vector at a general point on {{mvar|S}}, pointing ''out'' of{{mvar| V}}. Let {{mvar|n}} be the distance from {{mvar|S}} in the direction of&lt;math&gt;~\mathbf{\hat{n}}&lt;/math&gt; (positive outside {{mvar|V}}, negative inside), and let {{mvar|&amp;part;&lt;sub&gt;n&lt;/sub&gt;}} be an abbreviation for{{mvar| {{sfrac|&amp;part;|&amp;part;n}}&amp;#8202;}}, where the derivative&amp;mdash;commonly called the '''normal derivative'''&amp;mdash;is tacitly assumed to exist.

In {{mvar|V}}, and on {{mvar|S}}, let {{mvar|p}} be a scalar field (e.g., pressure in a fluid, or temperature), and let {{math|'''q'''}} be a vector field (e.g., flow velocity, or heat-flow density), and let {{mvar|&amp;psi;}} be a generic field which may be a scalar or a vector. Let a general ''element'' (small segment) of the surface {{mvar|S}} have area {{mvar|dS}}, and let it be small enough to allow &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;, {{mvar|p}}, {{math|'''q'''}}, and {{mvar|&amp;part;&lt;sub&gt;n&lt;/sub&gt;&amp;#8202;&amp;psi;}} to be considered uniform over the element (making a tacit assumption of local continuity). Then, for every element, the following four products are well defined:
{{NumBlk|:|&lt;math&gt;\mathbf{\hat{n}}            ~\!p\,dS ~,\qquad
                 \mathbf{\hat{n}}\times\mathbf{q}\,dS ~,\qquad
                 \mathbf{\hat{n}         \cdot q}\,dS ~,\qquad
                 \part_n                     \psi\;dS \,.
          &lt;/math&gt;|{{EquationRef|1}}}}
If {{mvar|p}} is pressure in a non-viscous fluid, the first of these products is the force exerted by the fluid in {{mvar|V}}&amp;#8202; through the area {{mvar|dS}}. The second product does not have such an obvious physical interpretation; but if{{math| '''q'''}} is ''circulating'' clockwise about an axis directed through{{mvar| V}}, the cross-product will be exactly tangential to{{mvar| S}} and will tend to have a component in the direction of that axis. The third product is the ''flux'' of{{math| '''q'''}} through the surface element; if{{math| '''q'''}} is flow velocity, the third product is the volumetric flow rate (volume per unit time) ''out'' of{{mvar| V}}&amp;#8202; through{{mvar| dS&amp;#8202;}}; or if {{math|'''q'''}} is heat-flow density, the third product is the heat transfer rate (energy per unit time) ''out'' of{{mvar| V}}&amp;#8202; through{{mvar| dS}}. The fourth product, by analogy with the third, might be called the flux of the normal derivative of{{mvar| &amp;psi;}} through the surface element, but is equally well defined whether {{mvar|&amp;psi;}} is a scalar or a vector&amp;mdash;or, for that matter, a matrix, or a tensor of any order, or anything else that we can differentiate w.r.t.{{mvar| n}}.

If we add up each of the four products over all the elements of the surface {{mvar|S}}, we obtain, respectively, the four '''surface integrals'''
{{NumBlk|:|&lt;math&gt;\iint_S \!\mathbf{\hat{n}}            ~\!p\,dS \,,~
                 \iint_S \!\mathbf{\hat{n}}\times\mathbf{q}\,dS \,,~
                 \iint_S \!\mathbf{\hat{n}         \cdot q}\,dS \,,~
                 \iint_S \!\part_n                     \psi\;dS \,,
          &lt;/math&gt;|{{EquationRef|2}}}}
in which the double integral sign indicates that the range of integration is two-dimensional. The first surface integral takes a scalar field and yields a vector; the second takes a vector field and yields a vector; the third takes a vector field and yields a scalar; and the fourth takes (e.g.) a scalar field yielding a scalar, or a vector field yielding a vector. If{{mvar| p}} is pressure in a non-viscous fluid, the first integral is the force exerted by the fluid in {{mvar|V}}&amp;#8202; on the fluid outside {{mvar|V}}. The second integral may be called the ''skew'' surface integral of{{math| '''q'''}} over {{mvar|S&amp;#8202;}},&lt;ref&gt;[[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;#8239;56.&lt;/ref&gt; or, for the reason hinted above, the ''circulation'' of{{math| '''q'''}} over {{mvar|S}}.&amp;#8201; The third integral, commonly called the ''flux integral'' (or simply the surface integral) of{{math| '''q'''}} over {{mvar|S}}, is the total ''flux'' of{{math| '''q'''}} out of{{mvar| V}}. And the fourth integral is the surface integral of the outward normal derivative of{{mvar| &amp;psi;}}.

Let the volume {{mvar|V}}&amp;#8202; be divided into elements. Let a general volume element have the volume {{mvar|dV}} and be enclosed by the surface {{mvar|&amp;delta;S}}&amp;#8201;&amp;mdash;not to be confused with the area {{mvar|dS}} of a surface ''element'', which may be an element of{{mvar| S}} or of{{mvar| &amp;delta;S}}. Then consider what happens if, instead of evaluating each of the above surface integrals over {{mvar|S}}, we evaluate it over each {{mvar|&amp;delta;S}} and add up the results for all the volume elements. In the ''interior'' of{{mvar| V}}, each surface element of area {{mvar|dS}} is on the boundary between two volume elements, for which the unit normals &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; at {{mvar|dS}}, and the respective values of{{mvar| &amp;part;&lt;sub&gt;n&lt;/sub&gt;&amp;#8202;&amp;psi;}}, are equal and opposite. Hence when we add up the integrals over the surfaces {{mvar|&amp;delta;S}}, the contributions from the elements {{mvar|dS}} cancel in pairs, except on the original surface {{mvar|S}}, so that we are left with the original integral over {{mvar|S}}. So, for the four surface integrals in ({{EquationNote|2}}), we have respectively
{{NumBlk|:|&lt;math&gt;\begin{align}
    \iint_S \mathbf{\hat{n}}~\!p \,dS
    &amp; \,= \sum_V\iint_{\delta S} \mathbf{\hat{n}}~\!p \,dS \,, \\
    \iint_S \mathbf{\hat{n}}\times\mathbf{q} \,dS
    &amp; \,= \sum_V\iint_{\delta S} \mathbf{\hat{n}}\times\mathbf{q} \,dS \,, \\
    \iint_S \mathbf{\hat{n}\cdot q} \,dS
    &amp; \,= \sum_V\iint_{\delta S} \mathbf{\hat{n}\cdot q} \,dS \,, \\
    \iint_S \part_n \psi \;dS
    &amp; \,= \sum_V\iint_{\delta S} \part_n \psi \;dS \,.
  \end{align}&lt;/math&gt;|{{EquationRef|3}}}}

Now comes a big &quot;if&quot;:&amp;#8201; ''if''&amp;#8202; we define the '''gradient''' of{{mvar| p}} (pronounced &quot;grad {{mvar|p}}&quot;) inside {{mvar|dV}}&amp;#8202; as
{{NumBlk|:|&lt;math&gt;\nabla p
    \,:=\, \frac{1}{dV}\iint_{\delta S} \mathbf{\hat{n}}~\!p \,dS
  &lt;/math&gt;|{{EquationRef|4g}}}}
and the '''curl''' of {{math|'''q'''}} inside {{mvar|dV}}&amp;#8202; as
{{NumBlk|:|&lt;math&gt;\operatorname{curl}\mathbf{q}
    \,:=\, \frac{1}{dV}\iint_{\delta S} \mathbf{\hat{n}}\times\mathbf{q} \,dS
  &lt;/math&gt;|{{EquationRef|4c}}}}
and the '''divergence''' of {{math|'''q'''}} inside {{mvar|dV}}&amp;#8202; as
{{NumBlk|:|&lt;math&gt;\operatorname{div}\mathbf{q}
    \,:=\, \frac{1}{dV}\iint_{\delta S} \mathbf{\hat{n}\cdot q} \,dS
  &lt;/math&gt;|{{EquationRef|4d}}}}
and the '''Laplacian''' of {{mvar|&amp;psi;}} inside {{mvar|dV}}&amp;#8202; as
{{NumBlk|:|&lt;math&gt;\triangle\psi
    \,:=\, \frac{1}{dV}\iint_{\delta S} \part_n \psi \;dS
  &lt;/math&gt;|{{EquationRef|4L}}}}
(where the letters after the equation number stand for ''gradient'', ''curl'', ''divergence'', and ''Laplacian'', respectively), then equations ({{EquationNote|3}}) can be rewritten
:&lt;math&gt;\begin{align}
    \iint_S \mathbf{\hat{n}}~\!p \,dS
    &amp; \,= \sum_V \nabla p ~dV \,, \\
    \iint_S \mathbf{\hat{n}}\times\mathbf{q} \,dS
    &amp; \,= \sum_V \operatorname{curl}\mathbf{q} ~dV \,, \\
    \iint_S \mathbf{\hat{n}\cdot q} \,dS
    &amp; \,= \sum_V \operatorname{div}\mathbf{q} ~dV \,, \\
    \iint_S \part_n \psi \;dS
    &amp; \,= \sum_V \triangle\psi ~dV \,.
  \end{align}&lt;/math&gt;
(For the Laplacian operator, we have used the broad triangle symbol{{math| (&amp;#9651;)}} rather than the narrower Greek Delta{{math| (&amp;Delta;)}}; the latter would more readily be misinterpreted as &quot;change in&amp;hellip;&quot;)&amp;#8201; But because each term in each sum above has a factor {{mvar|dV}}, we call the sum an integral; and because the range of integration is three-dimensional, we use a triple integral sign. Thus we obtain the following four theorems relating integrals over an enclosing surface {{mvar|S}}&amp;#8202; to integrals over the enclosed volume {{mvar|V&amp;#8202;}}:
{{NumBlk|:|&lt;math&gt;~~~~~\!\iint_S \mathbf{\hat{n}}~\!p \,dS
    \,= \iiint_V \nabla p ~dV \,;
  &lt;/math&gt;|{{EquationRef|5g}}}}
{{NumBlk|:|&lt;math&gt;\iint_S \mathbf{\hat{n}}\times\mathbf{q} \,dS
    \,= \iiint_V \operatorname{curl}\mathbf{q} ~dV \,;
  &lt;/math&gt;|{{EquationRef|5c}}}}
{{NumBlk|:|&lt;math&gt;~~\!\iint_S \mathbf{\hat{n}\cdot q} \,dS
    \,= \iiint_V \operatorname{div}\mathbf{q} ~dV \,;
  &lt;/math&gt;|{{EquationRef|5d}}}}
{{NumBlk|:|&lt;math&gt;~~\iint_S \part_n \psi \;dS
    \,= \iiint_V \triangle\psi ~dV \,.
  &lt;/math&gt;|{{EquationRef|5L}}}}
Of the above four results, only the third ({{EquationNote|5d}}) seems to have a standard name; it is called the '''divergence theorem''' (or ''Gauss's theorem'' or, more properly, ''[[w:Mikhail Ostrogradsky|Ostrogradsky]]'s theorem''&lt;ref&gt;[[#katz-79|Katz, 1979]], pp.&amp;#8239;146–9.&lt;/ref&gt;), and is indeed the best known of the four&amp;mdash;although the other three, having been derived in parallel with it, may be said to be equally fundamental.

As each of the operators {{math|&amp;nabla;,}} {{math|curl,}} and {{math|div}} calls for an integration w.r.t. area and then a division by volume, the ''dimension'' (or unit of measurement) of the result is the dimension of the operand divided by the dimension of length, as if the operation were some sort of differentiation w.r.t. position. Moreover, in each of equations ({{EquationNote|5g}}) to ({{EquationNote|5d}}), there is a triple integral on the right but only a double integral on the left, so that each of the operators {{math|&amp;nabla;,}} {{math|curl,}} and {{math|div}} appears to compensate for a single integration. For these reasons, and for convenience, we shall describe them as '''differential operators'''. By comparison, the {{math|&amp;#9651; }}operator in ({{EquationNote|4L}}) or ({{EquationNote|5L}}) calls for a further differentiation w.r.t.{{mvar| n&amp;#8202;}}; we shall therefore describe {{math|&amp;#9651;}} as a ''2nd-order'' differential operator. (An additional reason for these descriptions will emerge later.) As promised, the four definitions ({{EquationNote|4g}}) to ({{EquationNote|4L}}) are &quot;obvious variations on the same theme&quot; (although the fourth is somewhat less obvious than the others).

But remember the &quot;if&quot;: Theorems ({{EquationNote|5g}}) to ({{EquationNote|5L}}) depend on definitions ({{EquationNote|4g}}) to ({{EquationNote|4L}}) and are therefore only as definite as those definitions! Equations ({{EquationNote|3}}), without assuming anything about the shapes and relative sizes of the closed surfaces {{mvar|&amp;delta;S}} (except, tacitly, that&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is piecewise well-defined), indicate that the surface integrals are ''additive with respect to volume''. But this additivity, by itself, does not guarantee that the surface integrals are shared among neighboring volume elements ''in proportion'' to their volumes, as envisaged by &quot;definitions&quot; ({{EquationNote|4g}}) to ({{EquationNote|4L}}). Each of these &quot;definitions&quot; is unambiguous if, and only if, the ratio of the surface integral to{{mvar| dV}}&amp;#8202; is insensitive to the shape and size of{{mvar| &amp;delta;S}}&amp;#8202; for a sufficiently small {{mvar|&amp;delta;S}}. Notice that the issue here is ''not'' whether the ratios specified in equations ({{EquationNote|4g}}) to ({{EquationNote|4L}}) are true vectors or scalars, independent of the coordinates; all of the operations needed in those equations have coordinate-free definitions. Rather, the issue is whether the resulting ratios are unambiguous ''notwithstanding the ambiguity of'' {{mvar|&amp;delta;S}}, provided only that {{mvar|&amp;delta;S}} is sufficiently small. That is the advertised &quot;caveat&quot;, which must now be addressed.

Our proofs of the unambiguity of the differential operators will rest on a few [[w:thought experiment|thought experiments]], each of which applies an operator to a physical field, say{{mvar| f}}, and obtains another physical field whose unambiguity is beyond dispute, provided only that it can be considered uniform over the (small) volume element. The conclusion of the thought experiment is then applicable to any operand field whose ''mathematical'' properties are consistent with the physical interpretation; the loss of generality, if any, is only what is incurred by that interpretation.

{{cob}}

=== Unambiguity of the gradient ===

{{cot}}

Suppose that a fluid with density {{mvar|&amp;rho;}} (a scalar field) flows with velocity{{math| '''v'''}} (a vector field) under the influence of the internal pressure {{mvar|p}} (a scalar field). Then the integral in ({{EquationNote|4g}}) is the force exerted by the pressure of the fluid inside {{mvar|&amp;delta;S}} on the fluid outside, so that ''minus'' the integral is the force exerted ''on'' the fluid inside{{mvar| &amp;delta;S}}&amp;#8202; by the pressure of the fluid outside. Dividing by {{mvar|dV}}, we find that {{math|&amp;minus;&amp;nabla;''p''}}, as defined by ({{EquationNote|4g}}), is the force per unit volume, due to the pressure outside the volume.&lt;ref&gt;In [[#feynman-63|Feynman, 1963]],&amp;#8201; {{math|&amp;minus;&amp;nabla;''p'' }}as the &quot;pressure force per unit volume&quot; eventually appears in the 3rd-last lecture of Volume 2 (&amp;sect;40-1).&lt;/ref&gt; If this is the ''only'' force per unit volume acting ''on'' the volume (e.g., because the fluid is non-viscous and in a weightless environment, and the volume element is not in contact with the container), then it is equal to the acceleration times the mass per unit volume; that is,
{{NumBlk|:|&lt;math&gt;
    \rho\,\frac{d\mathbf{v}}{dt} = -\nabla p \,.
  &lt;/math&gt;|{{EquationRef|6g}}}}
Now provided that the left-hand expression can be considered uniform inside the small {{mvar|&amp;delta;S}}, it is unambiguous, whence&amp;#8202; {{math|&amp;nabla;''p'' }}''is also unambiguous''. If there are additional forces on the fluid element, e.g. due to gravity and&amp;#10744;or viscosity, then {{math|&amp;minus;&amp;nabla;''p''}} is not the sole contribution to density-times-acceleration, but is still the contribution due to pressure, which is still unambiguous.

By showing the unambiguity of definition ({{EquationNote|4g}}), we have confirmed theorem ({{EquationNote|5g}}). In the process we have seen that the volume-based definition of the gradient is useful for the modeling of fluids, and intuitive in that it formalizes the common notion that a pressure &quot;gradient&quot; gives rise to a force.

{{cob}}

=== Unambiguity of the divergence ===

{{cot}}

In the aforesaid fluid, in a short time{{mvar| dt}}, the volume that flows out of fixed closed surface {{mvar|&amp;delta;S}}&amp;#8202; through a fixed surface element of area {{mvar|dS}}&amp;#8202; is &lt;math&gt;\mathbf{v}~\!dt\!\cdot\!\mathbf{\hat{n}}\,dS&lt;/math&gt;&amp;#8239; (i.e., the displacement normal to the surface element, times the area).&amp;#8201; Multiplying this by density and integrating over {{mvar|&amp;delta;S}}, we find that the mass flowing out of{{mvar| &amp;delta;S}}&amp;#8202; in time{{mvar| dt}} is&amp;#8201; &lt;math&gt;\textstyle\iint_{\delta S}\rho\mathbf{v}~\!dt\cdot\mathbf{\hat{n}}\,dS&lt;/math&gt;.  Dividing this by {{mvar|dV}}, and then by {{mvar|dt}}, we get the rate of reduction of density inside {{mvar|&amp;delta;S&amp;#8202;}}; that is,
:&lt;math&gt;\tfrac{1}{dV}\!\iint_{\delta S}\mathbf{\hat{n}}\cdot\rho\mathbf{v}\,dS
       \,= -\frac{\part\rho}{\part t} \,,&lt;/math&gt;
where the derivative w.r.t. time is evaluated at a fixed location (because {{mvar|&amp;delta;S}} is fixed), and is therefore written as a ''partial'' derivative (because other variables on which {{mvar|&amp;rho;}} might depend&amp;mdash;namely spatial coordinates&amp;mdash;are held constant). Provided that the right-hand side can be considered uniform inside {{mvar|&amp;delta;S}}, it is unambiguous, so that the left side is likewise unambiguous. But the left side is simply{{math| div&amp;#8239;''&amp;rho;'''''v'''}}&amp;#8201; as defined by ({{EquationNote|4d}}),{{efn|There is no need for parentheses around{{math| ''&amp;rho;'''''v'''&amp;#8202;,}} because {{math|div&amp;#8239;''&amp;rho;'''''v'''&amp;#8202;}} cannot mean {{math|(div&amp;#8239;''&amp;rho;'')'''v'''&amp;#8202;,}} because the divergence of a scalar field is not defined.}} which is therefore also unambiguous,&lt;ref&gt;A demonstration like the foregoing is outlined by Gibbs ([[#gibbs-1881-4|1881]], &amp;sect;&amp;#8239;55).&lt;/ref&gt; confirming theorem ({{EquationNote|5d}}). In short, the divergence operator is that which maps {{math|''&amp;rho;'''''v'''}} to the rate of reduction of density at a fixed point:
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\rho\mathbf{v} = -\frac{\part\rho}{\part t} \,.
  &lt;/math&gt;|{{EquationRef|7d}}}}
This result, which expresses ''conservation of mass'', is a form of the so-called '''equation of continuity'''.

The partial derivative {{mvar|{{sfrac|&amp;part;&amp;rho;|&amp;part;t}}&amp;#8202;}} in ({{EquationNote|7d}}) must be distinguished from the '''material derivative''' {{mvar|{{sfrac|d&amp;rho;|dt}}&amp;#8202;}}, which is evaluated at a point that moves ''with the fluid''.{{efn|The material derivative operator {{mvar|{{sfrac|d|dt}}}} is also called the ''substantive'' derivative, and is sometimes written {{mvar|{{sfrac|D|Dt}}}} if the result is meant to be understood as a field rather than simply a function of time ([[#kemmer-77|Kemmer, 1977]], pp.&amp;#8239;184–5).}} [Similarly, {{math|{{sfrac|''d''&amp;#8202;'''v'''|''dt''}}}} in ({{EquationNote|6g}}) is the ''material'' acceleration, because it is the acceleration of the mobile mass&amp;mdash;not of a fixed point!&amp;#8239;]&amp;#8239; To re-derive the equation of continuity in terms of the ''material'' derivative, the volume &lt;math&gt;\mathbf{v}~\!dt\!\cdot\!\mathbf{\hat{n}}\,dS~\!,&lt;/math&gt; which flows out through{{mvar| dS}} in time{{mvar| dt}} (as above), is integrated over {{mvar|&amp;delta;S}} to obtain the increase in volume of the mass ''initially'' contained in {{mvar|dV}}. Dividing this by the mass, {{mvar|&amp;rho;&amp;#8239;dV}}, gives the increase in ''[[w:specific volume|specific volume]]'' {{math|(1&amp;#10744;''&amp;rho;'')}} of that mass, and then dividing by {{mvar|dt}} gives the rate of change of specific volume; that is,
:&lt;math&gt;\tfrac{1}{\rho~\!dV}\!\iint_{\delta S}\mathbf{\hat{n}}\cdot\mathbf{v}\,dS
       \,= \tfrac{d}{dt}\big(\rho^{-1}\big)
       = -\rho^{-2\,}\tfrac{d\rho}{dt} \,.&lt;/math&gt;
Multiplying by {{math|''&amp;rho;''&amp;sup2;}} and comparing the left side with ({{EquationNote|4d}}), we obtain
{{NumBlk|:|&lt;math&gt;
    \rho\operatorname{div}\mathbf{v} = -\frac{d\rho}{dt} \,.
  &lt;/math&gt;|{{EquationRef|7d'}}}}
Whereas ({{EquationNote|7d}}) shows that {{math|div&amp;#8239;''&amp;rho;'''''v'''&amp;#8202;}} is unambiguous, ({{EquationNote|7d'}}) shows that {{math|div&amp;#8239;'''v'''&amp;#8202;}} is unambiguous (provided that other things are locally continuous). In accordance with the everyday meaning of &quot;divergence&quot;, ({{EquationNote|7d'}}) also shows that {{math|div&amp;#8239;'''v'''&amp;#8202;}} is positive if the fluid is expanding ({{mvar|&amp;rho; }}decreasing), negative if it is contracting ({{mvar|&amp;rho; }}increasing), and zero if it is incompressible. In the last case, the equation of continuity reduces to
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\mathbf{v} = 0
  \qquad&lt;/math&gt;[&amp;#8202;for an incompressible fluid&amp;#8202;].|{{EquationRef|7i}}}}

For incompressible flow, any tubular surface tangential to the flow velocity, and consequently with no flow in or out of the &quot;tube&quot;, has the same volumetric flow rate across all cross-sections of the &quot;tube&quot;, as if the surface were the wall of a pipe full of liquid (except that the surface is not necessarily stationary). Accordingly, ''a vector field with zero divergence is described as '''solenoidal''''' (from the Greek word for &quot;pipe&quot;). More generally, a solenoidal vector field has the property that for any tubular surface tangential to the field, the flux integrals across any two cross-sections of the &quot;tube&quot; are the same&amp;mdash;because otherwise there would be a net flux integral out of the closed surface comprising the two cross-sections and any segment of tube between them, in which case, by the divergence theorem ({{EquationNote|5d}}), the divergence would have to be non-zero somewhere inside, contrary to ({{EquationNote|7i}}).

{{cob}}

=== Unambiguity of the curl (and gradient) ===

{{cot}}

The unambiguity of the curl ({{EquationNote|4c}}) follows from the unambiguity of the divergence. Let {{math|'''b'''}} be any ''uniform'' vector (i.e., any vector that is independent of location&amp;mdash;e.g. a uniform vector field, possibly time-dependent). Taking dot-products of ({{EquationNote|4c}}) with{{math| '''b''',}} we get
:&lt;math&gt;\begin{align}
    \mathbf{b}\cdot\operatorname{curl}\mathbf{q}\,
    &amp;=\, \mathbf{b}\cdot\tfrac{1}{dV}\!\iint_{\delta S}
           \mathbf{\hat{n}}\times\mathbf{q}
         \,dS \\[.5ex]
    &amp;=\, \tfrac{1}{dV}\!\iint_{\delta S}
           \mathbf{b}\cdot\mathbf{\hat{n}}\!\times\!\mathbf{q}
         \,dS \\[1ex]
    &amp;=\, \tfrac{1}{dV}\!\iint_{\delta S}
           \mathbf{\hat{n}}\cdot\mathbf{q}\!\times\!\mathbf{b}
         \,dS \,;
  \end{align}&lt;/math&gt;
that is, by ({{EquationNote|4d}}),
{{NumBlk|:|&lt;math&gt;\operatorname{curl}\mathbf{q} \cdot \mathbf{b}
    = \operatorname{div}(\mathbf{q}\!\times\!\mathbf{b})
  \qquad&lt;/math&gt;[&amp;#8202;for uniform {{math|'''b'''}}].|{{EquationRef|8c}}}}
(The parentheses around&amp;#8202; {{math|'''q'''&amp;#8202;&amp;times;&amp;#8202;'''b'''}}&amp;#8202; on the right, although helpful because of the spacing, are not strictly necessary, because the alternative binding would be {{math|(div&amp;#8201;'''q''')}}, which is a scalar, whose cross-product with the vector {{math|'''b'''}} is not defined. And the left-hand expression does not need parentheses, because it can only mean the dot-product of a curl with the vector {{math|'''b'''}}; it cannot mean the curl of a dot-product, because the curl of a scalar field is not defined.) Equation ({{EquationNote|8c}}) is an identity for ''uniform''{{math| '''b'''}}. If we make {{math|'''b'''}} a ''unit'' vector in any fixed direction, the left-hand side of the identity is the (scalar) component of {{math|curl&amp;#8239;'''q'''}} in that direction, and the right-hand side is unambiguous. Thus ''the curl is unambiguous because its component in any direction is unambiguous''. This confirms theorem ({{EquationNote|5c}}).

Similarly, the unambiguity of the divergence implies the unambiguity of the gradient. Starting with ({{EquationNote|4g}}), taking dot-products with an arbitrary uniform vector {{math|'''b''',}} and proceeding as above, we obtain
{{NumBlk|:|&lt;math&gt;\nabla p \cdot \mathbf{b}
    = \operatorname{div} p\mathbf{b}
  \qquad&lt;/math&gt;[&amp;#8202;for uniform {{math|'''b'''}}].|{{EquationRef|8g}}}}
(The left-hand side does not need parentheses, because it can only mean the dot-product of a gradient with the vector {{math|'''b'''}}; it cannot mean the gradient of the dot-product of a scalar field with a vector field, because that dot-product would not be defined.) If we make {{math|'''b'''}} a ''unit'' vector, this result ({{EquationNote|8g}}) says that the (scalar) component of{{math| &amp;nabla;''p''}} in the direction of{{math|&amp;#8202; '''b'''}} is given by the right-hand side, which again is unambiguous. So here we have a second explanation of the unambiguity of the gradient: like the curl, it is unambiguous because its component in any direction is unambiguous.

We might well ask what happens if we take ''cross''-products with {{math|'''b'''}} on the left, instead of dot-products. If we start with ({{EquationNote|4g}}), the process is straightforward: in the end we can switch the order of the cross-product on the left, and change the sign on the right, obtaining
{{NumBlk|:|&lt;math&gt;\nabla p \times \mathbf{b}
    = \operatorname{curl} p\mathbf{b}
  \qquad&lt;/math&gt;[&amp;#8202;for uniform {{math|'''b'''}}].|{{EquationRef|8p}}}}
(Again no parentheses are needed.) If we start with ({{EquationNote|4c}}) instead, and take {{math|'''b'''}} inside the integral, we get a vector triple product to expand, which leads to
:&lt;math&gt;\mathbf{b} \times \operatorname{curl}\mathbf{q}
    = \tfrac{1}{dV}\!\iint_{\delta S}\!\mathbf{\hat{n}}\,\mathbf{b{\cdot}q}\,dS
    - \tfrac{1}{dV}\!\iint_{\delta S}\!\mathbf{b{\cdot}\hat{n}\,q}\,dS \,,
  &lt;/math&gt;
in which the first term on the right is simply&amp;#8202; {{math|&amp;nabla;&amp;#8239;'''b&amp;sdot;q'''}}&amp;#8201; (the gradient of the dot-product). The second term is more problematic. ''If''&amp;#8202; we had a scalar {{mvar|p}} instead of the vector {{math|'''q''',}} we could take {{math|'''b'''}} outside the second integral, so that the second term would be (minus) {{math|'''b&amp;#8239;&amp;sdot;'''&amp;#8239;&amp;nabla;''p''}}. This suggests that the actual second term should be (minus) {{math|'''b&amp;#8239;&amp;sdot;'''&amp;#8239;&amp;nabla;'''q'''}}.&amp;#8201; Shall we therefore adopt the second term (without the sign) as the ''definition'' of{{math|&amp;#8202; '''b&amp;sdot;'''&amp;nabla;&amp;#8239;'''q'''}} for a ''vector'' {{math|'''q'''}} (treating {{math|'''b&amp;sdot;'''&amp;nabla;}} as an operator), and write
{{NumBlk|:|&lt;math&gt;\mathbf{b} \times \operatorname{curl}\mathbf{q} ~\!=
    \nabla\,\mathbf{b{\cdot}q} - \mathbf{b}~\!{\cdot}\nabla\,\mathbf{q}
  \qquad&lt;/math&gt;[&amp;#8202;for uniform {{math|'''b'''}}] ?|{{EquationRef|8q}}}}
The proposal would be open to the objection that&amp;#8202; {{math|'''b&amp;sdot;'''&amp;nabla;&amp;#8239;'''q'''}}&amp;#8202; had been defined only for ''uniform''{{math| '''b'''&amp;#8202;,}} whereas&amp;#8202; {{math|'''b&amp;#8239;&amp;sdot;'''&amp;#8239;&amp;nabla;''p''&amp;#8202;}} (for scalar{{mvar| p}}) is defined whether {{math|'''b'''}} is uniform or not.&amp;#8201; So, for the moment, let us put ({{EquationNote|8q}}) aside and run with ({{EquationNote|8c}}), ({{EquationNote|8g}}), and ({{EquationNote|8p}}).

{{cob}}

=== Another meaning of the gradient ===

{{cot}}

Let {{math|'''s&amp;#770;'''}} be a unit vector in a given direction, and let {{mvar|s}} be a parameter measuring distance (arc length) along a path in that direction. By equation ({{EquationNote|8g}}) and definition ({{EquationNote|4d}}), we have
:&lt;math&gt;\nabla p \cdot \mathbf{\hat{s}} =
    \operatorname{div} p\mathbf{\hat{s}} =
    \tfrac{1}{dV}\!\iint_{\delta S}\mathbf{\hat{n}}\cdot p\mathbf{\hat{s}}\,dS\,,
  &lt;/math&gt;
where, by the unambiguity of the divergence, the shape of the closed surface {{mvar|&amp;delta;S}} enclosing {{mvar|dV}}&amp;#8202; can be chosen for convenience. So let {{mvar|&amp;delta;S}} be a right cylinder with cross-sectional area {{mvar|&amp;alpha;}}&amp;#8201; and perpendicular height {{mvar|ds&amp;#8202;,}} with the path passing perpendicularly through the end-faces at parameter-values {{mvar|s}} and {{mvar|s+ds&amp;#8202;,}} where the outward unit normal &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; consequently takes the values {{math|&amp;minus;'''s&amp;#770;'''}} and {{math|'''s&amp;#770;'''&amp;#8202;,}} respectively. And let the cross-sectional dimensions be small compared with {{mvar|ds}}&amp;#8202; so that the values of{{mvar| p}} at the end-faces, say {{mvar|p}} and {{mvar|p+dp}}, can be taken to be the same as where the end-faces cut the path. Then&amp;#8202; {{mvar|dV&amp;#8201;{{=}}&amp;#8201;&amp;alpha;&amp;#8239;ds&amp;#8202;}}, and the surface integral over{{mvar| &amp;delta;S}} includes only the contributions from the end-faces (because &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is perpendicular to {{math|'''s&amp;#770;'''}} elsewhere); those contributions are respectively&amp;#8201; &lt;math&gt;-\mathbf{\hat{s}}\!\cdot\!p\mathbf{\hat{s}}\,\alpha\,&lt;/math&gt; and&amp;#8201; &lt;math&gt;\mathbf{\hat{s}}\!\cdot\!(p\!+\!dp)\mathbf{\hat{s}}\,\alpha~\!,&lt;/math&gt;  i.e.&amp;#8239; &lt;math&gt;-p\alpha\,&lt;/math&gt; and &lt;math&gt;(p\!+\!dp)\alpha&lt;/math&gt;.  With these substitutions the above equation becomes
:&lt;math&gt;\begin{align}
    \nabla p \cdot \mathbf{\hat{s}}
    &amp;= \tfrac{1}{\alpha\,ds}\Big({-}p\alpha + (p\!+\!dp)\alpha\Big) \\[1ex]
    &amp;= \frac{\,p\!+\!dp ~-~ p\,}{ds} = \frac{\part p}{\part s} ~;
  \end{align}&lt;/math&gt;
that is,
{{NumBlk|:|&lt;math&gt;
    \nabla p \cdot \mathbf{\hat{s}} = \part_s p \,,
  &lt;/math&gt;|{{EquationRef|9g}}}}
where the right-hand side, commonly called the '''directional derivative''' of{{mvar| p}} in the {{math|'''s&amp;#770;'''}} direction,&lt;ref&gt;[[#wilson-1901|Wilson, 1901]], pp.&amp;#8239;147–8; [[#borisenko-tarapov-68|Borisenko &amp;amp; Tarapov, 1968]], pp.&amp;#8239;147–8 (again); [[#hsu-84|Hsu, 1984]], p.&amp;#8239;92; [[#kreyszig-62-|Kreyszig, 1988]], pp.&amp;#8239;485–6; [[#wrede-spiegel-10|Wrede &amp;amp; Spiegel, 2010]], p.&amp;#8239;198.&lt;/ref&gt; is the derivative of{{mvar| p}} w.r.t. distance in that direction. Although ({{EquationNote|9g}}) has been obtained by taking that direction as fixed, the equality is evidently maintained if {{mvar|s}} measures arc length along any path ''tangential''&amp;#8202; to{{math| '''s&amp;#770;'''}} at the point of interest.

Equation ({{EquationNote|9g}}) is an alternative definition of the gradient: it says that ''the gradient of&lt;math&gt;~p&lt;/math&gt; is the vector whose scalar component in any direction is the directional derivative of&lt;math&gt;~p&lt;/math&gt; in that direction''. For ''real&lt;math&gt;~p&lt;/math&gt;'', this component has its maximum, namely {{math|{{abs|&amp;nabla;''p''}}&amp;#8202;,}} in the direction of{{math| &amp;nabla;''p''&amp;#8202;}}; thus ''the gradient of&lt;math&gt;~p&lt;/math&gt; is the vector whose direction is that in which the derivative of&lt;math&gt;~p&lt;/math&gt; w.r.t. distance is a maximum, and whose magnitude is that maximum''. This is the usual conceptual definition of the gradient.&lt;ref&gt;Gibbs ([[#gibbs-1881-4|1881]], &amp;sect;&amp;#8239;50) ''introduces'' the gradient with this definition, except that he calls {{math|&amp;nabla;''u''}} simply the ''derivative'' of{{mvar| u}}, and {{mvar|u}} the ''primitive'' of{{math| &amp;nabla;''u''}}. Use of the term ''gradient'' as an alternative to ''derivative'' is reported by Wilson ([[#wilson-1901|1901]], p.&amp;#8239;138).&lt;/ref&gt; Sometimes it is convenient to work directly from this definition. For example, in Cartesian coordinates {{math|(''x'',&amp;#8239;''y'',&amp;#8239;''z''),}} if a scalar field is given by {{mvar|x&amp;#8202;,}} its gradient is obviously the unit vector in the direction of the {{mvar|x }}axis, usually called {{math|'''i'''&amp;#8202;}}; that is, {{math|&amp;nabla;''x''&amp;#8201;{{=}}&amp;#8201;'''i'''}}. Similarly, if&amp;#8201; &lt;math&gt;\mathbf{r}=r\mathbf{\hat{r}}&lt;/math&gt;&amp;#8201; is the position vector, then &lt;math&gt;\nabla r = \mathbf{\hat{r}}&lt;/math&gt;.

If&amp;#8202; {{math|'''s&amp;#770;'''}} is ''tangential''&amp;#8202; to a '''level surface''' of{{mvar| p}} (a surface of constant{{mvar| p}}), then {{mvar|&amp;part;&lt;sub&gt;s&lt;/sub&gt;&amp;#8201;p}}&amp;#8202; in that direction is zero, in which case ({{EquationNote|9g}}) says that {{math|&amp;nabla;''p''}} (if not zero) is orthogonal to{{math| '''s&amp;#770;'''}}.&amp;#8201; So&lt;math&gt;\,\nabla p&lt;/math&gt; ''is orthogonal to the surfaces of constant&lt;math&gt;~p\,&lt;/math&gt;'' (as we would expect, having just shown that the direction of{{math| &amp;nabla;''p''}} is that in which {{mvar|p}} varies most steeply). This result leads to a method of finding a vector normal to a curved surface at a given point: if the equation of the surface is&amp;#8239; {{math|''f''&amp;#8239;('''r''')&amp;#8201;{{=}}&amp;#8201;''C''&amp;#8202;,}}&amp;#8202; where {{math|'''r''' }}is the position vector and {{mvar|C&amp;#8202; }}is a constant (possibly zero), a suitable vector is {{math|&amp;nabla;''f''}}&amp;#8202; evaluated at the given point.

If {{mvar|p}} is ''uniform''&amp;#8202;&amp;mdash;that is, if it has no spatial variation&amp;mdash;then its derivative w.r.t. distance in every direction is zero; that is, the component of{{math| &amp;nabla;''p''}} in every direction is zero, so that {{math|&amp;nabla;''p''}} must be the zero vector. In short, ''the gradient of a uniform scalar field is zero''. Conversely, if {{mvar|p}} is ''not'' uniform, there must be some location and some direction in which its derivative w.r.t. distance, if defined at all, is non-zero, so that its gradient, if defined at all, is also non-zero. Thus ''a scalar field with zero gradient in some region is uniform in that region''.

{{cob}}

=== Unambiguity of the Laplacian ===

{{cot}}

Armed with our new definition of the gradient ({{EquationNote|9g}}), we can revisit our definition of the Laplacian ({{EquationNote|4L}}). If{{mvar| &amp;psi;}} is a ''scalar'' field, then, by ({{EquationNote|9g}}),  &lt;math&gt;\part_n \psi&lt;/math&gt; can be replaced by &lt;math&gt;\nabla\psi\cdot\mathbf{\hat{n}}\,&lt;/math&gt; in ({{EquationNote|4L}}), which then becomes
{{NumBlk|:|&lt;math&gt;\triangle\psi \,=\,
    \tfrac{1}{dV}\!\iint_{\delta S} \mathbf{\hat{n}}\cdot\nabla\psi \;dS \,;
  &lt;/math&gt;|{{EquationRef|9L}}}}
that is, by definition ({{EquationNote|4d}}),
{{NumBlk|:|&lt;math&gt;\triangle\psi
    \,=\, \operatorname{div}\nabla\psi
  \qquad&lt;/math&gt;[&amp;#8202;for scalar {{mvar|&amp;psi;}}].|{{EquationRef|9L'}}}}
So ''the Laplacian of a scalar field is the divergence of the gradient''. This is the usual ''introductory'' definition of the Laplacian&amp;mdash;and on its face is applicable only in the case of a scalar field. The unambiguity of the Laplacian, in this case, follows from the unambiguity of the divergence and the gradient.

If, on the contrary, {{mvar|&amp;psi;}} in definition ({{EquationNote|4L}}) is a ''vector'' field, then we can again take dot-products with a uniform vector {{math|'''b''',}} obtaining
:&lt;math&gt;(\triangle\psi)\cdot\mathbf{b} \,=\,
    \tfrac{1}{dV}\!\iint_{\delta S} \part_n(\psi\!\cdot\!\mathbf{b}) \,dS \,.
  &lt;/math&gt;
If we make {{math|'''b'''}} a ''unit'' vector, this says that ''the scalar component of the Laplacian of a vector field, in any direction, is the Laplacian of the scalar component of that vector field in that direction''. As we have just established that the latter is unambiguous, so is the former.

But the unambiguity of the Laplacian can be generalized further. If
:{{big|{{math|''&amp;psi;'' {{=}} &amp;sum;&lt;sub&gt;''i'' &lt;/sub&gt;''&amp;alpha;&lt;sub&gt;i &lt;/sub&gt;&amp;phi;&lt;sub&gt;i&lt;/sub&gt;''}}}}
where each {{mvar|&amp;phi;&lt;sub&gt;i&lt;/sub&gt;}} is a scalar field, and each {{mvar|&amp;alpha;&lt;sub&gt;i&lt;/sub&gt;}} is a constant, and the counter {{mvar|i}} ranges from (say) 1 to{{mvar| k&amp;#8202;}}, then it is clear from ({{EquationNote|4L}}) that
{{NumBlk|:|{{big|{{math|&amp;#9651;{{big|(}}&amp;sum;&lt;sub&gt;''i'' &lt;/sub&gt;''&amp;alpha;&lt;sub&gt;i &lt;/sub&gt;&amp;phi;&lt;sub&gt;i&lt;/sub&gt;''{{big|)}} {{=}} &amp;sum;&lt;sub&gt;''i'' &lt;/sub&gt;{{big|(}}''&amp;alpha;&lt;sub&gt;i &lt;/sub&gt;''&amp;#9651;''&amp;phi;&lt;sub&gt;i&lt;/sub&gt;''{{big|)}}}} .}}|{{EquationRef|10}}}}
In words, this says that ''the Laplacian of a linear combination of fields is the same linear combination of the Laplacians of the same fields''&amp;mdash;or, more concisely, that ''the Laplacian is '''[[w:linearity|linear]]'''''. I say &quot;it is clear&quot; because the Laplacian as defined by ({{EquationNote|4L}}) is itself a linear combination, so that ({{EquationNote|10}}) merely asserts that we can regroup the terms of a nested linear combination; the gradient, curl, and divergence as defined by ({{EquationNote|4g}}) to ({{EquationNote|4d}}) are likewise linear. It follows from ({{EquationNote|10}}) that ''the Laplacian of a linear combination of fields is unambiguous if the Laplacians of the separate fields are unambiguous''. Now we have supposed that the fields {{mvar|&amp;phi;&lt;sub&gt;i&lt;/sub&gt;}} are scalar and that the coefficients {{mvar|&amp;alpha;&lt;sub&gt;i&lt;/sub&gt;}} are constants. But the same logic applies if the &quot;constants&quot; are uniform basis vectors (e.g.,{{math| '''i''',&amp;#8202;'''j''','''k'''}}), so that the &quot;linear combination&quot; can represent any vector field, whence the Laplacian of any vector field is unambiguous. And the same logic applies if the &quot;constants&quot; are chosen as a &quot;basis&quot; for a space of tensors of any order, so that the Laplacian of any tensor field of that order is unambiguous, and so on. In short, ''the Laplacian of any field that we can express with a uniform basis is unambiguous''.

{{cob}}

=== The dot-del, del-cross, and del-dot operators ===

{{cot}}

The gradient operator {{math|&amp;nabla;}} is also called {{mvar|'''del'''}}.{{efn|Or ''nabla'', because it allegedly looks like the ancient Phoenician harp that the Greeks called by that name.}} If it simply denotes the gradient, we tend to pronounce it &quot;grad&quot; in order to emphasize the result. But it can also appear in combination with other operators to give other results, and in those contexts we tend to pronounce it &quot;del&quot;.

One such combination is &quot;dot del&quot;&amp;mdash;&amp;#8202;as in &quot;{{math|&amp;#8202;'''b&amp;sdot;'''&amp;nabla;&amp;#8202;}}&quot;, which we proposed for ({{EquationNote|8q}}), but did not quite manage to define satisfactorily for a vector operand. With our new definition of the gradient ({{EquationNote|9g}}), we can now make a second attempt. A general vector field {{math|'''q'''}} can be written &lt;math&gt;|\mathbf{q}|\,\mathbf{\hat{q}}~\!,&lt;/math&gt; so that
:&lt;math&gt;\mathbf{q}\cdot\nabla\psi
    \,=\, |\mathbf{q}| \,\mathbf{\hat{q}}\cdot\nabla\psi \,.
  &lt;/math&gt;
If {{mvar|&amp;psi;}} is a ''scalar'' field, we can apply ({{EquationNote|9g}}) to the right-hand side, obtaining
:{{big|&lt;math&gt;\mathbf{q}\cdot\nabla\psi
    ~\!=~\! |\mathbf{q}| \,\part_{s_q} \psi \,,
  &lt;/math&gt;}}
where {{mvar|s&lt;sub&gt;q&lt;/sub&gt;}} is distance in the direction of{{math| '''q'''}}. For ''scalar'' {{mvar|&amp;psi;}}, this result is an identity between previously defined quantities. For ''non-scalar'' {{mvar|&amp;psi;}}, we have not yet defined the left-hand side, but the right-hand side is still well-defined and self-explanatory (provided that we can differentiate {{mvar|&amp;psi;}} w.r.t.{{mvar| s&lt;sub&gt;q&lt;/sub&gt;}}). So we are free to adopt
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\psi \,:=\, |\mathbf{q}| \,\part_{s_q} \psi
  &lt;/math&gt;}}|{{EquationRef|11}}}}
(where {{mvar|s&lt;sub&gt;q&lt;/sub&gt;}} is distance in the direction of{{math| '''q'''}}) as the general definition of the ''operator'' {{math|'''q&amp;sdot;'''&amp;nabla;&amp;#8202;,}} and to interpret it as defining both a ''unary'' operator&amp;#8202; {{math|'''q&amp;sdot;'''&amp;nabla;}} which operates on a generic field, and a ''binary'' operator&amp;#8202; {{math|'''&amp;sdot;'''&amp;nabla;}} which takes a (possibly uniform) vector field on the left and a generic field on the right.

For any vector field {{math|'''q'''&amp;#8202;,}} it follows from ({{EquationNote|11}}) that ''if&lt;math&gt;~\psi&lt;/math&gt; is a uniform field, then&lt;math&gt;\,\,\mathbf{q}\;\!{\cdot}\nabla~\!\psi=0&lt;/math&gt;''.

For the special case in which {{math|'''q'''}} is a unit vector {{math|'''s&amp;#770;'''&amp;#8202;,}} with {{mvar|s}} measuring distance in the direction of{{math|&amp;#8202; '''s&amp;#770;'''&amp;#8202;,}} definition ({{EquationNote|11}}) reduces to
{{NumBlk|:|&lt;math&gt;\mathbf{\hat{s}}{\cdot}\nabla\,\psi = \part_s \psi \,,
  &lt;/math&gt;|{{EquationRef|12}}}}
which agrees with ({{EquationNote|9g}}) but now holds for a ''generic'' field {{mvar|&amp;psi;}} [whereas ({{EquationNote|9g}}) was for a ''scalar'' field, and was derived as a ''theorem'' based on earlier definitions]. So{{math| '''s&amp;#770;&amp;sdot;'''&amp;nabla;&amp;#8202;,}} with a unit vector {{math|'''s'''&amp;#8202;,}} is the '''directional-derivative operator''' on a generic field; and by ({{EquationNote|11}}),&amp;#8201; {{math|'''q&amp;sdot;'''&amp;nabla;}} is a '''scaled directional derivative''' operator on a generic field.

In particular, if&amp;#8202; {{math|'''s&amp;#770;'''}} is &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;,&amp;#8201; we have
:&lt;math&gt;\part_n \psi \,=\, \mathbf{\hat{n}}\;\!{\cdot}\nabla\,\psi \,,&lt;/math&gt;
which we may substitute into the original definition of the Laplacian ({{EquationNote|4L}}) to obtain
{{NumBlk|:|&lt;math&gt;\triangle\psi \,=\,
    \tfrac{1}{dV}\!\iint_{\delta S} \mathbf{\hat{n}}{\cdot}\nabla\,\psi \;dS \,,
  &lt;/math&gt;|{{EquationRef|13L}}}}
which is just ({{EquationNote|9L}}) again, except that it now holds for for a ''generic'' field.

If our general definition of the gradient ({{EquationNote|4g}}) is also taken as the general definition of the {{math|&amp;nabla;}} operator,&lt;ref&gt;''Cf''. [[#borisenko-tarapov-68|Borisenko &amp;amp; Tarapov, 1968]], p.&amp;#8239;157, eq.&amp;#8239;(4.43), quoted in [[#tai-95|Tai, 1995]], p.&amp;#8239;33, eq.&amp;#8239;(4.19).&lt;/ref&gt; then, comparing ({{EquationNote|4g}}) with ({{EquationNote|4c}}), ({{EquationNote|4d}}), and ({{EquationNote|13L}}), we see that
:&lt;math&gt;\begin{align}
    \operatorname{curl}\mathbf{q} ~\!&amp;= \nabla(\times\mathbf{q}) \\
    \operatorname{div}\mathbf{q}  ~\!&amp;= \nabla(\cdot\,\mathbf{q}) \\
    \triangle\psi                 ~\!&amp;= \nabla(\cdot\nabla\,\psi) \,,
  \end{align}&lt;/math&gt;
where the parentheses may seem to be required on account of the closing {{mvar|dS}}&amp;#8202; in ({{EquationNote|4g}}).&lt;ref&gt;The first two cases may be compared with Javid &amp;amp; Brown, 1963, cited in [[#tai-94|Tai, 1994]], p.&amp;#8239;15.&lt;/ref&gt; But if we write the factor {{mvar|dS}} ''before'' the integrand, the del operator in ({{EquationNote|4g}}) becomes
:&lt;math&gt;\nabla = \tfrac{1}{dV}\!\iint_{\delta S} dS\,\mathbf{\hat{n}} &lt;/math&gt;
&amp;mdash;''if''&amp;#8202; we insist that it is to be read as a operator looking for an operand, and not as a self-contained expression. Then, if we similarly bring forward the {{mvar|dS}} in ({{EquationNote|4c}}), ({{EquationNote|4d}}), and ({{EquationNote|13L}}), the respective operators become&lt;ref&gt;The first two cases may be compared with Neff, 1991, cited in [[#tai-94|Tai, 1994]], p.&amp;#8239;16.&lt;/ref&gt;
{{NumBlk|:|&lt;math&gt;\begin{align}
    \operatorname{curl} &amp;= \nabla\times \\
    \operatorname{div}  &amp;= \nabla~\!\boldsymbol{\cdot} \\
    \triangle           &amp;= \nabla\boldsymbol{\cdot}\nabla
  \end{align}&lt;/math&gt;|{{EquationRef|14}}}}
(pronounced &quot;del cross&quot;, &quot;del dot&quot;, and &quot;del dot del&quot;), of which the last is usually abbreviated as{{math| &amp;nabla;&lt;sup&gt;2&lt;/sup&gt;}}&amp;#8201; (&quot;del squared&quot;).&lt;ref&gt;But Gibbs ([[#gibbs-1881-4|1881]]) and Wilson ([[#wilson-1901|1901]]) were content to leave it as {{math|&amp;nabla;'''&amp;sdot;'''&amp;nabla;}}.  And they did not call it the ''Laplacian''; they used that term with a different meaning, which has apparently fallen out of fashion.&lt;/ref&gt; These notations are ubiquitous.

Another way to obtain the {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; operators (but ''not''{{math|&amp;#8202; &amp;nabla;&lt;sup&gt;2&lt;/sup&gt;}}), again inspired by ({{EquationNote|4g}}), is to define
{{NumBlk|:|&lt;math&gt;T(\nabla)
    \,:=\, \tfrac{1}{dV}\!\iint_{\delta S} T(\mathbf{\hat{n}}) \,dS \,,
  &lt;/math&gt;|{{EquationRef|14s}}}}
where {{mvar|T}}&amp;#8202; is any well-defined function that takes a vector argument. Setting{{math|&amp;#8202; ''T''&amp;#8202;(&amp;nabla;)}} to {{math|&amp;nabla;''p''&amp;#8202;,}} {{math|&amp;nabla;&amp;#8202;&amp;times;&amp;#8202;'''q'''&amp;#8202;,}} and{{math| &amp;nabla;'''&amp;sdot;&amp;#8239;q'''}}&amp;#8239; in ({{EquationNote|14s}}), we obtain respectively {{math|&amp;nabla;''p''&amp;#8202;,}}&amp;#8202; {{math|curl&amp;#8201;'''q'''&amp;#8202;,}} and&amp;#8202; {{math|div&amp;#8239;'''q'''}}&amp;#8239; as given by ({{EquationNote|4g}}) to ({{EquationNote|4d}}). But this approach has undesirable side-effects&amp;mdash;for example, that {{math|&amp;nabla;''p''}}&amp;#8202; becomes synonymous with{{math| ''p''&amp;nabla;}}.&amp;#8201; Accordingly, Chen-To Tai,&lt;ref&gt;[[#tai-fang-91|Tai &amp;amp; Fang, 1991]], pp.&amp;#8239;168–9.&lt;/ref&gt; on the left of ({{EquationNote|14s}}), replaces{{math| &amp;nabla;}} with his original symbol &lt;math&gt;\nabla\!\!\!\!^{\textstyle_-}~\!\!,\,&lt;/math&gt; which he calls the &quot;symbolic operator&quot; or the &quot;{{nowrap|''S''&amp;#8202;-operator}}&quot; or, later, the &quot;symbolic vector&quot; or the &quot;dummy vector&quot;. Tai in his later works (e.g.,&amp;#8239;[[#tai-94|1994]],&amp;#8239;[[#tai-95|1995]]) does not tolerate cross- or dot-products involving the del operator, but ''does'' tolerate such products involving his symbolic vector ([[#tai-95|1995]], pp.&amp;#8239;50–52).

There is a misconception that the operational equivalences in ({{EquationNote|14}}) apply ''only'' in Cartesian coordinates.&lt;ref&gt;Durney &amp;amp; Johnson, in ''Introduction to Modern Electromagnetics'' (1969, p.&amp;#8239;45, cited in [[#tai-94|Tai, 1994]], p.&amp;#8239;12), make the absurd statement that &quot;a{{math| &amp;nabla;}} operator cannot be defined in the other coordinate systems&amp;hellip;&quot; In the context, they apparently meant to say that&amp;#8202; {{math|div&amp;#8202;'''A'''}} isn't&amp;#8202; {{math|&amp;nabla;'''&amp;sdot;A'''}}&amp;#8202; in other coordinate systems. Robert S. Elliott, in ''Electromagnetics'' (1966, p.&amp;#8239;606, cited in [[#tai-94|Tai, 1994]], p.&amp;#8239;13), says that &quot;only in Cartesian coordinates&amp;hellip; do the gradient and divergence operators turn out to be identical.&quot; Apparently he meant to say that only in Cartesian coordinates do the two operators differ by a dot. But what these authors apparently meant to say is still wrong, as shown with counterexamples by Kemmer (next citation).&lt;/ref&gt; Tai does not accept them even in that case. But, because these equivalences have been derived from ''coordinate-free'' definitions of the operators, they must remain valid in any coordinate system ''provided that they are expressed correctly''&amp;mdash;without (e.g.) inadvertently taking dependent variables inside or outside differentiations.&lt;ref&gt;The perception that they are restricted to Cartesian coordinates arises partly from failure to allow for the variability of the basis vectors in curvilinear coordinate systems; ''cf''. [[#kemmer-77|Kemmer, 1977]], pp.&amp;#8239;163–5, 172–3 (Exs.&amp;#8239;2,&amp;#8239;3,&amp;#8239;5), 230–33 (sol'ns). From the del operator and the derivatives of the basis vectors w.r.t. the coordinates, Kemmer finds the curl and divergence in cylindrical coordinates, notes that we can do the same &quot;with a little greater effort&quot; in spherical coordinates (p.&amp;#8239;230), and finds the Laplacian of a scalar in both coordinate systems (p.&amp;#8239;231). He further reports that the method works for the Laplacian of a vector in cylindrical and spherical coordinates and is relatively convenient for the former (p.&amp;#8239;232), for which &quot;differentiation of the unit vectors is very simple&quot; (p.&amp;#8239;165).&lt;/ref&gt; That does ''not'' mean that they are always convenient, or easily verified, or conducive to the avoidance of error. But they sometimes make useful mnemonics; e.g., they let us rewrite identities ({{EquationNote|8c}}), ({{EquationNote|8g}}), and ({{EquationNote|8p}}) as
{{NumBlk|:|&lt;math&gt;\left.\begin{align}
    \nabla\!\times\!\mathbf{q}\cdot\mathbf{b}
    &amp;\,=\, \nabla\cdot\mathbf{q}\!\times\!\mathbf{b}\\[.5ex]
    \nabla p \cdot \mathbf{b}
    &amp;\,=\, \nabla\cdot\;\! p\mathbf{b}\\[.5ex]
    \nabla p \times \mathbf{b}
    &amp;\,=\, \nabla \times p\mathbf{b}
  \end{align}~\right\}\quad&lt;/math&gt;for uniform {{math|'''b'''}}.
    |{{EquationRef|15}}}}
These would be basic ''algebraic'' vector identities if&amp;#8202; {{math|&amp;nabla;}} were an ordinary vector, and one could try to derive them from the &quot;algebraic&quot; behavior of{{math| &amp;nabla;}}; but they're not, because it isn't, so we didn't&amp;#8239;!  Moreover, these simple &quot;algebraic&quot; rules are for a uniform {{math|'''b''',}} and do not of themselves tell us what to do if&amp;#8202; {{math|'''b'''}} is spatially variable; for example, ({{EquationNote|8g}}) is not applicable to ({{EquationNote|7d}}).

{{cob}}

=== The advection operator ===

{{cot}}

Variation or transportation of a property of a medium due to motion with the medium is called '''advection''' (which, according to its Latin roots, means &quot;carrying to&quot;). Suppose that a medium (possibly a fluid) moves with a velocity field {{math|'''v'''}} in some inertial reference frame. Let {{mvar|&amp;psi;}} be a field (possibly a scalar field or a vector field) expressing some property of the medium (e.g., density, or acceleration, or stress,{{efn|Stress is a second-order tensor, and the origin of the term &quot;tensor&quot;; but, for present purposes, it's just another possible example of a field called{{mvar| &amp;psi;}}.}}&amp;hellip; or even {{math|'''v''' }}itself). We have seen that the time-derivative of{{mvar| &amp;psi;}} may be specified in two different ways: as the ''partial'' derivative {{mvar|{{sfrac|&amp;part;&amp;psi;|&amp;part;t}}&amp;#8202;,}} evaluated at a fixed point (in the chosen reference frame), or as the ''material'' derivative {{mvar|{{sfrac|d&amp;psi;|dt}}&amp;#8202;}}, evaluated at a point moving at velocity {{math|'''v'''}} (i.e., ''with the medium''). The difference&amp;#8202; {{mvar|{{sfrac|d&amp;psi;|dt}}&amp;#8202;&amp;minus;&amp;#8202;{{sfrac|&amp;part;&amp;psi;|&amp;part;t}}&amp;#8202;}} is due to motion with the medium. To find another expression for this difference, let {{mvar|s}} be a parameter measuring distance along the path traveled by a particle of the medium. Then, for points along the path, the surface-plot of the small change in {{mvar|&amp;psi;}} (or any component thereof) as a function of small changes in {{mvar|t}} and {{mvar|s&amp;#8202;}} (plotted on perpendicular axes) can be taken as a plane through the origin, so that
:{{big|&lt;math&gt;d\psi
    = \tfrac{\part\psi}{\part t}~\!dt + \tfrac{\part\psi}{\part s}~\!ds \;;
  &lt;/math&gt;}}
that is, the change in {{mvar|&amp;psi;}} is the sum of the changes due to the change in {{mvar|t}} and the change in {{mvar|s&amp;#8202;}}. Dividing by {{mvar|dt}} gives
:{{big|&lt;math&gt;\begin{align}\tfrac{d\psi}{dt}
  &amp;= \tfrac{\part\psi}{\part t}+\tfrac{\part\psi}{\part s}~\!\tfrac{ds}{dt}\\[1ex]
  &amp;= \tfrac{\part\psi}{\part t}+\tfrac{\part\psi}{\part s}~\!|\mathbf{v}| \,;
  \end{align}&lt;/math&gt;}}
i.e.,
:{{big|&lt;math&gt;\tfrac{d\psi}{dt}
    = \tfrac{\part\psi}{\part t} + |\mathbf{v}|\,\part_s \psi
  &lt;/math&gt;}}
(and the first term on the right could have been written {{mvar|&amp;part;&lt;sub&gt;t&lt;/sub&gt;&amp;#8239;&amp;psi;}}). So the second term on the right is the contribution to the material derivative due to motion with the medium; it is called the '''advective term''', and is non-zero wherever a particle of the medium moves along a path on which {{mvar|&amp;psi;}} varies with location&amp;mdash;even if {{mvar|&amp;psi;}} at ''each'' location is constant over time.&amp;#8201; So the operator&amp;#8202; {{math|{{abs|'''v'''}}&amp;#8201;''&amp;part;&lt;sub&gt;s&lt;/sub&gt;''&amp;#8202;,}} where {{mvar|s}} measures distance along the path, is the ''advection operator''&amp;#8239;: it maps a property of a medium to the advective term in the time-derivative of that property. If{{mvar| &amp;psi;}} is {{math|'''v''' }}itself, the above result becomes
:{{big|&lt;math&gt;\tfrac{d\mathbf{v}}{dt}
    = \tfrac{\part\mathbf{v}}{\part t} + |\mathbf{v}|\,\part_s \mathbf{v} \,,
  &lt;/math&gt;}}
where the left-hand side (the ''material'' acceleration) is as given by Newton's second law, and the first term on the right (which we might call the &quot;partial&quot; acceleration) is the time-derivative of velocity in the chosen reference frame, and the second term on the right (the ''advective'' term) is the correction that must be added to the &quot;partial&quot; acceleration in order to obtain the material acceleration. This term is non-zero wherever velocity is non-zero and varies along a path, even if the velocity at each point on the path is constant over time (as when water speeds up while flowing at a constant volumetric rate into a nozzle). Paradoxically, while the material acceleration and the &quot;partial&quot; acceleration are apparently linear (first-degree) in {{math|'''v''',}} their difference (the advective term) is not. Thus the distinction between {{mvar|{{sfrac|&amp;part;&amp;psi;|&amp;part;t}}}} and {{mvar|{{sfrac|d&amp;psi;|dt}}}}&amp;#8202; has the far-reaching implication that ''fluid dynamics is non-linear''.

Applying ({{EquationNote|11}}) to the last two equations, we obtain respectively
{{NumBlk|:|{{big|&lt;math&gt;\tfrac{d\psi}{dt}
    = \tfrac{\part\psi}{\part t} + \mathbf{v}{\cdot}\nabla\,\psi
  &lt;/math&gt;}}|{{EquationRef|16}}}}
and
{{NumBlk|:|{{big|&lt;math&gt;\tfrac{d\mathbf{v}}{dt}
    = \tfrac{\part\mathbf{v}}{\part t} + \mathbf{v}{\cdot}\nabla\,\mathbf{v} \,,
  &lt;/math&gt;}}|{{EquationRef|16v}}}}
where, in each case, the second term on the right is the advective term. So ''the '''advection operator''' can also be written''&amp;#8239;{{math| '''v&amp;sdot;'''&amp;nabla;&amp;#8202;}}.

When the generic {{mvar|&amp;psi;&amp;#8202;}} in ({{EquationNote|16}}) is replaced by the density {{mvar|&amp;rho;&amp;#8202;}}, we get a relation between {{mvar|{{sfrac|&amp;part;&amp;rho;|&amp;part;t}}&amp;#8202;}} and {{mvar|{{sfrac|d&amp;rho;|dt}}&amp;#8202;}}, both of which we have seen before&amp;mdash;in equations ({{EquationNote|7d}}) and ({{EquationNote|7d'}}) above. Substituting from those equations then gives
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\rho\mathbf{v}
    \,=\, \rho\operatorname{div}\mathbf{v} \,+\, \mathbf{v}\cdot\nabla\rho \,,
  &lt;/math&gt;|{{EquationRef|17}}}}
where {{math|&amp;nabla;''&amp;rho;''}} can be taken as a gradient since {{mvar|&amp;rho;}} is scalar. This result is in fact an identity&amp;mdash;a ''product rule for the divergence''&amp;mdash;as we shall eventually confirm by another method.

{{cob}}

=== Generalized volume-integral theorem ===

{{cot}}

We can rewrite the fourth integral theorem ({{EquationNote|5L}}) in the &quot;dot del&quot; notation as
{{NumBlk|:|&lt;math&gt;\iint_S \mathbf{\hat{n}}\;\!{\cdot}\nabla\,\psi \;dS
    \,= \iiint_V \triangle\psi ~dV \,.
  &lt;/math&gt;|{{EquationRef|18L}}}}
Then, using notations ({{EquationNote|14}}), we can condense ''all four'' integral theorems ({{EquationNote|5g}}), ({{EquationNote|5c}}), ({{EquationNote|5d}}), and ({{EquationNote|18L}}) into the single equation
{{NumBlk|:|&lt;math&gt;\iint_S \mathbf{\hat{n}} * \psi \;dS
    \,= \iiint_V \nabla * \psi ~dV \,,
  &lt;/math&gt;|{{EquationRef|19}}}}
where the wildcard {{math|&amp;lowast;}} (conveniently pronounced &quot;star&quot;) is a generic binary operator which may be replaced by a null (direct juxtaposition of the operands) for theorem ({{EquationNote|5g}}), or a cross for ({{EquationNote|5c}}), or a dot for ({{EquationNote|5d}}), or&amp;#8202; {{math|'''&amp;sdot;'''&amp;nabla;}} for ({{EquationNote|18L}}); and the operand {{mvar|&amp;psi;}} is of a kind that makes the operator meaningful. This single equation is a ''generalized volume-integral theorem'', relating an integral over a volume to an integral over its enclosing surface.&lt;ref&gt;Kemmer ([[#kemmer-77|1977]], p.&amp;#8239;98, eq.&amp;#8239;4) gives an equivalent result for our first three integral theorems ({{EquationNote|5g}} to {{EquationNote|5d}}) only, and calls it the ''generalized divergence theorem'' because the divergence theorem is its most familiar special case.&lt;/ref&gt;

Theorem ({{EquationNote|19}}) is based on the following definitions, which have been found unambiguous:
* the ''gradient'' of a scalar field {{mvar|p}} is the closed-surface integral of&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}p\,&lt;/math&gt; per unit volume, where &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is the outward unit normal;
* the ''curl'' of a vector field is the skew surface integral per unit volume, also called the surface circulation per unit volume;
* the ''divergence'' of a vector field is the outward flux integral per unit volume; and
* the ''Laplacian'' is the closed-surface integral of the outward normal derivative, per unit volume.
The gradient maps a scalar field to a vector field; the curl maps a vector field to a vector field; the divergence maps a vector field to a scalar field; and the Laplacian maps a scalar field to a scalar field, or a vector field to a vector field, etc.

The ''gradient'' of {{mvar|p}}, as defined above, has been shown to be also
* the vector whose (scalar) component in any direction is the ''directional derivative'' of{{mvar| p}} in that direction (i.e. the derivative of{{mvar| p}} w.r.t. distance in that direction), and
* the vector whose direction is that in which the directional derivative of{{mvar| p}} is a maximum, and whose magnitude is that maximum.
Consistent with these alternative definitions of the gradient, we have defined the&amp;#8202;{{math| '''&amp;sdot;'''&amp;nabla;}} operator so that&amp;#8202; {{math|'''s&amp;#770;&amp;sdot;'''&amp;nabla;}} (for a ''unit'' vector {{math|'''s&amp;#770;'''}}) is the operator yielding the directional derivative in the direction of&amp;#8202; {{math|'''s&amp;#770;'''&amp;#8202;,}} and we have used that notation to bring theorem ({{EquationNote|5L}}) under theorem ({{EquationNote|19}}).

So far, we have said comparatively little about the curl. That imbalance will now be rectified.

{{cob}}

== Closed-circuit integrals per unit area ==

=== Instant integral theorems (on a condition) ===

{{cot}}

Theorems ({{EquationNote|5g}}) to ({{EquationNote|5L}}) are three-dimensional: each of them relates an integral over a volume {{mvar|V}}&amp;#8202; to an integral over its enclosing surface{{mvar| S}}. We now seek analogous ''two''-dimensional theorems, each of which relates an integral over a surface segment to an integral around its enclosing curve. For maximum generality, the surface segment should be allowed to be curved into a third dimension.{{efn|In mathematical jargon, it should be a two-dimensional ''manifold'' embedded in 3D Euclidean space.}} Theorems of the latter kind can be obtained as special cases of theorems ({{EquationNote|5g}}) to ({{EquationNote|5L}}) by suitably choosing {{mvar|V}} and {{mvar|S&amp;#8202;}}; this is another advantage of our &quot;volume first&quot; approach.

Let {{mvar|&amp;Sigma;}} be a surface segment enclosed by a curve {{mvar|C}} (a ''circuit'' or ''closed contour''), and let {{mvar|l}} be a parameter measuring arc length around {{mvar|C&amp;#8202;}}, so that a general element of{{mvar| C}}&amp;#8202; has length{{mvar| dl&amp;#8202;}}; and let a general element of the surface {{mvar|&amp;Sigma;}}&amp;#8202; have area {{mvar|d&amp;Sigma;}}. Let&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt; be the unit normal vector at a general point on {{mvar|&amp;Sigma;&amp;#8202;}}, and let &lt;math&gt;\mathbf{\hat{t}}&lt;/math&gt; be the unit ''tangent'' vector to{{mvar| C}} at a general point on {{mvar|C}}&amp;#8202; in the direction of increasing{{mvar| l}}. In the original case of a surface enclosing a volume, we had to decide whether the unit normal pointed into or out of the volume (we chose the latter). In the present case of a circuit enclosing a surface segment, we have to decide whether {{mvar|l}} is measured clockwise or counterclockwise as seen when looking in the direction of the unit normal, and we choose clockwise. So {{mvar|l }}''is measured clockwise about&lt;math&gt;~\boldsymbol{\hat{\nu}},&lt;/math&gt;'' and {{mvar|C }}is ''traversed'' clockwise about&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;.

From {{mvar|&amp;Sigma;}}&amp;#8202; we can construct obvious candidates for {{mvar|V}} and{{mvar| S}}. From every point on {{mvar|&amp;Sigma;&amp;#8202;}}, erect a perpendicular with a uniform ''small''&amp;#8202; height {{mvar|h}} in the direction of&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;. Then simply let {{mvar|V}} be the volume occupied by all the perpendiculars, and let {{mvar|S}} be its enclosing surface. Thus {{mvar|V}} is a (generally curved) thin slab of uniform thickness{{mvar| h}}, whose enclosing surface {{mvar|S}} consists of two close parallel (generally curved) broad faces connected by a perpendicular ''edge-face'' of uniform height{{mvar| h&amp;#8202;}}; and we can treat&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt; as a vector ''field''&amp;#8202; by extrapolating it perpendicularly from{{mvar| &amp;Sigma;}}. If we can arrange for {{mvar|h}} to cancel out, the volume{{mvar| V}}&amp;#8202; will serve as a 3D representation of the surface segment{{mvar| &amp;Sigma;}}&amp;#8202; while the ''edge-face'' will serve as a 2D representation of the curve{{mvar| C&amp;#8202;}}, so that our four theorems will relate an integral around {{mvar|C}}&amp;#8202; to an integral over {{mvar|&amp;Sigma;}}&amp;#8201; ''provided that there is no contribution from the broad faces to the integral over''{{mvar| S}}. For brevity, let us call this proviso the '''2D condition'''.

''If''&amp;#8202; the 2D condition is satisfied, an integral over the new {{mvar|S}}&amp;#8202; reduces to an integral over the edge-face, on which
:&lt;math&gt;dS = h\,dl \,,&lt;/math&gt;
so that the cancellation of{{mvar| h}} will leave an integral over {{mvar|C}}&amp;#8202; w.r.t. length. Meanwhile, in an integral over the new{{mvar| V}}, regardless of the 2D condition, we have
:&lt;math&gt;dV = h\,d\varSigma \,,&lt;/math&gt;
so that the cancellation of{{mvar| h}} will leave an integral over {{mvar|&amp;Sigma;}}&amp;#8202; w.r.t. area. So, substituting for {{mvar|dS}} and {{mvar|dV}}&amp;#8202; in ({{EquationNote|5g}}) to ({{EquationNote|5L}}), and canceling {{mvar|h}} as planned, we obtain respectively
{{NumBlk|:|&lt;math&gt;~~~~~\!\oint_C \mathbf{\hat{n}}~\!p \,dl
    \,= \iint_{\varSigma} \nabla p ~d\varSigma \qquad(?),
  &lt;/math&gt;|{{EquationRef|20g}}}}
{{NumBlk|:|&lt;math&gt;\oint_C \mathbf{\hat{n}}\times\mathbf{q} \,dl
    \,= \iint_{\varSigma} \operatorname{curl}\mathbf{q} ~d\varSigma \qquad(?),
  &lt;/math&gt;|{{EquationRef|20c}}}}
{{NumBlk|:|&lt;math&gt;~~\!\oint_C \mathbf{\hat{n}\cdot q} \,dl
    \,= \iint_{\varSigma} \operatorname{div}\mathbf{q} ~d\varSigma \qquad(?),
  &lt;/math&gt;|{{EquationRef|20d}}}}
{{NumBlk|:|&lt;math&gt;~~\oint_C \part_n \psi \;dl
    \,= \iint_{\varSigma} \triangle\psi ~d\varSigma \qquad(?),
  &lt;/math&gt;|{{EquationRef|20L}}}}
''all subject to the 2D condition'' (hence the question marks). In each equation, the circle on the left integral sign acknowledges that the integral is around a closed loop. The unit vector &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;, which ''was'' normal to the edge-face, is now normal to both &lt;math&gt;\mathbf{\hat{t}}&lt;/math&gt; and&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;; that is, &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is tangential to the surface segment {{mvar|&amp;Sigma;}}&amp;#8202; and projects perpendicularly outward from its bounding curve.

On the left side of ({{EquationNote|20g}}), the 2D condition is satisfied if (but not only if)&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}p&lt;/math&gt; takes equal-and-opposite values at any two opposing points on opposing broad faces of{{mvar| S&amp;#8202;,}} i.e. if {{mvar|p}} takes the ''same'' value at such points, i.e. if {{mvar|p}} has a zero directional derivative normal to{{mvar| &amp;Sigma;}}.

Skipping forward to ({{EquationNote|20L}}), we see that the 2D condition is satisfied if&lt;math&gt;~\part_n \psi&lt;/math&gt; takes equal-and-opposite values at any two opposing points on opposing broad faces of{{mvar| S&amp;#8202;,}} i.e. if&lt;math&gt;~\part_{\nu}\psi&lt;/math&gt; (where &lt;math&gt;\nu&lt;/math&gt; measures distance in the direction of&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;) takes the ''same'' value at such points, i.e. if&lt;math&gt;~\part^2_{\nu}\psi\!=\!0&lt;/math&gt;.

For ({{EquationNote|20c}}) and ({{EquationNote|20d}}), the 2D condition can be satisfied by construction, with more useful results&amp;mdash;as explained under the next two headings. To facilitate this process, we first make a minor adjustment to {{mvar|&amp;Sigma;}} and{{mvar| C}}. Noting that any curved surface segment can be approximated to any desired accuracy by a ''polyhedral'' surface enclosed by a ''polygon'', we shall indeed consider {{mvar|&amp;Sigma;}}&amp;#8202; to be a polyhedral surface made up of small planar elements, {{mvar|d&amp;Sigma;}}&amp;#8202; being the area of a general element, and we shall indeed consider {{mvar|C}} to be a polygon with short sides, {{mvar|dl}} being the length of a general side.{{efn|If any part of our argument requires {{mvar|&amp;Sigma;}} or {{mvar|C}} to be ''smooth'', this is not an impediment, because having approximated {{mvar|&amp;Sigma;}} or{{mvar| C}} to any desired accuracy by a polyhedron or polygon, we can then approximate the polyhedron or polygon to any desired ''higher'' accuracy by a smooth surface or curve!}} The benefit of this trick, as we shall see, is to make the unit normal &lt;math&gt;\boldsymbol{\hat{\nu}}&lt;/math&gt; uniform over each surface element, without forcing us to treat {{math|'''q'''}} (or any other field) as uniform over the same element. But, as the elements of{{mvar| C}}&amp;#8202; can ''independently'' be made as short as we like (dividing straight sides into shorter elements if necessary!), we can still consider &lt;math&gt;\boldsymbol{\hat{\nu}},&lt;/math&gt; {{math|'''q'''&amp;#8202;,}} and &lt;math&gt;\mathbf{\hat{t}}&lt;/math&gt; to be uniform over each element of{{mvar| C}}.

{{cob}}

=== Special case for the gradient ===

{{cot}}

In ({{EquationNote|20c}}), the 2D condition is satisfied by&lt;math&gt;~\mathbf{q}\!=\!p\boldsymbol{\hat{\nu}}&lt;/math&gt; (where {{mvar|p}} is a scalar field), because then the integrand on the left is zero on the broad faces of{{mvar| S&amp;#8202;}}, where {{math|'''n'''}} is parallel to&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;. Equation ({{EquationNote|20c}}) then becomes
{{NumBlk|:|&lt;math&gt;
    \oint_C \mathbf{\hat{n}}{\times}\boldsymbol{\hat{\nu}}~\!p \;dl \,=
    \iint_{\varSigma}\operatorname{curl}p\boldsymbol{\hat{\nu}}\;d\varSigma \,.
  &lt;/math&gt;|{{EquationRef|21n}}}}
Now on the left,&amp;#8201; &lt;math&gt;\mathbf{\hat{n}}\!\times\!\boldsymbol{\hat{\nu}}\!=\!-\mathbf{\hat{t}}~\!;\,&lt;/math&gt; and on the right, over each surface element, the unit normal &lt;math&gt;\boldsymbol{\hat{\nu}}&lt;/math&gt; is uniform so that, by ({{EquationNote|8p}}),&amp;#8201; &lt;math&gt;\operatorname{curl}p\boldsymbol{\hat{\nu}}=~\!\!\nabla p \!\times\!\boldsymbol{\hat{\nu}}=-\boldsymbol{\hat{\nu}}\!\times\!\nabla p&lt;/math&gt;.  With these substitutions, the minus signs cancel and we get
{{NumBlk|:|&lt;math&gt;
    \oint_C p\mathbf{\hat{t}} \,dl \,=
    \iint_{\varSigma} \boldsymbol{\hat{\nu}}\times\nabla p \;d\varSigma
  &lt;/math&gt;|{{EquationRef|21g}}}}
or, if we write&amp;#8201; &lt;math&gt;d\mathbf{r}\!=\!\mathbf{\hat{t}}~\!dl&lt;/math&gt;&amp;#8201; and&amp;#8201; &lt;math&gt;\boldsymbol{d\varSigma}\!=\!\boldsymbol{\hat{\nu}}\,d\varSigma~\!,&lt;/math&gt;
{{NumBlk|:|&lt;math&gt;
    \oint_C p \,d\mathbf{r} \,=
    \iint_{\varSigma} \big(\boldsymbol{d\varSigma}\times\!\nabla p\big) \,.
  &lt;/math&gt;|{{EquationRef|21r}}}}
This result, although well attested in the literature,&lt;ref&gt;E.g., [[#gibbs-1881-4|Gibbs, 1884]], &amp;sect;&amp;#8239;165, eq.&amp;#8239;(1); [[#wilson-1901|Wilson, 1901]], p.&amp;#8239;255, Ex.&amp;#8239;1; [[#kemmer-77|Kemmer, 1977]], p.&amp;#8239;99, eq.&amp;#8239;(6); [[#hsu-84|Hsu, 1984]], p.&amp;#8239;146, eq.&amp;#8239;(7.31).&lt;/ref&gt; does not seem to have a name&amp;mdash;unlike the next result.

{{cob}}

=== Special case for the curl ===

{{cot}}

In ({{EquationNote|20d}}), the 2D condition is satisfied if {{math|'''q'''}} is replaced by&lt;math&gt;~\boldsymbol{\hat{\nu}}{\times}\mathbf{q}~\!,\,&lt;/math&gt; because then (again) the integrand on the left is zero on the broad faces of{{mvar| S&amp;#8202;}}, where {{math|'''n'''}} is parallel to&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;. Equation ({{EquationNote|20d}}) then becomes
{{NumBlk|:|&lt;math&gt;
    \oint_C \mathbf{\hat{n}}\cdot\boldsymbol{\hat{\nu}}{\times}\mathbf{q} \;dl
    \,=
    \iint_{\varSigma}
      \operatorname{div}(\boldsymbol{\hat{\nu}}\!\times\!\mathbf{q})
    \,d\varSigma \,.
  &lt;/math&gt;|{{EquationRef|22n}}}}
Now on the left, the integrand can be written&amp;#8201; &lt;math&gt;\mathbf{\hat{n}}{\times}\boldsymbol{\hat{\nu}}\!\cdot\!\mathbf{q}\!=\!-\mathbf{\hat{t}}\!\cdot\!\mathbf{q}~\!;\,&lt;/math&gt; and on the right,&amp;#8201; &lt;math&gt;\operatorname{div}(\boldsymbol{\hat{\nu}}\!\times\!\mathbf{q})\!=\!-\operatorname{div}(\mathbf{q}\!\times\!\boldsymbol{\hat{\nu}})\!=\!-\operatorname{curl}\mathbf{q}\cdot\boldsymbol{\hat{\nu}}\,&lt;/math&gt; by identity ({{EquationNote|8c}}), since &lt;math&gt;\boldsymbol{\hat{\nu}}&lt;/math&gt; is uniform over each surface element.  With these substitutions, the minus signs cancel and we get
{{NumBlk|:|&lt;math&gt;
    \oint_C \mathbf{q} \cdot \mathbf{\hat{t}} \,dl \,=
    \iint_{\varSigma}
      \operatorname{curl}\mathbf{q}\cdot\boldsymbol{\hat{\nu}}
    \,d\varSigma
  &lt;/math&gt;|{{EquationRef|22c}}}}
or, if we again write&amp;#8201; &lt;math&gt;d\mathbf{r}\!=\!\mathbf{\hat{t}}~\!dl&lt;/math&gt;&amp;#8201; and&amp;#8201; &lt;math&gt;\boldsymbol{d\varSigma}\!=\!\boldsymbol{\hat{\nu}}\,d\varSigma~\!,&lt;/math&gt;
{{NumBlk|:|&lt;math&gt;
    \oint_C \mathbf{q} \cdot d\mathbf{r} \,=
    \iint_{\varSigma}
      \operatorname{curl}\mathbf{q}\cdot\boldsymbol{d\varSigma} \,.
  &lt;/math&gt;|{{EquationRef|22r}}}}
This result&amp;mdash;the best-known theorem relating an integral over a surface segment to an integral around its enclosing curve, and the best-known theorem involving the curl&amp;mdash;is called ''[[w:Sir George Stokes, 1st Baronet|Stokes]]' theorem'' or, more properly, the '''[[w:Lord Kelvin|Kelvin]]&amp;ndash;Stokes theorem''',&lt;ref&gt;''Cf''. [[#katz-79|Katz, 1979]], pp.&amp;#8239;149–50.&lt;/ref&gt; or simply the ''curl theorem''.&lt;ref&gt;Although Hsu ([[#hsu-84|1984]], p.&amp;#8239;141) applies that name to our theorem ({{EquationNote|5c}}).&lt;/ref&gt;

The integral on the left of ({{EquationNote|22c}}) or ({{EquationNote|22r}}) is called the '''circulation''' of the vector field {{math|'''q'''}} around the closed curve{{mvar| C}}. So, &lt;span id=&quot;kelvin-stokes-verbal&quot;&gt;in words&lt;/span&gt;, the Kelvin&amp;ndash;Stokes theorem says that ''the circulation of a vector field around a closed curve is equal to the flux of the curl of that vector field through any surface spanning that closed curve''.

Now let a general element of {{mvar|&amp;Sigma;}} (with area {{mvar|d&amp;Sigma;&amp;#8202;}}) be enclosed by the curve {{mvar|&amp;delta;C}}, traversed in the same direction as the outer curve {{mvar|C}}. Then, applying ({{EquationNote|22c}}) to the single element, we have
:&lt;math&gt;
    \oint_{\delta C} \!\mathbf{q} \cdot \mathbf{\hat{t}} \,dl \,=\,
    \operatorname{curl}\mathbf{q}\cdot\boldsymbol{\hat{\nu}} \,d\varSigma \,;
  &lt;/math&gt;
that is,
{{NumBlk|:|&lt;math&gt;
    \operatorname{curl}\mathbf{q}\cdot\boldsymbol{\hat{\nu}} \,=\,
    \frac{1}{d\varSigma}\oint_{\delta C}\!\mathbf{q}\cdot\mathbf{\hat{t}}\,dl\,,
  &lt;/math&gt;|{{EquationRef|23c}}}}
where the right-hand side is simply the ''circulation per unit area''.

Equation ({{EquationNote|23c}}) is an alternative definition of the curl: it says that ''the curl of''{{math| '''q'''}} ''is the vector whose scalar component in any direction is the circulation of''{{math| '''q'''}} ''per unit area of a surface whose normal points in that direction''. For ''real''{{math| '''q''',}} this component has its maximum, namely {{math|{{abs|curl&amp;#8201;'''q'''}}&amp;#8202;,}} in the direction of{{math| curl&amp;#8239;'''q'''&amp;#8202;}}; thus ''the curl of''{{math| '''q'''}} ''is the vector whose direction is that which a surface must face if the circulation of''{{math| '''q'''}} ''per unit area of that surface is to be a maximum, and whose magnitude is that maximum''. This is the usual conceptual definition of the curl.&lt;ref&gt;E.g., [[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;#8239;61; [[#hsu-84|Hsu, 1984]], pp.&amp;#8239;117–18.&lt;/ref&gt;

[Notice, however, that our original volume-based definition ({{EquationNote|4c}}) is more succinct: the curl is the closed-surface circulation per unit volume, i.e. the skew surface integral per unit volume.]

It should now be clear where the curl gets its name (coined by [[w:James Clerk Maxwell|Maxwell]]), and why it is also called the ''rotation'' (indeed the {{math|curl}} operator is sometimes written &quot;{{math|rot}}&quot;, especially in Continental languages, in which &quot;rot&quot; does not have the same unfortunate everyday meaning as in English).

[[File:Vorticity_Figure_03_a-m.gif|thumb|Animation of a non-vortex-like velocity field whose curl (like its circulation around the red loop) is non-zero due to shear.]]
[[File:Vorticity_Figure_02_a-m.gif|thumb|Animation of a vortex-like velocity field whose curl is zero because the shear compensates for the rotation.]]

And it should now be unsurprising that ''a vector field with zero curl is described as '''irrotational''''' (which one must carefully pronounce differently from &quot;{{nowrap|irr''i&amp;#8202;''tational}}&quot;!), and that the curl of the velocity of a medium is called the '''vorticity'''.

However, a field does not need to be vortex-like in order to have a non-zero curl. For example, by identity ({{EquationNote|8p}}), in Cartesian coordinates, the velocity field {{math|''x'''''j'''}} has a curl equal to&amp;#8201; {{math|&amp;nabla;''x''&amp;#8201;&amp;times;&amp;#8239;'''j'''&amp;#8201;{{=}}&amp;#8201;'''i'''&amp;#8202;&amp;times;&amp;#8202;'''j'''&amp;#8201;{{=}}&amp;#8201;'''k'''&amp;#8202;,}}&amp;#8201; although it describes a ''shearing'' motion rather than a rotating motion. This is understandable because if you hold a pencil between the palms of your hands and slide one palm over the other (a shearing motion), the pencil rotates.

Conversely, we can have a vortex-like field whose curl is zero everywhere except on or near the axis of the vortex. For example, the '''Maxwell&amp;ndash;Amp&amp;egrave;re law''' in magnetostatics says that&amp;#8201; {{math|curl&amp;#8201;'''H'''&amp;#8201;{{=}}&amp;#8201;'''J'''&amp;#8202;,}} where {{math|'''H'''}} is the '''magnetizing field''' and {{math|'''J'''}} is the current density.{{efn|In the general case, there is an extra term {{math|{{sfrac|''&amp;part;''&amp;#8202;'''D'''|''&amp;part;t''}}}} on the right; but this term is zero in the magneto''static'' case.}} So if the current is confined to a wire, {{math|curl&amp;#8201;'''H'''&amp;#8202;}} is zero outside the wire&amp;mdash;although, as is well known, the field lines circle the wire. The resolution of the paradox is that {{math|'''H'''}} gets stronger as we approach the wire, making a shearing pattern, whose effect on the curl counteracts that of the rotation.

{{cob}}

=== The curl-grad and div-curl operators ===

{{cot}}

We have seen from ({{EquationNote|9L}}) that the Laplacian of a scalar field is the divergence of the gradient. Four more such second-order combinations make sense, namely the curl of the gradient (of a scalar field), and the divergence of the curl, the gradient of the divergence, and the curl of the curl (of a vector field). The first two&amp;#8239;&amp;mdash;&quot;curl grad&quot; and &quot;div curl&quot;&amp;mdash;&amp;#8239;can now be disposed of.

Let the surface segment {{mvar|&amp;Sigma;}} enclosed by the curve{{mvar| C}}&amp;#8202; be a segment of the closed surface {{mvar|S}} surrounding the volume{{mvar| V}}, and let {{mvar|&amp;Sigma;}} expand across {{mvar|S}} until it engulfs{{mvar| V}}, so that {{mvar|C}} shrinks to a point on the far side of{{mvar| S}}. Then, in the nameless theorem ({{EquationNote|21g}}) and the Kelvin&amp;ndash;Stokes theorem ({{EquationNote|22c}}), the integral on the left becomes zero while {{mvar|&amp;Sigma;}} and &lt;math&gt;\boldsymbol{\hat{\nu}}&lt;/math&gt; on the right become {{mvar|S}} and &lt;math&gt;\mathbf{\hat{n}},&lt;/math&gt; so that the theorems respectively reduce to
:&lt;math&gt;
    \iint_S \mathbf{\hat{n}}\times\nabla p \;dS
     \,=\, \mathbf{0}
  &lt;/math&gt;
and
:&lt;math&gt;
    \iint_S
      \mathbf{\hat{n}}\cdot\operatorname{curl}\mathbf{q}
    \;dS \,=\, 0 \,.
  &lt;/math&gt;
Applying theorem ({{EquationNote|5c}}) to the first of these two equations, and the divergence theorem ({{EquationNote|5d}}) to the second, we obtain respectively
:&lt;math&gt;\iiint_V \operatorname{curl}\nabla p \;dV ~\!=\, \mathbf{0} \,,&lt;/math&gt;
and
:&lt;math&gt;
    \iiint_{V}
      \operatorname{div}\operatorname{curl}\mathbf{q}
    \;dV ~\!=\, 0 \,.
  &lt;/math&gt;
As the integrals vanish for ''any'' volume {{mvar|V}}&amp;#8202; in which the integrands are defined, the integrands must be zero wherever they are defined; that is,
{{NumBlk|:|&lt;math&gt;
    \operatorname{curl}\nabla p \equiv \mathbf{0}
  &lt;/math&gt;|{{EquationRef|24c}}}}
and
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\operatorname{curl}\mathbf{q} \equiv 0 \,.
  &lt;/math&gt;|{{EquationRef|24d}}}}
In words, ''the curl of the gradient is zero'', and ''the divergence of the curl is zero''; or, more concisely, ''any gradient is irrotational'', and ''any curl is solenoidal''.

We might well ask whether the converses are true. Is every irrotational vector field the gradient of something? And is every solenoidal vector field the curl of something? The answers are affirmative, but the proofs require more preparation.

Meanwhile we may note, as a mnemonic aid, that when the left-hand sides of the last two equations are rewritten in the del-cross and del-dot notations, they become&amp;#8201; {{math|&amp;nabla;&amp;#8201;&amp;times;&amp;#8201;&amp;nabla;''p''}}&amp;#8201; and&amp;#8201; {{math|&amp;nabla;&amp;#8201;'''&amp;sdot;'''&amp;#8201;&amp;nabla;&amp;#8202;&amp;times;&amp;#8202;'''q'''&amp;#8202;,}} respectively. The former ''looks like'' (but isn't) a cross-product of two parallel vectors, and the latter ''looks like'' (but isn't) a scalar triple product with a repeated factor, so that each expression ''looks like'' it ought to be zero (and it is). But such appearances can lead one astray, because {{math|&amp;nabla;}} is an operator, not a self-contained vector quantity; for example,&amp;#8201; {{math|&amp;nabla;''p''&amp;#8201;&amp;times;&amp;#8201;&amp;nabla;''&amp;phi;''}}&amp;#8201; is ''not'' identically zero, because two gradients are not necessarily parallel.&lt;ref&gt;''Cf''. [[#feynman-63|Feynman, 1963]], vol.&amp;#8239;2, &amp;sect;2-8.&lt;/ref&gt;

We should also note, to tie a loose end, that identity ({{EquationNote|24d}}) was to be expected from our [[#kelvin-stokes-verbal|verbal statement]] of the Kelvin&amp;ndash;Stokes theorem ({{EquationNote|22c}}). That statement implies that the flux of the curl through any two surfaces spanning the same closed curve is the same. So if we make a ''closed'' surface from two spanning surfaces, the flux into one spanning surface is equal to the flux out of the other, i.e. the net flux out of the closed surface is zero, i.e. the integral of the divergence over the enclosed volume is zero; and since ''any'' simple volume in which the divergence is defined can be enclosed this way, the divergence itself (of the curl) must be zero wherever it is defined.

{{cob}}

== Change per unit length ==

{{cot}}

Continuing (and concluding) the trend of reducing the number of dimensions, we now seek ''one''-dimensional theorems, each of which relates an integral over a ''path'' to values at the endpoints of the path. For maximum generality, the path should be allowed to be curved into a second and a third dimension.

We ''could'' do this by further specializing theorems ({{EquationNote|5g}}) to ({{EquationNote|5L}}). We could take a curve {{math|&amp;Gamma;}} with a unit tangent vector{{math| '''s&amp;#770;'''}}. At every point on{{math| &amp;Gamma;}} we could mount a circular disk with a uniform ''small'' area{{mvar| &amp;alpha;&amp;#8202;,}} centered on{{math| &amp;Gamma;}} and orthogonal to it. We could let {{mvar|V}} be the volume occupied by all the disks and let {{mvar|S}} be its enclosing surface; thus {{mvar|V}} would be a thin right circular cylinder, except that its axis could be curved. If we could arrange for {{mvar|&amp;alpha;}} to cancel out, our four theorems would indeed be reduced to the desired form, ''provided'' that there were no contribution from the curved face of the &quot;cylinder&quot; to the integral over{{mvar| S}} (the &quot;1D proviso&quot;). But, as it turns out, this exercise yields only one case in which the &quot;1D proviso&quot; can be satisfied by a construction involving {{math|'''s&amp;#770;'''}} and a general field, and we have already ''almost'' discovered that case by a simpler and more conventional argument&amp;mdash;which we shall now continue.

{{cob}}

=== Fundamental theorem ===

{{cot}}

Equation ({{EquationNote|9g}}) is applicable where {{math|''p''('''r''')}} is a scalar field,&amp;#8201; {{mvar|s}} is a parameter measuring arc length along a curve{{math| &amp;Gamma;,}} and {{math|'''s&amp;#770;'''}} is the unit tangent vector to{{math| &amp;Gamma;}} in the direction of increasing{{mvar| s}}. Let {{mvar|s}} take the values {{math|''s''&lt;sub&gt;1&lt;/sub&gt;}} and {{math|''s''&lt;sub&gt;2&lt;/sub&gt;}} at the endpoints of{{math| &amp;Gamma;,}} where the position vector {{math|'''r'''}} takes the values {{math|'''r'''&lt;sub&gt;1&lt;/sub&gt;}} and {{math|'''r'''&lt;sub&gt;2&lt;/sub&gt;}} respectively. Then, integrating ({{EquationNote|9g}}) w.r.t.{{mvar| s}} from {{math|''s''&lt;sub&gt;1&lt;/sub&gt;}} to {{math|''s''&lt;sub&gt;2&lt;/sub&gt;}} and applying the fundamental theorem of calculus, we get
{{NumBlk|:|&lt;math&gt;
    \int_{s_1}^{s_2} \nabla p \cdot \mathbf{\hat{s}} \,ds
      \,=\, p(\mathbf{r}_2) - p(\mathbf{r}_1) \,.
  &lt;/math&gt;|{{EquationRef|25g}}}}
This is our third integral theorem involving the gradient, and the best-known of the three: it is commonly called simply the '''[[w:gradient theorem|gradient theorem]]''',&lt;ref&gt;Although Hsu ([[#hsu-84|1984]], p.&amp;#8239;141) applies that name to our theorem ({{EquationNote|5g}}).&lt;/ref&gt; or the ''fundamental theorem of the gradient'', or the ''fundamental theorem of line integrals''; it generalizes the fundamental theorem of calculus to a curved path.&lt;ref&gt;''Cf''. [[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;sect;&amp;#8239;50,&amp;#8239;59; presumably this is one reason why Gibbs called the gradient simply the ''derivative''.&lt;/ref&gt; If we write {{math|''d'''''r'''}}&amp;#8202; for&amp;#8239; {{math|'''s&amp;#770;'''&amp;#8239;''ds''}} (the change in the position vector), we get the theorem in the alternative form
{{NumBlk|:|&lt;math&gt;
    \int_{\mathbf{r}_1}^{\mathbf{r}_2} \nabla p \cdot d\mathbf{r}
      \,=\, p(\mathbf{r}_2) - p(\mathbf{r}_1) \,.
  &lt;/math&gt;|{{EquationRef|25r}}}}
As the right-hand side of ({{EquationNote|25g}}) or ({{EquationNote|25r}}) obviously depends on the endpoints but ''not on the path in between'', so does the integral on the left. This integral is commonly called the '''work integral''' of{{math| &amp;nabla;''p''}} over the path&amp;mdash;because if {{math|&amp;nabla;''p''}} is a force, the integral is the work done by the force over the path. So, in words, the gradient theorem says that ''the change in value of a scalar field from one point to another is the work integral of the gradient of that field field over any path from the one to the other''.

Applying ({{EquationNote|25r}}) to a single element of the curve, we get
{{NumBlk|:|&lt;math&gt;\nabla p \cdot d\mathbf{r} = dp \,,
  &lt;/math&gt;|{{EquationRef|26g}}}}
which is reminiscent of&amp;#8201; &lt;math&gt;y'(x)~\!dx\,{=}\,dy\,&lt;/math&gt; in elementary calculus.&lt;ref&gt;''Cf''. [[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;sect;&amp;#8239;50,&amp;#8239;51; presumably this is another reason why Gibbs called the gradient the ''derivative''.&lt;/ref&gt; Alternatively, we could have obtained ({{EquationNote|26g}}) by multiplying both sides of ({{EquationNote|9g}}) by{{mvar| ds}}, and then obtained ({{EquationNote|25r}}) by adding ({{EquationNote|26g}}) over all the elemental displacements{{math| ''d'''''r'''}} on any path from {{math|'''r'''&lt;sub&gt;1&lt;/sub&gt;}} to{{math| '''r'''&lt;sub&gt;2&lt;/sub&gt;}}.

If we ''close'' the path by setting&amp;#8201; {{math|'''r'''&lt;sub&gt;2 &lt;/sub&gt;{{=}} '''r'''&lt;sub&gt;1&lt;/sub&gt;&amp;#8202;,}} the gradient theorem reduces to
{{NumBlk|:|&lt;math&gt;\oint \nabla p \cdot d\mathbf{r} \,=\, 0 \,,
  &lt;/math&gt;|{{EquationRef|27g}}}}
where the integral is around ''any'' closed loop. Applying the Kelvin&amp;ndash;Stokes theorem then gives
{{NumBlk|:|&lt;math&gt;
    \iint_{\varSigma}
      \operatorname{curl}\nabla p \cdot \boldsymbol{\hat{\nu}}
    \,d\varSigma \,=\, 0 \,,
  &lt;/math&gt;|{{EquationRef|28g}}}}
where {{mvar|&amp;Sigma;}}&amp;#8202; is any surface spanning the loop and&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt; is the unit normal to{{mvar| &amp;Sigma;}}.&amp;#8201; As this applies to any loop spanned by any surface on which the integrand is defined,&amp;#8201; {{math|curl&amp;#8201;&amp;nabla;''p''}}&amp;#8202; must be zero wherever it is defined. This is a second proof (more conventional than the first) of theorem ({{EquationNote|24c}}).

{{cob}}

=== Scalar potential: field with given gradient ===

{{cot}}

'''Lemma''':  If&amp;#8201; {{math|curl&amp;#8239;'''q'''&amp;#8201;{{=}}&amp;#8201;'''0'''}}&amp;#8201; in a simply connected region{{mvar| V}},&amp;#8201; then&amp;#8239; &lt;math&gt;\textstyle\int\!\mathbf{q}\!\cdot\!d\mathbf{r}\,&lt;/math&gt; over any path in{{mvar| V}}&amp;#8239; depends only on the endpoints of the path.

''Proof:''  Suppose, on the contrary, that there are two paths {{math|&amp;Gamma;}} and {{math|&amp;Lambda;}} in{{mvar| V}},&amp;#8202; with a common starting point and a common finishing point, such that
:&lt;math&gt;\textstyle\int_{\Gamma}\mathbf{q}\cdot d\mathbf{r}
       \,\neq \textstyle\int_{\Lambda}\mathbf{q}\cdot d\mathbf{r} \,.&lt;/math&gt;
Let&amp;#8201; {{math|&amp;minus;&amp;Lambda;}} denote {{math|&amp;Lambda;}} traversed backwards. Then for every {{math|''d'''''r'''}} on {{math|&amp;Lambda;}}&amp;#8201; there is an equal and opposite{{math| ''d'''''r'''}} on&amp;#8202; {{math|&amp;minus;&amp;Lambda;&amp;#8202;,}} and vice versa, so that we have
:&lt;math&gt;\textstyle\int_{\Gamma}\mathbf{q}\cdot d\mathbf{r} \,\neq\,
       \textstyle-\!\int_{-\Lambda}\mathbf{q}\cdot d\mathbf{r} \,,&lt;/math&gt;
i.e.
:&lt;math&gt;\textstyle\int_{\Gamma}\mathbf{q}\cdot d\mathbf{r} \,+
  \textstyle\int_{-\Lambda}\mathbf{q}\cdot d\mathbf{r} \,\neq\, 0 \,,&lt;/math&gt;
where the left-hand side is now a work integral of{{math| '''q'''}} around a closed loop in{{mvar| V}}.&amp;#8201; By the simple connectedness of{{mvar| V}},&amp;#8202; this loop is spanned by some surface{{mvar| &amp;Sigma;}} in{{mvar| V}}.&amp;#8201; So we can apply the Kelvin&amp;ndash;Stokes theorem and conclude that the flux integral of&amp;#8202; {{math|curl&amp;#8239;'''q'''}}&amp;#8202; through{{mvar| &amp;Sigma;}}&amp;#8202; is non-zero, in which case&amp;#8202; {{math|curl&amp;#8239;'''q'''}}&amp;#8202; must be non-zero somewhere on{{mvar| &amp;Sigma;&amp;#8202;,}} hence somewhere in{{mvar| V}}&amp;#8201;&amp;mdash;&amp;#8239;contradicting the hypothesis of the lemma. &amp;#9724;

'''Corollary''':  If&amp;#8201; {{math|curl&amp;#8239;'''q'''&amp;#8201;{{=}}&amp;#8201;'''0'''}}&amp;#8201; in a simply connected region{{mvar| V}},&amp;#8202; there exists a scalar field {{mvar|p}} such that&amp;#8239; {{math|'''q'''&amp;#8201;{{=}}&amp;#8201;&amp;nabla;''p''}}&amp;#8201; in{{mvar| V}}.

''Proof:''  We shall show that a suitable candidate is
:&lt;math&gt;p(\mathbf{r}) \,=
    \int_{\mathbf{r}_0}^{\mathbf{r}} \!\mathbf{q}\cdot d\boldsymbol{\rho} \,,
  &lt;/math&gt;
where {{math|'''r'''&lt;sub&gt;0&lt;/sub&gt;}} is the position vector of any fixed point in{{mvar| V}},&amp;#8202; and {{mvar|'''&amp;rho;'''}} is the position vector of a general point on the path of integration, which may be any path in{{mvar| V}}. First note that {{math|''p''('''r''')}} is unambiguous because, by the preceding lemma, it is independent of the path for given {{math|'''r'''&lt;sub&gt;0&lt;/sub&gt;}} and{{math| '''r''',}} provided that the path is in{{mvar| V}}.&amp;#8201; Now to find&amp;#8202; {{math|&amp;nabla;''p''('''r'''),}}&amp;#8202; let {{mvar|&amp;sigma;}} be the arc length along the path from {{math|'''r'''&lt;sub&gt;0&lt;/sub&gt;}} to{{mvar| '''&amp;rho;'''&amp;#8202;}}, so that {{mvar|&amp;sigma;}} ranges from 0 to (say){{mvar| s}}&amp;#8201; as {{mvar|'''&amp;rho;'''}} ranges from {{math|'''r'''&lt;sub&gt;0&lt;/sub&gt;}} to{{math| '''r'''&amp;#8202;}}; and let {{math|'''s&amp;#770;'''}} be the unit vector tangential to the path at{{mvar| '''&amp;rho;'''&amp;#8202;}}, in the direction of increasing{{mvar| &amp;sigma;}}.&amp;#8201; Then&amp;#8239; {{math|''d'''&amp;rho;'''''&amp;#8201;{{=}}&amp;#8201;'''s&amp;#770;'''&amp;#8239;''d&amp;sigma;''&amp;#8202;,}} so that the above equation becomes
:&lt;math&gt;p\big(\mathbf{r}(s)\big) \,=
    \int_0^s \!\mathbf{q}\cdot\mathbf{\hat{s}} \,d\sigma \,.
  &lt;/math&gt;
Differentiating w.r.t.{{mvar| s}} gives
:&lt;math&gt;\part_s p = \mathbf{q}\cdot\mathbf{\hat{s}} \,,&lt;/math&gt;
where {{math|'''s&amp;#770;'''}} is evaluated at&amp;#8201; {{mvar|&amp;sigma;&amp;#8201;{{=}}&amp;#8201;s}}&amp;#8201; and is therefore in the direction in which the path reaches{{math| '''r'''}}.&amp;#8201; By the generality of the path, this can be ''any'' direction. So the last equation says that {{math|'''q'''}} is the vector whose (scalar) component in any direction is the derivative of{{mvar| p}} w.r.t. arc length in that direction; that is, {{math|'''q'''&amp;#8201;{{=}}&amp;#8201;&amp;nabla;''p''&amp;#8202;,}} as required. &amp;#9724;

This is the promised converse of theorem ({{EquationNote|24c}}). ''But'', given an irrotational vector field {{math|'''q'''&amp;#8202;,}} we usually prefer to find a scalar field whose ''negative'' gradient is{{math| '''q'''&amp;#8202;}};&amp;#8201; that is, we usually prefer a scalar field &lt;math&gt;\varphi&lt;/math&gt; such that&amp;#8201;
&lt;math&gt;\mathbf{q}~\!\!=\!-\nabla\varphi&lt;/math&gt;.&amp;#8201; Such a field &lt;math&gt;\varphi&lt;/math&gt; is called a '''scalar potential''' for{{math| '''q'''}}.&amp;#8201; From the above expression for {{math|''p''('''r'''),}} a suitable candidate is
{{NumBlk|:|&lt;math&gt;\varphi(\mathbf{r}) \,=\,
    -\!\int_{\mathbf{r}_0}^{\mathbf{r}} \!\mathbf{q}\cdot d\boldsymbol{\rho} \,.
  &lt;/math&gt;|{{EquationRef|29}}}}

A scalar field has zero gradient if and only if it is uniform, so that adding a uniform field, but ''only'' a uniform field, to a given scalar field leaves its gradient unchanged. Thus ''the scalar potential is determined up to an arbitrary additive uniform field''. This would be the case with or without the minus sign in front of the gradient. The reason for preferring the minus sign appears next.

{{cob}}

=== Conservative fields ===

{{cot}}

An irrotational vector field&amp;mdash;or, equivalently, a field that is (plus or minus) the gradient of something&amp;mdash;is described as '''conservative''', because if the field is a force, it does zero work around a closed loop, and consequently ''conserves energy'' around the loop (at least if the field does not change during traversal of the loop).

If the only force acting on a particle is&amp;#8201; {{math|'''F'''&amp;#8201;{{=}}&amp;#8201;&amp;minus;&amp;nabla;''U'',}}&amp;#8201; then, by the gradient theorem, the work done on the particle over a path is the increase in {{mvar|&amp;minus;U}},&amp;#8201; i.e. the ''decrease'' in{{mvar| U&amp;#8202;}}; and this work is the increase in the particle's kinetic energy{{mvar| T}}.&amp;#8201; Hence, if we identify {{mvar|U}} with the ''potential'' energy, the total energy&amp;#8202; {{mvar|U&amp;#8202;+&amp;#8202;T}}&amp;#8201; is conserved. This interpretation of the scalar potential is possible only if the force is ''minus'' the gradient of the potential.

The minus sign is also used if the conservative vector field is an '''electric field''' (force per unit charge) or a gravitational acceleration (force per unit mass); the scalar potential is potential energy per unit charge, or potential energy per unit mass, respectively.

{{cob}}

== Some special fields ==

=== The 1/''r'' scalar potential ===

{{cot}}

For the potential energy field
{{NumBlk|:|&lt;math&gt;U = \frac{1}{\,r\,} \,&lt;/math&gt;|{{EquationRef|30}}}}
where {{mvar|r}} is the distance from the origin (and {{math|''r''&amp;#8201;&amp;ne;&amp;#8201;0}}), let us find the corresponding force&amp;#8201; {{math|'''F'''&amp;#8201;{{=}}&amp;#8201;&amp;minus;&amp;nabla;''U''}}.&amp;#8201; The direction of&amp;#8202; {{math|&amp;nabla;''U''}}&amp;#8202; is that of the steepest increase of{{mvar| U}}, which, by the spherical symmetry, can only be parallel or antiparallel to &lt;math&gt;\mathbf{\hat{r}}&lt;/math&gt; (the unit vector pointing away from the origin). So
:&lt;math&gt;\nabla U = \big(\nabla U \cdot \mathbf{\hat{r}}\big)~\!\mathbf{\hat{r}}
                = \part_r U \,\mathbf{\hat{r}}
                = \frac{d}{dr}\Big(\!\frac{1}{\,r\,}\!\Big)~\!\mathbf{\hat{r}}
                = -\frac{1}{\,r^2}~\!\mathbf{\hat{r}} \,,&lt;/math&gt;
whence
{{NumBlk|:|&lt;math&gt;\mathbf{F} = \frac{\mathbf{\hat{r}}}{\,r^2} \,.
  &lt;/math&gt;|{{EquationRef|31}}}}
So the negative gradient of the {{math|1/''r''}}&amp;#8202; scalar potential ({{EquationNote|30}}) is the unit '''inverse-square radial vector field'''. Multiplying the numerator and denominator by {{mvar|r}} gives the alternative form
:&lt;math&gt;\mathbf{F} = \frac{\mathbf{r}}{\,r^3} \,,&lt;/math&gt;
which is convenient if the center of the force is shifted from the origin to position{{math| '''r&amp;prime;'''}}: in that case we simply replace {{math|'''r'''}} by {{math|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;''',}} and {{mvar|r}} by {{math|{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}},}} so that the force becomes
:&lt;math&gt;\mathbf{F} = \frac{\mathbf{r}\!-\!\mathbf{r}'}
                         {|\mathbf{r}\!-\!\mathbf{r}'|^3}&lt;/math&gt;
and the corresponding scalar potential becomes
:&lt;math&gt;U = \frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|} \,.&lt;/math&gt;

{{cob}}

=== Inverse-square radial vector field ===

{{cot}}

We derived the vector field ({{EquationNote|31}}) as the negative gradient of the scalar potential ({{EquationNote|30}}). Conversely, given the inverse-square radial vector field ({{EquationNote|31}}), we could derive its scalar potential from ({{EquationNote|29}}). At a general point on the path, let the position vector be&amp;#8201; &lt;math&gt;\boldsymbol{\rho}~\!\!=\!\rho\boldsymbol{\hat{\rho}}\,&lt;/math&gt; so that, by ({{EquationNote|31}}),&amp;#8201; &lt;math&gt;\mathbf{F}\!=\!\boldsymbol{\hat{\rho}}/\rho^2&lt;/math&gt;.&amp;#8201; Then ({{EquationNote|29}}) becomes
:&lt;math&gt;\begin{align}U(\mathbf{r})
    \,&amp;=\, -\!\int_{\mathbf{r}_0}^{\mathbf{r}} \!
               \mathbf{F} \cdot d\boldsymbol{\rho} \\[1ex]
      &amp;=\, -\!\int_{\mathbf{r}_0}^{\mathbf{r}} \!
               \tfrac{1}{\,\rho^2}~\!
              \boldsymbol{\hat{\rho}} \!\cdot\! d\boldsymbol{\rho} \\[1ex]
      &amp;=\, -\!\int_{r_0}^r \! \tfrac{1}{\,\rho^2} \,d\rho
     \ =\, \tfrac{1}{\,\rho\,}\bigg|^r_{r_0}
     \,=\, \frac{1}{\,r\,}-\frac{1}{\,r_0} \,,
  \end{align}&lt;/math&gt;
so that, if we choose&amp;#8201; {{math|''r''&lt;sub&gt;0&lt;/sub&gt;&amp;#8202;&amp;rarr;&amp;#8239;&amp;infin;&amp;#8202;,}} we recover ({{EquationNote|30}}).

Because {{math|'''F''',}} given by ({{EquationNote|31}}), has a scalar potential,&amp;#8201; {{math|curl&amp;#8201;'''F'''}}&amp;#8202; must be zero. This is independently obvious in that the spherical symmetry of{{math|&amp;#8202; '''F'''}} seems to rule out any resemblance of rotation or shear&amp;mdash;even at the origin, where {{math|'''F'''}} becomes infinite. On the last point, let us check whether&amp;#8201; {{math|curl&amp;#8201;'''F'''}}&amp;#8202; has a meaningful integral over a volume containing the origin. If the volume {{mvar|V}}&amp;#8202; is enclosed by the surface {{mvar|S}}&amp;#8202; whose outward unit normal is &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;, then, by theorem ({{EquationNote|5c}}),
:&lt;math&gt;\iiint_V \operatorname{curl}\mathbf{F} ~dV
    \,= \iint_S \mathbf{\hat{n}}\times\mathbf{F} \,dS
    \,= \iint_S \mathbf{\hat{n}}\times\frac{\mathbf{\hat{r}}\,}{r^2} \,dS \,.
  &lt;/math&gt;
If {{mvar|V}} contains the origin, then, because&amp;#8201; {{math|curl&amp;#8201;'''F'''}}&amp;#8239; is zero everywhere ''except'' at the origin, the volume {{mvar|V}}&amp;#8202; can be replaced by any ''element'' of{{mvar| V}}&amp;#8202; containing the origin, whatever the shape of that element may be. If we choose that element to be a spherical ball centered on the origin, then &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is parallel to &lt;math&gt;\mathbf{\hat{r}}&lt;/math&gt;, so that the cross-product in the integrand on the right is zero. Thus the volume integral on the left is not only meaningful, but is ''zero'', even if the volume contains the point where the integrand is infinite. In this sense, the field {{math|'''F'''}} is ''so'' irrotational that its curl may be taken as zero even where the field itself is undefined!

The situation concerning the ''divergence'' of{{math|&amp;#8202; '''F'''}} is more complicated. Again, let the volume {{mvar|V}}&amp;#8202; be enclosed by the surface {{mvar|S}} whose outward unit normal is &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;.&amp;#8201; By the divergence theorem ({{EquationNote|5d}}),
:&lt;math&gt;\begin{align}\iiint_V \operatorname{div}\mathbf{F} ~dV
    \,= \iint_S \mathbf{\hat{n}}\cdot\mathbf{F} \,dS
   \,&amp;= \iint_S \mathbf{\hat{n}}\cdot\frac{\mathbf{\hat{r}}\,}{r^2} \,dS\\[1ex]
     &amp;= \iint_S \frac{\mathbf{\hat{r}}\cdot\mathbf{\hat{n}}\,dS}{\,r^2} \\[1ex]
     &amp;= \iint_S d\Omega \,,
  \end{align}&lt;/math&gt;
where {{math|''d''&amp;Omega;}} is the ''solid angle'' subtended at the origin by the surface element of area{{mvar| dS&amp;#8202;}}, and is taken as positive if the outward unit normal &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; has a positive component ''away from'' the origin &lt;math&gt;(\mathbf{\hat{r}}\!\cdot\!\mathbf{\hat{n}}&gt;0)&lt;/math&gt;, and negative if&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; has a positive component ''toward'' the origin &lt;math&gt;(\mathbf{\hat{r}}\!\cdot\!\mathbf{\hat{n}}&lt;0)&lt;/math&gt;. If the volume enclosed by {{mvar|S}} does ''not'' include the origin, then for every positive contribution {{math|''d''&amp;Omega;}}&amp;#8201; there is a compensating negative contribution, so that the integral of&amp;#8201; {{math|div&amp;#8201;'''F'''}}&amp;#8202; over the volume is zero. As this applies to every such volume,&amp;#8201; {{math|div&amp;#8201;'''F'''}}&amp;#8202; must be zero everywhere except at the origin. If, on the contrary, the volume ''does'' include the origin, then the contributions {{math|''d''&amp;Omega;}} add up to the total solid angle subtended by the enclosing surface, which is{{math| 4''&amp;pi;''}}. In summary,
{{NumBlk|:|&lt;math&gt;\mathrm{div}\Big(\frac{\mathbf{\hat{r}}}{\,r^2}\Big)
  =~\! 4\pi~\!\delta(\mathbf{r}) \,,&lt;/math&gt;|{{EquationRef|32d}}}}
where {{math|''&amp;delta;''('''r'''),}} the 3D '''unit delta function''', is zero everywhere except at the origin, but has an integral of&amp;#8202; {{math|1}} over any volume that includes the origin. For example, a unit point-mass at the origin has the density {{math|''&amp;delta;''('''r''')}}, and a point-mass {{mvar|m}} at position {{math|'''r&amp;prime;'''}} has the density&amp;#8202; {{math|''m&amp;delta;''('''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;''')}}. As the argument of&amp;#8202; {{math|div}}&amp;#8202; in ({{EquationNote|32d}}) is&amp;#8202; {{math|&amp;minus;&amp;nabla;(1/''r''),}} we also have
{{NumBlk|:|&lt;math&gt;\triangle\Big(\frac{1}{\,r\,}\Big)
  = -4\pi~\!\delta(\mathbf{r}) \,.&lt;/math&gt;|{{EquationRef|32L}}}}
If we shift the centers from the origin to {{math|'''r&amp;prime;''',}} the last two results become
{{NumBlk|:|&lt;math&gt;
    \mathrm{div}\bigg(\frac{ \mathbf{r}\!-\!\mathbf{r}'}
                           {|\mathbf{r}\!-\!\mathbf{r}'|^3} \!\bigg)
    =~\! 4\pi~\!\delta(\mathbf{r}\!-\!\mathbf{r}')
  &lt;/math&gt;|{{EquationRef|33d}}}}
and
{{NumBlk|:|&lt;math&gt;
    \triangle\bigg(\frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|}\bigg)
    = -4\pi~\!\delta(\mathbf{r}\!-\!\mathbf{r}') \,.
  &lt;/math&gt;|{{EquationRef|33L}}}}

{{cob}}

=== Field with given divergence (and zero curl) ===

{{cot}}

It follows from '''Coulomb's law''' that the electric field due to a point-charge {{mvar|Q}} at the origin, in a vacuum, is
:&lt;math&gt;\mathbf{E}
  = \frac{Q}{4\pi\epsilon_0 r^2} ~\!\mathbf{\hat{r}} \,,&lt;/math&gt;
where {{math|''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;}} is a physical constant (called the '''vacuum permittivity''' or simply the '''electric constant'''). In a ''vacuum'', the '''electric displacement field''', denoted by{{math| '''D'''&amp;#8202;,}} is {{math|''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;'''E'''}}.&amp;#8201; So it is convenient to multiply the above equation by {{math|''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8202;,}} obtaining
:&lt;math&gt;\mathbf{D}
  = \frac{Q}{4\pi} ~\!\frac{\mathbf{\hat{r}}\,}{r^2} \,.&lt;/math&gt;
This is a inverse-square radial vector field and therefore has zero curl.

Now suppose that, instead of a charge {{mvar|Q}} at the origin, we have a static ''charge density'' {{math|''&amp;rho;''('''r&amp;prime;''')}} in a general elemental volume {{mvar|dV&amp;prime;}}&amp;#8202; at position{{math| '''r&amp;prime;'''}} (the standard symbol for ''charge'' density being unfortunately the same as for ''mass'' density). Then the contribution from that element to the field{{math| '''D'''}} at position{{math| '''r'''}}&amp;#8202; is
:&lt;math&gt;d\mathbf{D}(\mathbf{r}) ~\!=~\!
    \frac{\,\rho(\mathbf{r}')\,dV'}{4\pi}\,
    \frac{\mathbf{r}\!-\!\mathbf{r}'}{|\mathbf{r}\!-\!\mathbf{r}'|^3}
  &lt;/math&gt;
provided that, for each {{math|'''r''',}} the dimensions of each volume element are small compared with {{math|{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}}}}. This contribution likewise has zero curl. The total field due to static charges is then the sum of the contributions:
{{NumBlk|:|&lt;math&gt;\mathbf{D}(\mathbf{r}) \,= \iiint
    \frac{\,\rho(\mathbf{r}')}{4\pi}\,
    \frac{\mathbf{r}\!-\!\mathbf{r}'}{|\mathbf{r}\!-\!\mathbf{r}'|^3} \,dV'
  &lt;/math&gt;|{{EquationRef|34}}}}
where the integral is over all space. And {{math|'''D'''('''r''')}} has zero curl because all the contributions have zero curl.

Independently of the physical significance of&amp;#8202; {{math|'''D'''('''r'''),}} we can take its divergence &quot;term by term&quot; (or &quot;under the integral sign&quot;), obtaining
:&lt;math&gt;\begin{align}\operatorname{div}\mathbf{D}(\mathbf{r})
  \,&amp;= \iiint \frac{\,\rho(\mathbf{r}')}{4\pi}~\!\mathrm{div}\bigg(
         \frac{\mathbf{r}\!-\!\mathbf{r}'}{|\mathbf{r}\!-\!\mathbf{r}'|^3}
       \!\bigg) ~\!dV' \\[3pt]
    &amp;= \iiint \frac{\,\rho(\mathbf{r}')}{4\pi}\,
         4\pi~\!\delta(\mathbf{r}\!-\!\mathbf{r}') \,dV'
         \quad \big[\mathsf{by~eq.(33d)}\big] \\[3pt]
    &amp;= \iiint \rho(\mathbf{r}')\,\delta(\mathbf{r}\!-\!\mathbf{r}')\,dV'\\[3pt]
    &amp;= \iiint \rho(\mathbf{r})\,\delta(\mathbf{r}\!-\!\mathbf{r}') \,dV'
           ~~~ \begin{bmatrix}~\!\!
                 \mathsf{since}~\delta(\mathbf{r}\!-\!\mathbf{r}')\!=\!0\\
                 \mathsf{unless}~\,\mathbf{r}'{=}~\!\mathbf{r}
               ~\!\!\end{bmatrix} \\
    &amp;= \,\rho(\mathbf{r})\!\iiint\delta(\mathbf{r}\!-\!\mathbf{r}')\,dV'\\[3pt]
    &amp;= \,\rho(\mathbf{r})\!\iiint\delta(\mathbf{r}'{-}~\!\mathbf{r})\,dV',
  \end{align}&lt;/math&gt;
where the last step is permitted because the volume integral of the delta function of{{math| '''r&amp;prime;'''}} is not changed by a &quot;[[w:point reflection|point reflection]]&quot; (inversion) across {{math|'''r'''}}.&amp;#8201; As the volume of integration (all space) includes the shifted origin of the delta function, the integral is simply{{math| 1&amp;#8202;,}} so that
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\mathbf{D}=\rho \,,
  &lt;/math&gt;|{{EquationRef|35}}}}
where both sides are evaluated at{{math| '''r'''}}.

Mathematically, this result is an identity which applies if&amp;#8202; {{math|'''D'''}} is given by ({{EquationNote|34}}); substituting for{{math| '''D'''&amp;#8202;,}} we can write the identity in full as
{{NumBlk|:|&lt;math&gt;\rho(\mathbf{r}) \,\equiv\, \mathrm{div}\iiint
    \frac{\,\rho(\mathbf{r}')}{4\pi}\,
    \frac{\mathbf{r}\!-\!\mathbf{r}'}{|\mathbf{r}\!-\!\mathbf{r}'|^3}
  \,dV',&lt;/math&gt;|{{EquationRef|36}}}}
where the integral is over all space, or at least all of the space in which {{mvar|&amp;rho;}} may be non-zero. Subject to the convergence of the integral, this shows that ''we can construct an irrotational vector field whose divergence is a given scalar field''{{math| ''&amp;rho;''('''r''')}}. And of course, by theorem ({{EquationNote|24d}}), ''any curl'' can be added to that vector field without changing its divergence.

In ''electrostatics'', ({{EquationNote|34}}) is a generalization of Coulomb's law; and ({{EquationNote|35}}), which follows from ({{EquationNote|34}}), is '''Gauss's law''' expressed in ''differential form''. If we integrate ({{EquationNote|35}}) over a volume enclosed by a surface{{mvar| S}} (with outward unit normal &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;) and apply the divergence theorem on the left, we get the ''integral form'' of Gauss's law:
{{NumBlk|:|&lt;math&gt;
    \iint_S \mathbf{D}\cdot\mathbf{\hat{n}}\,dS \,=\, Q_{\mathrm{e}} \,,
  &lt;/math&gt;|{{EquationRef|37}}}}
where {{math|''Q''&lt;sub&gt;e&lt;/sub&gt;}} is the total charge ''enclosed''&amp;#8202; by{{mvar| S}}.

{{cob}}

=== Field with given Laplacian ===

{{cot}}

In ({{EquationNote|36}}), we can recognize the {{math|'''r'''}}-dependent factor&amp;#8202; {{math|{{sfrac|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''|{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}}&lt;sup&gt;3&lt;/sup&gt;}}}}&amp;#8201; as&amp;#8202; {{math|&amp;minus;&amp;nabla;{{sfrac|1|&amp;#8202;{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}}&amp;#8202;}}}}&amp;#8201; and take the gradient operator outside the integral, obtaining
&lt;div style=&quot;margin-top: 1em&quot;&gt;
:&lt;math&gt;\rho(\mathbf{r}) \,\equiv\, \mathrm{div}\bigg(\!{-}\nabla\!\iiint
    \frac{\,\rho(\mathbf{r}')}{4\pi}\,
    \frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|}
  \,dV' \!\bigg) \,,&lt;/math&gt;
&lt;/div&gt;
i.e.
{{NumBlk|:|&lt;math&gt;\rho(\mathbf{r}) \,\equiv\, \triangle\bigg(\!{-}\!\iiint
    \frac{\,\rho(\mathbf{r}')}{4\pi}\,
    \frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|}
  \,dV' \!\bigg) \,,&lt;/math&gt;|{{EquationRef|38}}}}
where again the integral is over all space, or at least all of the space in which {{mvar|&amp;rho;}} may be non-zero. Subject to the convergence of the integral, this shows that ''we can construct a field whose Laplacian is a given field''. More precisely, it shows that we can construct a ''scalar'' field whose Laplacian is a given ''scalar''&amp;#8202; field{{math| ''&amp;rho;''('''r''')}}. But, due to the linearity of the Laplacian, the same applies to any given linear combination of scalar fields, including any combination whose coefficients are uniform vectors, uniform matrices, or uniform tensors of any order; that is, the same applies to any field that we can express with a uniform basis.

Mathematically, ({{EquationNote|38}}) is simply an identity. To find its significance in electrostatics, we can multiply it by&amp;#8202; {{math|&amp;minus;1&amp;#10744;''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8239;,}} obtaining
{{NumBlk|:|&lt;math&gt;-\frac{\rho(\mathbf{r})}{\epsilon_0} \,\equiv\,
  \triangle\iiint
    \frac{\,\rho(\mathbf{r}')}{4\pi\epsilon_0}\,
    \frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|}
  \,dV',&lt;/math&gt;|{{EquationRef|39}}}}
which is also an identity. But the negative gradient of the expression after the integral sign is
:&lt;math&gt;\frac{\,\rho(\mathbf{r}')\,dV'}{4\pi\epsilon_0}\,
  \frac{\mathbf{r}\!-\!\mathbf{r}'}{|\mathbf{r}\!-\!\mathbf{r}'|^3}\,,&lt;/math&gt;
which is the contribution to the electric field at position{{math| '''r'''}} due to a charge&amp;#8202; {{math|''&amp;rho;''('''r&amp;prime;''')&amp;#8239;''dV&amp;prime;''}}&amp;#8239; at position{{math| '''r&amp;prime;'''}} in a vacuum. So the expression after the integral sign is the corresponding contribution to the electrostatic potential, and the whole integral is the whole electrostatic potential. Denoting this by &lt;math&gt;\varphi~\!,\,&lt;/math&gt; we can rewrite ({{EquationNote|39}}) as
{{NumBlk|:|&lt;math&gt;
    \triangle\varphi = -\frac{\rho}{\,\epsilon_0} \,.
  &lt;/math&gt;|{{EquationRef|40}}}}
This is '''Poisson's equation''' in electrostatics, treating the medium as a vacuum (so that {{mvar|&amp;rho; }}must be taken as the ''total'' charge density, including any contributions caused by the effect of the field on the medium). In a region in which&amp;#8201; {{math|''&amp;rho;''&amp;#8201;{{=}}&amp;#8201;0&amp;#8202;,}}&amp;#8201; Poisson's equation ({{EquationNote|40}}) reduces to
{{NumBlk|:|&lt;math&gt;
    \triangle\varphi = 0 \,,
  &lt;/math&gt;|{{EquationRef|41}}}}
which is '''Laplace's equation''' in electrostatics.

{{cob}}

=== The wave equation ===

{{cot}}

It is an empirical fact that a compressible fluid, such as air, carries waves of a mechanical nature: sound waves. In establishing the unambiguity of the gradient and the divergence, we have already derived equations dealing with the inertia and continuity (mass-conservation) of non-viscous fluids. So, by introducing a relation describing the compressibility, and eliminating variables, we should be able to get ''one'' equation (the &quot;wave equation&quot;) in ''one'' scalar or vector field (the &quot;wave function&quot;), with recognizably &quot;wavelike&quot; solutions. And we should expect this equation to be analogous to equations describing other kinds of waves.

If we suppose, for simplicity, that the only force acting on an element of fluid is the pressure force, then the applicable equation of motion is ({{EquationNote|6g}}). But, for reasons which will soon be apparent, let us call the pressure{{mvar| P}}, so that ({{EquationNote|6g}}) becomes
:&lt;math&gt;
    \rho\,\frac{d\mathbf{v}}{dt} = -\nabla P \,.
  &lt;/math&gt;
Then at ''equilibrium'' we have
:&lt;math&gt;
    0 ~\!= -\nabla P_0 \,,
  &lt;/math&gt;
where {{math|''P''&lt;sub&gt;0&lt;/sub&gt;}} is the equilibrium pressure. Subtracting this equation from the previous one and defining
:&lt;math&gt;p = P - P_0 \,,&lt;/math&gt;
we get
:&lt;math&gt;
    \rho\,\frac{d\mathbf{v}}{dt} = -\nabla p \,,
  &lt;/math&gt;
which looks like ({{EquationNote|6g}}), except that {{mvar|p}} is now the '''sound pressure''' (also called &quot;acoustic pressure&quot;, or sometimes &quot;excess pressure&quot;), i.e. the pressure rise above equilibrium.

For the equation of continuity we can use ({{EquationNote|7d'}}), which we repeat for convenience:
:&lt;math&gt;
    \rho\operatorname{div}\mathbf{v} = -\frac{d\rho}{dt} \,.
  &lt;/math&gt;

Eliminating {{math|'''v'''}} between the last two equations is fraught because {{math|'''v''' }}is evaluated at a moving point in the former and at a fixed point in the latter; and introducing any relation between {{mvar|p}} and {{mvar|&amp;rho;}} is similarly fraught because {{mvar|p}} is evaluated at a fixed point and {{mvar|&amp;rho;}} at a moving point. The obvious remedy is to apply the advection rule ({{EquationNote|16}}) to the last two equations, obtaining respectively
:&lt;math&gt;\begin{align}
    \rho\Big(\tfrac{\part\mathbf{v}}{\part t}
             + \mathbf{v}\cdot\nabla\mathbf{v}\Big) \,&amp;=\, -\nabla p ~; \\
    \rho\operatorname{div}\mathbf{v} \,&amp;=\,
      -\tfrac{\part\rho}{\part t} - \mathbf{v}\cdot\nabla\rho \,.
  \end{align}&lt;/math&gt;
That gets all the variables evaluated at fixed points, at the cost of making the equations more complicated and more obviously non-linear. But the equations and be simplified and linearized by '''small-amplitude approximations'''. In the parentheses in the first equation, the first term is proportional to the amplitude of the vibrations while the second term is a product of ''two'' factors proportional to the amplitude, so that, for sufficiently small amplitudes, the second term is negligible. Similarly, in the second equation, for sufficiently small amplitudes and a ''homogeneous medium'', we can neglect the second term on the right. Then, on the left side of each equation, we are left with a factor proportional to the amplitude, multiplied by{{mvar| &amp;rho;}}. But {{mvar|&amp;rho;}} is not proportional to the amplitude; only its deviation from the equilibrium density is so proportional. Hence, for small amplitudes,&amp;#8201; {{mvar|&amp;rho; }}can be replaced by the equilibrium density, which we shall call{{math| ''&amp;rho;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8202;,}} which is independent of time and (in a homogeneous medium) independent of position. With these approximations, our equations of motion and continuity become
:&lt;math&gt;\begin{align}
    \rho_0 \mathbf{\dot{v}} &amp;= -\nabla p \,, \\[.5ex]
    \rho_0 \operatorname{div}\mathbf{v} &amp;= -\dot{\rho} \,,
  \end{align}&lt;/math&gt;
where, for brevity, we use an overdot to denote ''partial'' differentiation w.r.t. time (i.e., at a ''fixed'' point, not a point moving with the fluid).

Now we can eliminate {{math|'''v'''}}. Taking divergences in the first equation, and differentiating the second ''partially'' w.r.t. time (which can be done inside the {{math|div}} operator, which represents a linear combination), we get
:&lt;math&gt;\begin{align}
    \rho_0 \operatorname{div}\mathbf{\dot{v}} &amp;= -\triangle p \,, \\[.5ex]
    \rho_0 \operatorname{div}\mathbf{\dot{v}} &amp;= -\ddot{\rho} \,,
  \end{align}&lt;/math&gt;
so that we can equate the right-hand sides, obtaining
{{NumBlk|:|&lt;math&gt;\ddot{\rho} = \triangle p \,.&lt;/math&gt;|{{EquationRef|42}}}}

Maintaining the small-amplitude assumption, we can now consider compressibility. For ''small'' compressions in a ''homogeneous'' medium, we may suppose that the pressure change {{mvar|dp}} is some constant times the density change{{mvar| d&amp;rho;}}. It is readily verified that such a constant must have the dimension of velocity squared. So we can say&amp;#8201; {{math|''dp''&amp;#8201;{{=}}&amp;#8201;''c''&amp;sup2;&amp;#8202;''d&amp;rho;''&amp;#8202;,}} where {{mvar|c}} is a constant with the units of velocity.{{efn|When a gas is compressed, work is done on it, causing its temperature to rise, so that the ratio of {{mvar|dp}} to{{mvar| d&amp;rho;}} is higher than if the compression were isothermal. In sound waves, there is typically not enough time for a significant part of the heat of compression to be conducted away; that is, the compression is near enough to '''adiabatic'''. The words &quot;not enough time&quot; may suggest that the adiabatic approximation is a high-frequency approximation. But in fact, in free air, it is a ''low''-frequency approximation, because as the frequency is reduced, the equalization of temperature is hindered more by the longer wavelength than it is helped by the longer period. Only in a confined space, which limits the required distance of conduction, does the adiabatic assumption require the frequency to be ''above'' some lower limit. In a musical wind instrument, that lower limit tends to be far below the audible range. Meanwhile the upper limit, due to easier heat conduction within a shorter wavelength, tends to be very far above the audible range. Thus, under typical conditions, for the purpose of calculating{{mvar| c&amp;#8202;}}, the adiabatic assumption is reasonable. (See [[#fletcher-74|Fletcher, 1974]].)}} Dividing by {{mvar|dt}} gives&amp;#8201; &lt;math&gt;\dot{p}\!=\!c^2\dot{\rho}~\!,\,&lt;/math&gt; whence
{{NumBlk|:|&lt;math&gt;\ddot{p} = c^2~\!\ddot{\rho} \,.&lt;/math&gt;|{{EquationRef|43}}}}
Substituting from  ({{EquationNote|42}}) then gives the desired '''wave equation''':
{{NumBlk|:|&lt;math&gt;\ddot{p} = c^2 \triangle p \,.&lt;/math&gt;|{{EquationRef|44}}}}

This is the 3D classical wave equation with the sound pressure {{mvar|p}} as the wave function. For a generic wave function {{mvar|&amp;psi;&amp;#8202;,}} in a homogeneous isotropic medium, we would expect the equation to be
{{NumBlk|:|&lt;math&gt;\ddot{\psi} = c^2 \triangle\psi \,,&lt;/math&gt;|{{EquationRef|45}}}}
which may be written more compactly as
{{NumBlk|:|&lt;math&gt;\Box\psi =~\! 0 \,,&lt;/math&gt;|{{EquationRef|46}}}}
where {{math|&amp;#9744;,}} pronounced &quot;wave&quot; or &quot;box&quot;,{{efn|Or sometimes &quot;quabla&quot;, by analogy with &quot;nabla&quot;.}} is called the '''D'Alembertian''' operator and is defined by
{{NumBlk|:|&lt;math&gt;
    \Box\psi := \triangle\psi - \frac{1}{\,c^2}\frac{\part^2 \psi}{\part t^2}
  &lt;/math&gt;|{{EquationRef|47}}}}
in this paper, although other conventions exist.{{efn|In particular, some authorities change the sign, defining {{math|&amp;#9744;}} as&amp;#8201; &lt;math&gt;\tfrac{1}{\,c^2}\tfrac{\part^2}{\part t^2}\!-\!\triangle&lt;/math&gt;&amp;#8239;,&amp;#8201; and some write the operator (however defined) as{{math| &amp;#9744;&lt;sup&gt;2&lt;/sup&gt;}}.}}

In a ''static'' situation, the second term on the right of ({{EquationNote|47}}) is zero. So one advantage of definition ({{EquationNote|47}}), over any alternative definition that changes the sign or the scale factor, is that ''in the static case, the D'Alembertian is reduced to the Laplacian'', making it especially obvious that ''in the static case, the wave equation is reduced to Laplace's equation'' [compare ({{EquationNote|46}}) and ({{EquationNote|41}})]. Also notice that the D'Alembertian, being a linear combination of two linear operators, is itself ''linear''.

{{cob}}

=== Spherical waves ===

{{cot}}

Having established that there are wavelike time-dependent fields described by equation ({{EquationNote|45}}), in which the constant {{mvar|c}} has the units of velocity, we can now make an informed guess at an elementary solution of the equation. Consider the candidate
{{NumBlk|:|&lt;math&gt;
    \psi(\mathbf{r},t) = \tfrac{1}{\,r\,}~\!f\big(t-r/c\big) \,,
  &lt;/math&gt;|{{EquationRef|48}}}}
where&amp;#8201; &lt;math&gt;\mathbf{r}=r\mathbf{\hat{r}}&lt;/math&gt;&amp;#8201; is the position vector (so that {{mvar|r}} is distance from the origin),&amp;#8201; {{mvar|f}}&amp;#8239; is an arbitrary function (arbitrary except that it will need to be twice differentiable),&amp;#8201; {{mvar|t }}is time, and {{mvar|c }}is a constant (and obviously {{mvar|&amp;psi; }}is not defined at the origin even if {{mvar|f&amp;#8202; }}is.)

If, at the origin, the function {{mvar|f}}&amp;#8239; has a certain argument at time&amp;#8201; {{math|''t&amp;#8201;{{=}}&amp;#8201;&amp;tau;''&amp;#8202;,}}&amp;#8201; then at any distance{{mvar| r}}&amp;#8202; from the origin, it has the same argument at time&amp;#8201; {{math|''t&amp;#8201;{{=}}&amp;#8201;&amp;tau;&amp;#8239;+&amp;#8202;r''&amp;#10744;''c''&amp;#8202;,}}&amp;#8201; which is&amp;#8201; {{math|''r''&amp;#10744;''c'' }}&amp;#8239;''later''&amp;#8202; than at the origin. Hence, if {{mvar|f}}&amp;#8239; has a certain feature (e.g., a zero-crossing) at the origin, the time taken for that feature to reach any distance{{mvar| r}}&amp;#8239; is{{math| ''r''&amp;#10744;''c''&amp;#8202;,}}&amp;#8239; implying that the feature travels outward from the origin at speed{{mvar| c}}.&amp;#8201; Another way to perceive this is to set the argument of{{mvar| f}}&amp;#8201; equal to a constant (corresponding to some feature of the function) and differentiate w.r.t.{{mvar| t&amp;#8202;,}} obtaining&amp;#8201; &lt;math&gt;\dot{r}\!=\!c&lt;/math&gt;&amp;#8202; (the speed at which the feature recedes from the origin). Thus equation ({{EquationNote|48}}) describes ''waves''&amp;#8202; radiating outward from the origin with speed{{mvar| c}}. (The symbol {{mvar|c}} comes from a general-purpose Latin word for speed, but has become the usual symbol for ''wave'' speed.)

Equation ({{EquationNote|48}}) further implies that there are surfaces over which the wave function {{mvar|&amp;psi;}}&amp;#8239; is uniform&amp;mdash;namely surfaces of constant{{mvar| r}},&amp;#8201; i.e. spheres centered on the origin. These are the '''wavefronts'''. So ({{EquationNote|48}}) describes '''spherical waves'''.

Because the surface area of a sphere is proportional to the square of its radius, we should expect the radiated '''intensity''' (power per unit area) to satisfy an ''inverse-square law'' (if the medium is ''lossless''&amp;mdash;neither absorbing nor scattering the radiated power). That does ''not'' mean that the wave function itself should satisfy an inverse-square law. In a traveling wave in 3D space, there will be an &quot;effort&quot; variable (e.g., sound pressure) and a &quot;flow&quot; variable (e.g., fluid velocity), and the instantaneous intensity will be proportional to the product of the two. If the two are proportional to each other, the instantaneous intensity will be proportional to the square of one or the other. Hence if the instantaneous intensity falls off like{{math| 1/''r''&amp;#8202;&amp;sup2;,}} the effort and flow variables&amp;mdash;and the wave function, if it is proportional to one or the other&amp;mdash;will fall off like{{math| 1/''r''}}. That suggests the attenuation factor {{math|1/''r''}}&amp;#8202; in ({{EquationNote|48}}).

But there are big ''if''&amp;#8202;s in that argument. For all we know so far, the relation between effort and flow could involve a lag, so that the ''instantaneous'' product of the two could swing negative although it averages to something positive. And for all we know so far, the lag could vary with{{mvar| r}}, allowing at least one of the two (effort or flow) to depart from the {{math|1/''r''}}&amp;#8202; law, even if their average product still falls off like{{math| 1/''r''&amp;#8202;&amp;sup2;}}. The {{math|1/''r''}}&amp;#8202; factor in ({{EquationNote|48}}) is therefore only an &quot;informed guess&quot;. Notwithstanding these complications, we have also guessed that the form of the function {{mvar|f}}&amp;#8202; (the '''waveform''') does not change as {{mvar|r}} increases; we have not considered whether this behavior might depend on the medium, or the waveform, or the geometry of the wavefronts.

So let us carefully check whether ({{EquationNote|48}}) satisfies ({{EquationNote|45}}) or, equivalently, ({{EquationNote|46}}).

As a first step, and as a useful inquiry in its own right, we find {{math|&amp;#9651;''&amp;psi;''}} from definition ({{EquationNote|4L}}), given that {{mvar|&amp;psi;}} is a function of {{math|(''r'',&amp;#8202;''t'') }}only. For the surface {{mvar|&amp;delta;S}}&amp;#8202; let us start with
* a cone (''not'' a double cone) with its apex at the origin, subtending a ''small'' solid angle {{mvar|&amp;omega;}} at the origin,
* a sphere centered on the origin, with radius {{math|''r''}}, and
* a sphere centered on the origin, with radius {{mvar|r&amp;#8202;+&amp;#8202;dr&amp;#8202;}};
and let the volume element be the region inside the cone and between the spheres, so that its enclosing surface {{mvar|&amp;delta;S}}&amp;#8202; has three faces: a segment of the cone, a segment of the inner sphere with area{{math| ''r''&amp;#8202;&amp;sup2;''&amp;#8202;&amp;omega;''&amp;#8202;,}} and a segment of the outer sphere with area{{math| (''r&amp;#8202;+&amp;#8202;dr'')&lt;sup&gt;2&lt;/sup&gt;''&amp;omega;''&amp;#8202;}}. By the symmetry of{{mvar| &amp;psi;&amp;#8202;}}, the outward normal derivative {{mvar|&amp;part;&lt;sub&gt;n&lt;/sub&gt;&amp;#8202;&amp;psi;}}&amp;#8202; is equal to zero on the conical face,&amp;#8201; {{math|+''&amp;part;&lt;sub&gt;r&lt;/sub&gt;&amp;#8202;&amp;psi;''(''r&amp;#8202;+&amp;#8202;dr'',&amp;#8202;''t'')}} on the outer spherical face, and&amp;#8202; {{math|&amp;minus;''&amp;part;&lt;sub&gt;r&lt;/sub&gt;&amp;#8202;&amp;psi;''(''r'',&amp;#8202;''t'')}} on the inner spherical face. The volume of the element is&amp;#8201; {{math|''dV''&amp;#8201;{{=}}&amp;#8201;''r''&amp;#8202;&amp;sup2;''&amp;#8202;&amp;omega;&amp;#8239;dr''}}. So, assembling the pieces of definition ({{EquationNote|4L}}), we get
:&lt;!-- SUBSCRIPTS ENLARGED FOR LEGIBILITY: --&gt;&lt;math&gt;\begin{align}\triangle\psi
    &amp;= \frac{1}{r^2 \omega \,dr}\Big(\!
         (r\!+\!dr)^2 \omega ~\!\part_{\textstyle r} \psi(r\!+\!dr,t)
         - r^2 \omega ~\!\part_{\textstyle r} \psi(r,t)
       \!\Big) \\[1ex]
    &amp;= \frac{1}{\,r^2}~\!
       \frac{(r\!+\!dr)^2 \part_{\textstyle r}\psi(r\!+\!dr,t)
             - r^2 \part_{\textstyle r}\psi(r,t)}{dr} \\[.5ex]
    &amp;= \frac{1}{\,r^2}~\!
       \frac{\part}{\part r}\Big(r^2 \part_{\textstyle r}\psi(r,t)\Big) \,,
  \end{align}&lt;/math&gt;
i.e.
{{NumBlk|:|&lt;math&gt;
    \triangle\psi(r,t)
    \equiv \frac{1}{\,r^2}~\!\frac{\part}{\part r}
             \Big(r^2 \frac{\part\psi}{\part r}\Big)
             \qquad \big[\mathsf{if}\,\,r\!\neq~\!\!0\big]\,.
  &lt;/math&gt;|{{EquationRef|49}}}}

Now we can verify our &quot;informed guess&quot;. Differentiating ({{EquationNote|48}}) twice w.r.t.{{mvar| t}}&amp;#8202; by the chain rule gives
{{NumBlk|:|&lt;math&gt;
    \frac{\part^2\psi}{\part t^2} = \frac{1}{\,r\,}~\!f''\!\big(t-r/c\big) \,,
  &lt;/math&gt;|{{EquationRef|50}}}}
where each prime {{math|(&amp;prime;)}} denotes differentiation of the function w.r.t. its own argument. Differentiating ({{EquationNote|48}}) once w.r.t.{{mvar| r}}&amp;#8202; by the product rule and chain rule, we get
{{NumBlk|:|&lt;math&gt;
    \frac{\part\psi}{\part r}
    \,=\, -\frac{1}{cr}~\!f'\!\big(t-r/c\big)
          -\frac{1}{\,r^2}~\!f\big(t-r/c\big) \,.
  &lt;/math&gt;|{{EquationRef|51}}}}
Proceeding as specified in ({{EquationNote|49}}), we multiply this by {{math|''r''&amp;#8202;&amp;sup2;}}, differentiate again w.r.t.{{mvar| r}} (giving three terms, of which two cancel), and divide by {{math|''r''&amp;#8202;&amp;sup2;}}, obtaining
{{NumBlk|:|&lt;math&gt;
    \triangle\psi = \frac{1}{c^2 r}~\!f''\!\big(t-r/c\big) \,.
  &lt;/math&gt;|{{EquationRef|52}}}}
Then if we substitute ({{EquationNote|52}}) and ({{EquationNote|50}}) into ({{EquationNote|47}}), we obviously get&amp;#8201; {{math|&amp;#9744;''&amp;psi;''&amp;#8201;{{=}}&amp;#8201;0&amp;#8202;,}} satisfying ({{EquationNote|46}}). So we have guessed correctly.

Having shown that the D'Alembertian of{{mvar| &amp;psi;&amp;#8202;}}, as given by ({{EquationNote|48}}), is zero everywhere except at the origin (where it is not defined), let us now find its integral over a volume{{mvar| V}} (enclosed by a surface{{mvar| S}}) that includes the origin. From ({{EquationNote|47}}),
:&lt;math&gt;\begin{align}
    \iiint_V \Box\psi \,dV
      \,&amp;= \iiint_V \triangle\psi \,dV
           - \frac{1}{\,c^2}\iiint_V \frac{\part^2 \psi}{\part t^2}\,dV\\[.5em]
        &amp;= \iint_S \part_n \psi \,dS
           - \frac{1}{\,c^2}\iiint_V \frac{\part^2 \psi}{\part t^2}\,dV\,,
  \end{align}&lt;/math&gt;
where the second equality follows from theorem ({{EquationNote|5L}}). Now because the integrand on the left is zero except at the origin, ''any''{{mvar| V}} containing the origin will give the same integral. So for convenience, let {{mvar|V}} be a spherical ball of radius{{mvar| R}} centered on the origin. Then, by the spherical symmetry of{{mvar| &amp;psi;&amp;#8202;,}} integration over{{mvar| S}} reduces to multiplication by{{math| 4''&amp;pi;R''&amp;#8202;&lt;sup&gt;2&lt;/sup&gt;,}} and {{mvar|&amp;part;&lt;sub&gt;n&lt;/sub&gt;}} is equivalent to{{mvar| &amp;part;&lt;sub&gt;r&lt;/sub&gt;&amp;#8202;,}} and {{mvar|dV}} can be taken as{{math| 4''&amp;pi;r''&lt;sup&gt;&amp;#8202;2&lt;/sup&gt;''dr''}}. With these substitutions we have
:&lt;math&gt;
    \iiint_V \Box\psi \,dV \,=\,
      4\pi R^2 \frac{\part\psi}{\part r}\bigg|_{r=R} \!
      - \frac{1}{\,c^2}\!\int_0^R \!\frac{\part^2 \psi}{\part t^2}\,4\pi r^2\,dr
  &lt;/math&gt;
or, substituting from ({{EquationNote|51}}) and ({{EquationNote|50}}),
:&lt;math&gt;\begin{align}
    \iiint_V \!\Box\psi \,dV ~\!\!
     =\,&amp; 4\pi R^2 \!\Big(\!{-}\tfrac{1}{cR} f'\!\big(t\!-\!R/c\big)
                   - \tfrac{1\,}{R^2} f\big(t\!-\!R/c\big)\!\Big) \\
        &amp;- \tfrac{1}{\,c^2}\!\int_0^R \!\tfrac{1}{\,r\,}~\!
                                      f''\!\big(t-r/c\big)\,4\pi r^2\,dr \\[1ex]
     =\,&amp;-\tfrac{4\pi R}{\,c\,}~\!f'\!\big(t\!-\!R/c\big)
           - 4\pi f\big(t\!-\!R/c\big) \\
        &amp;- \tfrac{4\pi}{\,c^2}\!\int_0^R \!rf''\!\big(t-r/c\big) \,dr \,.
  \end{align}&lt;/math&gt;
Again noting that any {{mvar|V}} containing the origin will give the same volume integral, we can let {{mvar|R}} approach zero, with the result that the right-hand side approaches&amp;#8239;{{math| &amp;minus;4''&amp;pi;f''&amp;#8202;(''t'')}}. This is the integral of{{math| &amp;#9744;''&amp;psi;''}} over any volume containing the origin, for {{mvar|&amp;psi;}} given by ({{EquationNote|48}}). Meanwhile {{math|&amp;#9744;''&amp;psi;''}} is zero everywhere except that the origin. In summary,
{{NumBlk|:|&lt;math&gt;
    \Box~\!\Big\{\!\tfrac{1}{\,r\,}~\!f\big(t-r/c\big)\!\Big\}
    \equiv -4\pi f(t)\,\delta(\mathbf{r}) \,.
  &lt;/math&gt;|{{EquationRef|53}}}}
Shifting the center of the spherical waves from the origin to position{{math| '''r&amp;prime;''',}} we get
{{NumBlk|:|&lt;math&gt;
    \Box~\!\Big\{\tfrac{1}{|\mathbf{r}-\mathbf{r}'|}
                 ~\!f\big(t-\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big)\Big\}
    \equiv -4\pi f(t)\,\delta(\mathbf{r}\!-\!\mathbf{r}') \,.
  &lt;/math&gt;|{{EquationRef|54}}}}

We shall refer to the field given by ({{EquationNote|48}}) as the wave function due to a '''monopole''' source with '''strength''' {{math|''f''&amp;#8202;(''t'')}} at the origin. The D'Alembertian of this wave function is given by ({{EquationNote|53}}).&lt;ref&gt;Our definition of ''strength'' follows the old convention used by Baker &amp;amp; Copson ([[#baker-copson-39|1939, p.&amp;#8239;42]]), Born &amp;amp; Wolf ([[#born-wolf-02|2002, p.&amp;#8239;421]]), and Larmor ([[#larmor-1904|1904, p.&amp;#8239;5]]). The newer convention followed by Miller ([[#miller-91|1991, p.&amp;#8239;1371]]) would use the denominator {{math|4''&amp;pi;r''}} instead of our {{mvar|r}} in ({{EquationNote|48}}); this would have the advantage of eliminating the factor{{math| 4''&amp;pi;''}} from the D'Alembertian of the wave function, and the disadvantage of introducing that factor into the (denominator of the) wave function itself.&lt;/ref&gt; Hence the field whose D'Alembertian is given by ({{EquationNote|54}}) is the wave function due to a monopole source with strength {{math|''f''&amp;#8202;(''t'')}} at position{{math| '''r&amp;prime;'''}}. In each case, the D'Alembertian is zero everywhere except at the source; that is, the field satisfies the wave equation except at the source.

''A note in passing:&amp;#8202;'' The above verification that the field ({{EquationNote|48}}) satisfies the wave equation (except at the origin) did not depend on whether {{mvar|c}} was positive or negative. Nor did the demonstration that its D'Alembertian is given by ({{EquationNote|53}}). Hence we may replace{{mvar| c}} by{{mvar| &amp;minus;c}} in ({{EquationNote|54}}) and conclude that, even for positive{{mvar| c&amp;#8202;}}, the field
{{NumBlk|:|&lt;math&gt;
    \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}
    ~\!f\big(t+\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big)
  &lt;/math&gt;|{{EquationRef|54a}}}}
also satisfies the wave equation (except at{{math| '''r&amp;prime;'''}}), and has the same D'Alembertian as in ({{EquationNote|54}}) and the same limiting behavior as{{math| '''r'''&amp;rightarrow;'''r&amp;prime;'''}} and the argument of{{mvar| f}}&amp;#8201; approaches{{mvar| t}}. In this case, however, for positive{{mvar| c&amp;#8202;}}, we can hardly speak of a &quot;source&quot; at{{math| '''r&amp;prime;'''}}, because expression ({{EquationNote|54a}}) describes ''inward''-bound spherical waves converging on{{math| '''r&amp;prime;'''}}, which would violate causality if {{math| '''r&amp;prime;'''}} were the location of the source. But even if ({{EquationNote|54a}}) is dismissed as an &quot;acausal&quot; or &quot;unphysical&quot; solution of the wave equation, it nevertheless ''is'' a solution, and we are free to exploit this fact (such as it is) in derivations and proofs.

{{cob}}

=== Field with given D'Alembertian ===

{{cot}}

Now suppose that, instead of a monopole wave source with strength {{math|''f''&amp;#8202;(''t'')}} at the general position{{math| '''r&amp;prime;''',}} we have at that position a source strength ''density&lt;math&gt;~w(\mathbf{r}'\!,t)&lt;/math&gt;'' in an elemental volume {{mvar|dV&amp;prime;}}, whose (causal!) contribution to the wave function {{mvar|&amp;psi;}} at position{{math| '''r'''}}&amp;#8202; is therefore
:&lt;math&gt;d\psi(\mathbf{r},t)
    = \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}~\!
      w\big(\mathbf{r}',\,t-\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big)\,dV' ,
  &lt;/math&gt;
where for each {{math|'''r''',}} the dimensions of each volume element are small compared with {{math|{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}}}}. Then the total wave function is the sum of the contributions:
{{NumBlk|:|&lt;math&gt;\psi(\mathbf{r},t)
    = \iiint \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}~\!
      w\big(\mathbf{r}',\,t-\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big)\,dV' ,
  &lt;/math&gt;|{{EquationRef|55}}}}
where the integral is over all space.

Independently of the physical significance of{{math| ''&amp;psi;''('''r''',&amp;#8202;''t''),}} we can take its D'Alembertian &quot;under the integral sign&quot; by rule ({{EquationNote|54}}), obtaining
:&lt;math&gt;\begin{align}\Box\psi(\mathbf{r},t)
   &amp;= \iiint \Big({-}4\pi~\!w(\mathbf{r}'\!,t)\,
                  \delta(\mathbf{r}\!-\!\mathbf{r}')\Big)\,dV' \\[.5ex]
   &amp;= \iiint \Big({-}4\pi~\!w(\mathbf{r},t)\,
                  \delta(\mathbf{r}\!-\!\mathbf{r}')\Big)\,dV' \\[.5ex]
   &amp;= -4\pi~\!w(\mathbf{r},t)\!
       \iiint\delta(\mathbf{r}\!-\!\mathbf{r}')\,dV' \\[.5ex]
   &amp;= -4\pi~\!w(\mathbf{r},t)\!
       \iiint\delta(\mathbf{r}'{-}~\!\mathbf{r})\,dV' ;
  \end{align}&lt;/math&gt;
that is,
{{NumBlk|:|&lt;math&gt;
    \Box\psi(\mathbf{r},t) = -4\pi~\!w(\mathbf{r},t) \,.
  &lt;/math&gt;|{{EquationRef|56}}}}

Mathematically, equation ({{EquationNote|56}}) is an identity which applies if {{math|''&amp;psi;''('''r''',&amp;#8202;''t'')}} is given by ({{EquationNote|55}}). Substituting from ({{EquationNote|55}}) and solving for&lt;math&gt;~w,&lt;/math&gt; we can write the identity in full as
{{NumBlk|:|&lt;math&gt;w(\mathbf{r},t) \equiv \Box\bigg({-}\tfrac{1}{4\pi}\!\iiint
      \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}~\!
      w\big(\mathbf{r}',\,t-\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big)
    \,dV'\bigg) \,,
  &lt;/math&gt;|{{EquationRef|57}}}}
where the integral is over all space, or at least all of the space in which&lt;math&gt;~w&lt;/math&gt; may be non-zero. Subject to the convergence of the integral, this shows that ''we can construct a wave function with a given D'Alembertian''.

Physically, equation ({{EquationNote|56}}) gives the D'Alembertian of the wave function for a source density&amp;nbsp;&lt;math&gt;w&lt;/math&gt;. It is the ''inhomogeneous wave equation'', which applies in the presence of an arbitrary source density&amp;mdash;in contrast to the ''homogeneous wave equation'' ({{EquationNote|46}}), which applies in a region where the source density is zero. In this context the word ''homogeneous'' or ''inhomogeneous'' describes the equation, not the medium (which has been assumed homogeneous and isotropic).

In a ''static'' situation, in which the D'Alembertian is reduced to the Laplacian, the inhomogeneous wave equation ({{EquationNote|56}}) is reduced to the form of Poisson's equation ({{EquationNote|40}}). As written, equation ({{EquationNote|40}}) is Poisson's equation in electro''statics''; it applies to the charge density{{math| ''&amp;rho;''('''r''')}}, for which the scalar potential [in ({{EquationNote|39}})] is
:&lt;math&gt;\varphi(\mathbf{r}) = \tfrac{1}{4\pi\epsilon_0}
    \iiint \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}
    \rho(\mathbf{r}') \,dV'.
  &lt;/math&gt;
In electro''dynamics'', which takes time-dependence into account, the scalar potential due to the charge density{{math| ''&amp;rho;''('''r''',&amp;#8202;''t'')}} is
:&lt;math&gt;\varphi(\mathbf{r},t) = \tfrac{1}{4\pi\epsilon_0}
    \iiint \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}
    \rho\big(\mathbf{r}',\,t-\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big) \,dV',
  &lt;/math&gt;
where the wave speed {{mvar|c}} is the speed of light; this is the same as in the static case except for the delay {{math|{{sfrac|&amp;#8202;{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}}&amp;#8202;|''c''}}&amp;#8202;,}} indicating that the influence of the change density at{{math| '''r&amp;prime;'''}} travels outward from that point at the speed of light. In the dynamic case, by rule ({{EquationNote|57}}), the D'Alembertian of the scalar potential is
:&lt;math&gt;
    \Box\varphi = -\frac{\rho(\mathbf{r},t)}{\,\epsilon_0} \,.
  &lt;/math&gt;
This result is the inhomogeneous wave equation in the scalar potential&amp;mdash;the equation which, in the electro''static'' case, reduces to Poisson's equation ({{EquationNote|40}}).

In electro''dynamics'', however, the electric field&amp;#8202; {{math|'''E'''}} is ''not'' simply&lt;math&gt;\,\,{-}\nabla\varphi~\!,\,&lt;/math&gt; but&lt;math&gt;~\,{-}\nabla\varphi~\!\!-\!\tfrac{\part\mathbf{A}}{\part t}~\!,\,&lt;/math&gt; where {{math|'''A'''}} is the '''magnetic vector potential''', whose defining property is that its curl is the '''magnetic flux density''':
:&lt;math&gt;\mathbf{B} = \operatorname{curl}\mathbf{A} \,.&lt;/math&gt;
By identity ({{EquationNote|24d}}), this property implies
:&lt;math&gt;\operatorname{div}\mathbf{B} = 0 \,,&lt;/math&gt;
which is '''Gauss's law for magnetism'''. We have noted in passing&amp;mdash;but not yet proven&amp;mdash;that ({{EquationNote|24d}}) has a converse, whereby the solenoidality of{{math|&amp;#8202; '''B'''}} implies the ''existence'' of the vector potential{{math| '''A'''}}.  Precedents suggest we might be able to prove this by finding a vector field whose curl is a delta function&amp;mdash;perhaps through new identities relating it to a field whose divergence is a delta function&amp;mdash;and using it to construct a vector field with a given curl. In fact we shall prove our &quot;converse&quot; differently, but we shall still need some new identities for the purpose. And to obtain those identities (among others), we must take the detour that we have made a virtue of ''not'' taking until now&amp;hellip;

{{cob}}

== Cartesian coordinates ==

=== Indicial notation; implicit summation ===

{{cot}}

Considering that a scalar field is a function of three coordinates, while a vector field has three components each of which is a function of three coordinates, we can readily imagine that coordinate-based derivations of vector-analytic identities are likely to be excruciatingly repetitive&amp;mdash;unless perhaps we choose a notation that concisely specifies the repetition. So, instead of writing the Cartesian coordinates as {{math|''x'',&amp;#8239;''y'',&amp;#8239;''z''&amp;#8202;,}}&amp;#8239; we shall usually write them as {{mvar|x&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; where&amp;#8202; {{math|''i''&amp;#8201;{{=}}&amp;#8239;1,&amp;#8202;2,&amp;#8202;3&amp;#8202;,}}&amp;#8201; respectively;&amp;#8202; and instead of writing the unit vectors in the directions of the respective axes as {{math| '''i''',&amp;#8202;'''j''','''k'''&amp;#8202;,}}&amp;#8239; we shall usually write them as {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; And for partial differentiation w.r.t.{{math| ''x&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;,}} instead of writing {{mvar|{{sfrac|&amp;part;|&amp;part;x&lt;sub&gt;i&lt;/sub&gt;}}}} or even {{math|''&amp;part;&lt;sub&gt;x&lt;sub&gt;i&lt;/sub&gt;&lt;/sub&gt;''&amp;#8202;,}} we shall write {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;}}.

Now comes a stroke of genius for which we are indebted to Einstein (although he used it in a more sophisticated context!). Instead of writing the position vector as
:&lt;math&gt;\mathbf{r} = x_1\mathbf{e}_1 + x_2\mathbf{e}_2 + x_3\mathbf{e}_3&lt;/math&gt;
or even as
:{{big|&lt;math&gt;\mathbf{r} = \textstyle\sum_i x_i \mathbf{e}_i \,,&lt;/math&gt;}}
we shall write it simply as
:{{big|&lt;math&gt;\mathbf{r} = x_i \mathbf{e}_i \,,&lt;/math&gt;}}
where it is ''understood''&amp;#8202; that we ''sum over the repeated index''. More generally, we shall write the vector field {{math|'''q'''}} as
:{{big|&lt;math&gt;\mathbf{q} = q_i \mathbf{e}_i&lt;/math&gt;}}
with implicit summation, and the vector field {{math|'''v'''}} as
:{{big|&lt;math&gt;\mathbf{v} = v_i \mathbf{e}_i&lt;/math&gt;}}
with implicit summation, and so on. (By that nomenclature, the position vector in Cartesian coordinates should be, and often is, called {{math|'''x'''&amp;#8202;}}; but we called it {{math|'''r'''}} because we wanted to call its magnitude {{mvar|r}}, for ''radius''.)

Implicit summation not only avoids writing the {{big|{{math|&amp;Sigma;}}}} symbol and specifying the index of summation, but also allows a summation over ''two'' repeated indices, say {{mvar|i}} and {{mvar|j&amp;#8202;}}, to be considered as summed first over {{mvar|i}} and then over {{mvar|j}} or vice versa, removing the need for an explicit regrouping of terms. Of course, if we hide messy details behind a notation, we need to make sure that it handles those details correctly. In particular, when we perform an operation on an implicit sum, we implicitly perform it ''term-by-term'', and must therefore make sure that the operation is valid when interpreted that way.

{{cob}}

=== Formulation of operators ===

{{cot}}

'''Gradient''':  Putting&amp;#8201; {{mvar|s&amp;#8201;{{=}}&amp;#8201;x&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8201; in ({{EquationNote|9g}}), we find that the scalar component of{{math|&amp;#8202; &amp;nabla;''p''}} in the direction of each {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;}}&amp;#8202; is{{mvar|&amp;#8202; &amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8201;p}}.&amp;#8201; To obtain the vector component in that direction, we multiply by {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; Assembling the components, we have (with implicit summation)
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla p = \mathbf{e}_i ~\!\part_i p
  &lt;/math&gt;}}|{{EquationRef|58g}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla =~\! \mathbf{e}_i \part_i
  &lt;/math&gt;}}|{{EquationRef|58o}}}}
or, in traditional longhand notation,
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla =~\! \mathbf{i}~\!\tfrac{\part}{\part x}
           +~\! \mathbf{j}~\!\tfrac{\part}{\part y}
           +~\! \mathbf{k}~\!\tfrac{\part}{\part z} \,.
  &lt;/math&gt;}}|{{EquationRef|58t}}}}
It is also worth noting, from ({{EquationNote|58g}}), that the squared magnitude of{{math|&amp;#8202; &amp;nabla;''p'' &amp;#8239;}}is
{{NumBlk|:|{{big|&lt;math&gt;
    |\nabla p|^2 =~\! \part_i p \;\part_i p \,,
  &lt;/math&gt;}}|{{EquationRef|58s}}}}
where we write&amp;#8239; {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p&amp;#8201;&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p}}&amp;#8201; rather than {{math|(''&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p'')&lt;sup&gt;2&lt;/sup&gt;}}&amp;#8239; to ensure that implicit summation applies.

As reported by Tai ([[#tai-94|1994]]), there are unfortunately some textbooks in which the del operator is defined as
:{{big|&lt;math&gt;\nabla =~\!
    \tfrac{\part}{\part x}~\!\mathbf{i} +
    \tfrac{\part}{\part y}~\!\mathbf{j} +
    \tfrac{\part}{\part z}~\!\mathbf{k} \quad\qquad
  &lt;/math&gt;}}{{big|1=[''sic!''&amp;#8239;]}}
&amp;mdash;which, on its face, is not an operator at all, but a self-contained expression whose value is the zero vector (because it is a sum of derivatives of constant vectors). Among the offenders is Erwin Kreyszig, who, in the 6th edition of his bestselling ''Advanced Engineering Mathematics'' ([[#kreyszig-62-|1988]], p.&amp;#8239;486), misdefines the del operator thus and then rewrites the gradient of{{mvar|&amp;#8202; f}}&amp;#8239; as {{math|&amp;nabla;&amp;#8202;''f'',}} apparently imagining that the differentiation operators look ''through'' the constant vectors rather than ''at''&amp;#8202; them. Six pages later, he defines the divergence in Cartesian coordinates (which we shall do shortly) and then immediately informs us that &quot;Another common notation for the divergence of{{math| '''v'''}} is {{math|&amp;nabla;'''&amp;sdot;&amp;#8202;v'''}},&quot; where {{math|&amp;nabla;}} is defined as before, but the resulting {{math|&amp;nabla;'''&amp;sdot;&amp;#8202;v'''}} is apparently not identically zero!&lt;ref&gt;The latter passage, as it appears in the 5th edition (p.&amp;#8239;397), is the one cited by Tai ([[#tai-94|1994]], p.&amp;#8239;6).&lt;/ref&gt; These errors persist in the 10th edition ([[#kreyszig-62-|2011]], pp.&amp;#8239;396,&amp;#8239;402–3). Tai finds similar howlers in mathematics texts by Wilfred Kaplan, Ladis D. Kovach, and Merle C. Potter, and in electromagnetics texts by William H. Hayt and Martin A. Plonus.&lt;ref&gt;Quoted by Tai ([[#tai-94|1994]]), in alphabetical order within each category. For Kovach he could have added p.&amp;#8239;308.&amp;#8201; Potter he misnames as Porter.&lt;/ref&gt;&amp;#8201; Knudsen &amp;amp; Katz, in ''Fluid Dynamics and Heat Transfer'' (1958), avoid the misdefinition of{{math| &amp;nabla;,}} but implicitly define the divergence of{{math| '''V'''}} as {{math|1='''V&amp;sdot;'''&amp;nabla;}}&amp;#8202; (which, as we have seen, is actually an operator), and then somehow reduce it to the correct expression for{{math|&amp;#8202; div&amp;#8239;'''V'''}}.&amp;#8239;&lt;ref&gt;Quoted by Tai ([[#tai-94|1994]], p.&amp;#8239;23).&lt;/ref&gt;  But I digress.

'''Curl and divergence''':  Expressing the operand of the curl in components, and noting that the unit vectors are ''uniform'', we can apply ({{EquationNote|8p}}):
:{{big|&lt;math&gt;\begin{align}
    \operatorname{curl}\mathbf{q}
    &amp;= ~\!\mathrm{curl}(q_j ~\!\mathbf{e}_j) &amp;&amp; \\
    &amp;= \nabla q_j \times \mathbf{e}_j &amp;&amp; [\mathsf{\scriptstyle by~eq.(8p)}] \\
    &amp;= ~\!\mathbf{e}_i \part_i q_j \times \mathbf{e}_j
       &amp;&amp; [\mathsf{\scriptstyle by~eq.(58g)}] \\
    &amp;= ~\!\mathbf{e}_i ~\!\!\times \part_i ~\!q_j \mathbf{e}_j \,. &amp;&amp;
  \end{align}&lt;/math&gt;}}
If we sum over {{mvar|j}} first, this is
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}\mathbf{q} =~\! \mathbf{e}_i ~\!\!\times\part_i\mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|59c}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl} =~\! \mathbf{e}_i ~\!\!\times\part_i
  &lt;/math&gt;}}|{{EquationRef|59o}}}}
or, in traditional longhand,
:{{big|&lt;math&gt;
    \operatorname{curl}
    \,=\, \mathbf{i} \times ~\!\!\tfrac{\part}{\part x}
     +~\! \mathbf{j} \times ~\!\!\tfrac{\part}{\part y}
     +~\! \mathbf{k} \times ~\!\!\tfrac{\part}{\part z} \,.
  &lt;/math&gt;}}
For the ''divergence'' we proceed as for the curl except that, instead of ({{EquationNote|8p}}), we use ({{EquationNote|8g}}):
:{{big|&lt;math&gt;\begin{align}
    \operatorname{div}\mathbf{q}
    &amp;= ~\!\mathrm{div}(q_j ~\!\mathbf{e}_j) &amp;&amp; \\
    &amp;= \nabla q_j \cdot \mathbf{e}_j &amp;&amp; [\mathsf{\scriptstyle by~eq.(8g)}] \\
    &amp;= ~\!\mathbf{e}_i \part_i q_j \cdot \mathbf{e}_j
       &amp;&amp; [\mathsf{\scriptstyle by~eq.(58g)}] \\
    &amp;= ~\!\mathbf{e}_i ~\!\!\cdot \part_i ~\!q_j \mathbf{e}_j \,; &amp;&amp;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div}\mathbf{q} =~\! \mathbf{e}_i ~\!\!\cdot \part_i\mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|60d}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div} =~\! \mathbf{e}_i ~\!\!\cdot \part_i
  &lt;/math&gt;}}|{{EquationRef|60o}}}}
or, in traditional longhand,
:{{big|&lt;math&gt;
    \operatorname{div}
    \,=\, \mathbf{i} \cdot \tfrac{\part}{\part x}
     +~\! \mathbf{j} \cdot \tfrac{\part}{\part y}
     +~\! \mathbf{k} \cdot \tfrac{\part}{\part z} \,.
  &lt;/math&gt;}}
It follows from ({{EquationNote|59c}}) and ({{EquationNote|60d}}), if it was not already obvious, that ''a uniform vector field has zero curl and zero divergence''.

Although the above expressions for the divergence and curl will surprise many modern readers, they match the ''initial definitions'' of the divergence and curl given by the founder of vector analysis as we know it, [[w:Josiah Willard Gibbs|J.&amp;#8239;Willard Gibbs]] ([[#gibbs-1881-4|1881]], &amp;sect;&amp;#8239;54). Gibbs even uses the {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; notations on the left sides of the defining equations, and only ''after''&amp;#8202; the equations (albeit immediately after) does he announce that&amp;#8202; &quot;{{math|&amp;#8202;&amp;nabla;'''&amp;sdot;'''&amp;#8202;''&amp;omega;''}} is called the ''divergence'' of{{mvar| &amp;omega;}}&amp;#8239; and {{math|&amp;nabla;&amp;#8202;&amp;times;''&amp;omega;''}}&amp;#8239; its ''curl''.&quot; (He uses Greek letters for vectors.) Our notation and Cartesian expression for the gradient ({{EquationNote|58g}}) also match Gibbs ([[#gibbs-1881-4|1881]], &amp;sect;&amp;#8239;52). Hence, using the Gibbs notations, we can merge definitions ({{EquationNote|58g}}), ({{EquationNote|59c}}), and ({{EquationNote|60d}}) into the general Cartesian formula
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla~\!\! * \psi =~\! \mathbf{e}_i ~\!\! * \part_i \psi
  &lt;/math&gt;}}|{{EquationRef|60s}}}}
(with implicit summation), where the {{math|&amp;lowast;}} operator may be a null (for the gradient), a cross (for the curl), or a dot (for the divergence).

Gibbs does not offer any justification for the {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; notations, but nor is it difficult to find such a justification based on his definitions. As{{math| '''e'''&lt;sub&gt;''i''&lt;/sub&gt;}} is a ''uniform'' vector, we can rewrite ({{EquationNote|59c}}) ''rigorously'' as
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}\mathbf{q} = \part_i(\mathbf{e}_i ~\!\!\times\mathbf{q})
  &lt;/math&gt;}}|{{EquationRef|61c}}}}
and thence ''operationally'' as
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}\mathbf{q} =~\! \mathbf{e}_i\part_i \times \mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|61o}}}}
or, recalling ({{EquationNote|58o}}),
:&lt;math&gt;
    \operatorname{curl}\mathbf{q} ~\!= \nabla \times \mathbf{q} \,,
  &lt;/math&gt;
which can be evaluated in the usual manner as
:&lt;math&gt;
    \operatorname{curl}\mathbf{q} \,=\,
      \begin{vmatrix}
        \mathbf{i} &amp; \part_x &amp; q_x \\
        \mathbf{j} &amp; \part_y &amp; q_y \\
        \mathbf{k} &amp; \part_z &amp; q_z
      \end{vmatrix} \,,
  &lt;/math&gt;
where {{mvar|q&lt;sub&gt;x&lt;/sub&gt;}} is the {{mvar|x}} component of{{math| '''q'''&amp;#8202;}}, etc. This indeed is how one evaluates the curl of a given field in Cartesian coordinates, although we shall find ({{EquationNote|59c}}) more convenient for deriving identities. Similarly, we can rewrite ({{EquationNote|60d}}) ''rigorously'' as
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div}\mathbf{q} = \part_i(\mathbf{e}_i ~\!\!\cdot \mathbf{q})
  &lt;/math&gt;}}|{{EquationRef|62d}}}}
and thence ''operationally'' as
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div}\mathbf{q} =~\! \mathbf{e}_i\part_i \cdot \mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|62o}}}}
or, recalling ({{EquationNote|58o}}),
:&lt;math&gt;
    \operatorname{div}\mathbf{q} ~\!= \nabla \!\cdot \mathbf{q} ~.
  &lt;/math&gt;
For evaluating the divergence of a given field, however, we simplify ({{EquationNote|62d}}) to
:{{big|&lt;math&gt;
    \operatorname{div}\mathbf{q} = \part_i q_i
  &lt;/math&gt;}}
or, in traditional longhand,
:&lt;math&gt;
    \operatorname{div}\mathbf{q}
    ~\!= \frac{\part q_x}{\part x}
       + \frac{\part q_y}{\part y}
       + \frac{\part q_z}{\part z} \,,
  &lt;/math&gt;
although we shall find ({{EquationNote|60d}}) more convenient for deriving identities. But the longhand form makes it especially obvious that if{{math|&amp;#8202; '''r'''}} is the position vector,
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\mathbf{r} = 3 \,.
  &lt;/math&gt;|{{EquationRef|62r}}}}

Notice that we can get from ({{EquationNote|62o}}) back to ({{EquationNote|60d}}) by permuting the {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}} with the dot, and from ({{EquationNote|61o}}) back to ({{EquationNote|59c}}) by permuting the {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}} with the cross, as if the differentiation operator could, as it were, look through the dot or the cross&amp;mdash;or, as Gibbs's student [[w:Edwin Bidwell Wilson|Edwin B.&amp;#8201;Wilson]] puts it, &quot;pass by&quot; the dot and the cross, yielding Gibbs's original definitions.&lt;ref&gt;[[#wilson-1901|Wilson, 1901]], p.&amp;#8239;150.&lt;/ref&gt; Hence Wilson considers it helpful to regard Gibbs's {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; and {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; notations as &quot;the (formal) scalar product and the (formal) vector product of{{math|&amp;#8202; &amp;nabla;}} into&quot; the operand, or &quot;the symbolic scalar and vector products of{{math|&amp;#8202; &amp;nabla;}} into&quot; the operand, and to regard {{math|&amp;nabla;}} as a &quot;symbolic vector&quot;&lt;ref&gt;[[#wilson-1901|Wilson, 1901]], pp.&amp;#8239;150,&amp;#8239;152. Wilson does not announce this idea in his preface (p.&amp;#8239;xii), although Tai ([[#tai-95|1995, p.&amp;#8239;26]]) gets the contrary impression by omitting a comma from the relevant quote.&lt;/ref&gt; (not to be confused with Tai's symbolic vector&lt;math&gt;~\nabla\!\!\!\!^{\textstyle_-}&lt;/math&gt;).

Tai ([[#tai-94|1994]], [[#tai-95|1995]]) rejects Wilson's argument together with the entire tradition of treating {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; as compound operators. Of formal products, Tai says that the concept &quot;has had a tremendously detrimental effect upon the learning of vector analysis&quot;; he calls such a product a &quot;meaningless assembly&quot;.&lt;ref&gt;[[#tai-95|Tai, 1995]], pp.&amp;#8239;26,&amp;#8239;38.&lt;/ref&gt; Of the &quot;pass by&quot; step, he complains that &quot;standard books on mathematical analysis do not have such a theorem.&quot;&lt;ref&gt;[[#tai-95|Tai, 1995]], p.&amp;#8239;28.&lt;/ref&gt;

I submit, however, that the intermediate steps ({{EquationNote|61c}}) and ({{EquationNote|62d}}), after which we take the constant multiplier outside the operator (eqs. {{EquationNote|61o}} &amp;amp; {{EquationNote|62o}}), support Wilson's &quot;pass by&quot; argument. In any event the reader may write out the sums on the right-hand sides of ({{EquationNote|59c}}) and ({{EquationNote|60d}}) and verify that they agree with the formal products {{math|&amp;nabla;&amp;#8202;&amp;times;&amp;#8202;'''q'''}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;&amp;#8239;q'''}}&amp;#8202; respectively&amp;mdash;and may notice that in the evaluation of each formal product, the cross or dot is eventually eliminated, leaving nothing to &quot;pass by&quot;.&lt;ref&gt;The latter observation is made, or at least suggested, by Kemin et al. ([[#kemin-et-al-00|2000]], p.&amp;#8239;605).&lt;/ref&gt; I further submit that the great generality of our derivation of equations ({{EquationNote|14}}), above, compels us to treat the {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; notations as more than mere notations. But the kicker is that Tai himself, having found the form of the del operator in ''general'' coordinates ([[#tai-95|1995]], p.&amp;#8239;64, eq.&amp;#8239;9.33), derives original corresponding forms of the {{math|div}} and {{math|curl}} operators (his eqs.&amp;#8239;9.35 &amp;amp; 9.40) which, upon reversal of the forbidden &quot;pass by&quot;, become del-dot and del-cross! Indeed his three equations, just cited, are reminiscent of our ({{EquationNote|58o}}), ({{EquationNote|60o}}), and ({{EquationNote|59o}}) respectively. That being said, I shall find some points of agreement with Tai, and some reasons to criticize Wilson.

'''Laplacian''':  If {{mvar|&amp;psi;}} is a ''scalar'' field, then
:{{big|&lt;math&gt;\begin{align}\triangle\psi
    &amp;= \operatorname{div}\nabla\psi
       &amp;&amp; [\mathsf{\scriptstyle by~eq.(9L')}] \\
    &amp;= \part_i(\mathbf{e}_i \cdot \nabla\psi)
       &amp;&amp; [\mathsf{\scriptstyle by~eq.(62d)}] \\
    &amp;= \part_i(\part_i \psi) \,; &amp;&amp;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \triangle\psi = \part_i \part_i \psi \,,
  &lt;/math&gt;}}|{{EquationRef|63L}}}}
where we write&amp;#8202; {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;&amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; rather than {{math|''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&lt;sup&gt;2&lt;/sup&gt;}}&amp;#8202; in order to maintain implicit summation. In traditional longhand, ({{EquationNote|63L}}) becomes
:&lt;math&gt;\triangle\psi ~\!= \frac{\part^2 \psi}{\part x ^2}
                        + \frac{\part^2 \psi}{\part y ^2}
                        + \frac{\part^2 \psi}{\part z ^2}
  &lt;/math&gt;
or, in operational terms,
:&lt;math&gt;\triangle ~\!= \frac{\part^2}{\part x ^2}
                    + \frac{\part^2}{\part y ^2}
                    + \frac{\part^2}{\part z ^2}
  &lt;/math&gt;
or, by comparison with ({{EquationNote|58t}}),
:&lt;math&gt;\triangle = \nabla{\cdot}\nabla &lt;/math&gt;
&amp;mdash;as expected.

By the linearity of the Laplacian, the same applies if {{mvar|&amp;psi;}} is any field expressible in terms of a uniform basis. For example, if {{mvar|&amp;psi;}} is a ''vector'' field given by&amp;#8239; {{math|''&amp;psi;&lt;sub&gt;j&lt;/sub&gt;''&amp;#8239;'''e'''&lt;sub&gt;''j''&lt;/sub&gt;}}&amp;#8239; (with implicit summation), then
:{{big|&lt;math&gt;\begin{align}
    \triangle\psi
    &amp;= \triangle(\psi_j ~\!\mathbf{e}_j) \\
    &amp;= \mathbf{e}_j ~\!\triangle\psi_j \\
    &amp;= \mathbf{e}_j \part_i \part_i \psi_j \\
    &amp;= \part_i \part_i (\psi_j ~\!\mathbf{e}_j)
     = \part_i \part_i \psi \,,
  \end{align}&lt;/math&gt;}}
where the third line follows from ({{EquationNote|63L}}) as applied to a scalar field. Thus ({{EquationNote|63L}}) is quite general.

After listing theorems ({{EquationNote|5g}}) to ({{EquationNote|5L}}) above, we gave reasons for describing {{math|&amp;nabla;,}} {{math|curl,}} and {{math|div}} as ''differential operators'', and {{math|&amp;#9651;}} as a ''2nd-order'' differential operator&amp;mdash;the implication being that the others are only 1st-order. We now have the promised &quot;additional reason&quot; for these descriptions: when expressed in Cartesian coordinates, the {{math|&amp;#9651;}} operator involves second derivatives, while the others involve (only) first derivatives. In the meantime we have acquired the {{math|'''q&amp;sdot;'''&amp;nabla;}} operator, which is also 1st-order, as we shall now confirm.

'''Advection, directional derivative, etc.''':   If {{mvar|&amp;psi;}} is a ''scalar'' field, then
:{{big|&lt;math&gt;\begin{align}
    \mathbf{q}\;\!{\cdot}\nabla\,\psi
    &amp;= (\mathbf{q}) \cdot (\nabla\psi) \\
    &amp;= (q_i~\!\mathbf{e}_i) \cdot (\mathbf{e}_j ~\!\part_j \psi) \\
    &amp;= \;\!\mathbf{e}_i{\cdot}\;\!\mathbf{e}_j \;q_i \part_j \psi \,.
  \end{align}&lt;/math&gt;}}
In this double summation, the only non-zero terms are those for which&amp;#8202; {{mvar|j&amp;#8239;{{=}}&amp;#8201;i&amp;#8202;}},&amp;#8239; in which case&amp;#8239; {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;'''&amp;sdot;&amp;#8239;e'''&lt;sub&gt;''j''&lt;/sub&gt;&amp;#8201;{{=}}&amp;#8239;1}}.&amp;#8201; So we have
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\psi = q_i ~\!\part_i \psi
  &lt;/math&gt;}}|{{EquationRef|64}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla = q_i ~\!\part_i
  &lt;/math&gt;}}|{{EquationRef|64o}}}}
or, in traditional longhand,
:{{big|&lt;math&gt;\mathbf{q}\;\!{\cdot}\nabla
    =~\! q_x\tfrac{\part}{\part x}
    +~\! q_y\tfrac{\part}{\part y}
    +~\! q_z\tfrac{\part}{\part z} \,,
  &lt;/math&gt;}}
which indeed is the &quot;formal&quot; or &quot;symbolic&quot; dot-product of&amp;#8202; {{math|'''q'''}} and{{math| &amp;nabla;}}.&amp;#8202; By the linearity of the directional derivative in ({{EquationNote|11}}), the same result applies if {{mvar|&amp;psi;}} is a vector field or any field expressible in terms of a uniform basis. In particular, if{{math| '''r'''}} is the position vector, we have
:{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\mathbf{r}
    = q_i ~\!\part_i \mathbf{r}
    = q_i ~\!\mathbf{e}_i \,,
  &lt;/math&gt;}}
i.e.,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\mathbf{r} = \mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|64r}}}}
&amp;mdash;which is also deducible from ({{EquationNote|11}}).

For convenience in the following discussion, we shall refer to the scaled-directional-derivative operator {{math|'''q&amp;sdot;'''&amp;nabla;}} as an &quot;advection&quot; operator although, physically, it represents advection only if {{math|'''q'''}} is the material velocity.

{{cob}}

=== Identities without pain ===

{{cot}}

In deriving the Cartesian expressions for the gradient, curl, divergence, Laplacian, and advection operators, we used the preceding identities ({{EquationNote|9g}}), ({{EquationNote|8p}}), ({{EquationNote|8g}}), ({{EquationNote|9L'}}), and ({{EquationNote|11}}) respectively, the last being a definition generalizing ({{EquationNote|9g}}). Thus we could have derived the Cartesian expressions quite early in the exposition, although we did not find that option convenient. The other vector-analytic identities that we have previously mentioned are:
* ({{EquationNote|8c}}), which showed the unambiguity of the curl;
* ({{EquationNote|8q}}), which has a question mark after it;
* ({{EquationNote|17}}), a product rule for the divergence, which is yet to be proven as a general identity;
* ({{EquationNote|24c}}) and ({{EquationNote|24c}}), concerning &quot;curl grad&quot; and &quot;div curl&quot;; and
* the identities showing that we can construct a field with a given divergence ({{EquationNote|36}}), Laplacian ({{EquationNote|38}}), or D'Alembertian ({{EquationNote|57}}).

The above list exposes the following shortcomings:
* we have not yet investigated &quot;grad div&quot; and &quot;curl curl&quot;;
* we have only one ''product rule''&amp;#8202;&amp;mdash;the unverified identity ({{EquationNote|17}})&amp;mdash;in which ''both'' factors are spatially variable fields; this needs to be verified and identities ({{EquationNote|8c}}) and ({{EquationNote|8p}}) need to be generalized;
* our collection of product rules does not yet include the curl of a cross-product, or the gradient of a dot-product or of a product of scalars, or the advection of a product; and
* we do not yet have any ''chain rules'' involving {{math|&amp;nabla;,}} {{math|curl,}} or {{math|div}}.
With the aid of the Cartesian forms of the various operators, we may now fill these gaps.

&lt;br /&gt;

The &quot;'''grad div'''&quot; and &quot;'''curl curl'''&quot; operators turn out to be related:
:{{big|&lt;math&gt;\begin{align}
    \operatorname{curl}\operatorname{curl}\mathbf{q} \;\!
    &amp;= \mathbf{e}_i \times\part_i(\operatorname{curl}\mathbf{q}) \\
    &amp;= \mathbf{e}_i \times\part_i(\mathbf{e}_j \times\part_j\mathbf{q}) \\
    &amp;= \mathbf{e}_i \times(\mathbf{e}_j \times\part_i\part_j\mathbf{q}) \,,
  \end{align}&lt;/math&gt;}}
whence expanding the vector triple product gives
:{{big|&lt;math&gt;\operatorname{curl}\operatorname{curl}\mathbf{q} \;\!
    = \mathbf{e}_i \!\cdot~\!\!\part_i\part_j\mathbf{q} ~\mathbf{e}_j
      - \mathbf{e}_i {\cdot}~\!\mathbf{e}_j \,\part_i\part_j\mathbf{q} \,.
  &lt;/math&gt;}}
In the first term on the right, we can switch the order of partial differentiation; and in the second term&amp;mdash;which, like the first, is a double summation&amp;mdash;the only non-zero contributions are those for which&amp;#8202; {{mvar|j&amp;#8239;{{=}}&amp;#8201;i}}&amp;#8239; and&amp;#8239; {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;'''&amp;sdot;&amp;#8239;e'''&lt;sub&gt;''j''&lt;/sub&gt;&amp;#8201;{{=}}&amp;#8239;1}}.&amp;#8201; So we have
:{{big|&lt;math&gt;\begin{align}
    \operatorname{curl}\operatorname{curl}\mathbf{q} \;\!
    &amp;= \mathbf{e}_i \!\cdot~\!\!\part_j\part_i\mathbf{q} ~\mathbf{e}_j
       - \part_i\part_i\mathbf{q} \\
    &amp;= \mathbf{e}_j \,\part_j(\mathbf{e}_i \!\cdot~\!\!\part_i\mathbf{q})
      - \part_i\part_i\mathbf{q} \,;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}\operatorname{curl}\mathbf{q}
    ~\!\equiv \nabla\operatorname{div}\mathbf{q} - \triangle\mathbf{q} \,.
  &lt;/math&gt;}}|{{EquationRef|65}}}}
This result may be memorized as &quot;''curl curl is grad div minus del squared''&amp;#8239;&quot; and written as
{{NumBlk|:|{{big|{{math|&amp;nabla;&amp;#8239;&amp;times;&amp;#8201;(&amp;nabla;&amp;#8239;&amp;times;&amp;#8201;'''q''') &amp;equiv; &amp;nabla; &amp;nabla;'''&amp;sdot;&amp;#8239;q''' &amp;minus; &amp;nabla;&lt;sup&gt;2&lt;/sup&gt;&amp;#8202;'''q'''}}&amp;#8201;,}}|{{EquationRef|66}}}}
which ''looks like'' the expansion of a vector triple product; and the key step in the above derivation, based on the Gibbs definitions of the operators,  ''really is''&amp;#8202; the expansion of a vector triple product.

&lt;br /&gt;

We now turn to ''product rules'' in which neither factor is assumed uniform.

The '''curl of a cross-product''' is
:{{big|&lt;math&gt;\begin{align}
    &amp;\operatorname{curl}(\mathbf{a}\!\times\!\mathbf{b}) \\
    &amp;~= \mathbf{e}_i ~\!\!\times\part_i(\mathbf{a}\times\mathbf{b}) \\
    &amp;~= \mathbf{e}_i ~\!\!\times(\part_i\mathbf{a}\times\mathbf{b} +
                                 \mathbf{a}\times\part_i\mathbf{b}) \\
    &amp;~= \mathbf{e}_i ~\!\!\times(\part_i\mathbf{a}\times\mathbf{b}) +
        \mathbf{e}_i ~\!\!\times(\mathbf{a}\times\part_i\mathbf{b}) \\
    &amp;~= \mathbf{e}_i{\cdot}~\!\mathbf{b} \,\part_i\mathbf{a}
        - \mathbf{e}_i{\cdot}~\!\part_i\mathbf{a} \;\mathbf{b}
        + \mathbf{e}_i{\cdot}~\!\part_i\mathbf{b} \;\mathbf{a}
        - \mathbf{e}_i{\cdot}~\!\mathbf{a} \,\part_i\mathbf{b} \\
    &amp;~= b_i\part_i\mathbf{a} - (\operatorname{div}\mathbf{a})~\!\mathbf{b}
        + (\operatorname{div}\mathbf{b})~\!\mathbf{a} - a_i\part_i\mathbf{b} \\
    &amp;~= \mathbf{b}~\!{\cdot}\nabla\,\mathbf{a}
        - \mathbf{b}\operatorname{div}\mathbf{a}
        + \mathbf{a}\operatorname{div}\mathbf{b}
        - \mathbf{a}~\!{\cdot}\nabla\,\mathbf{b} \,,
  \end{align}&lt;/math&gt;}}
i.e.,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}(\mathbf{a}\!\times\!\mathbf{b})
    \equiv \mathbf{a}\operatorname{div}\mathbf{b}
         - \mathbf{b}\operatorname{div}\mathbf{a}
         + \mathbf{b}~\!{\cdot}\nabla\,\mathbf{a}
         - \mathbf{a}~\!{\cdot}\nabla\,\mathbf{b} \,.
  &lt;/math&gt;}}|{{EquationRef|67c}}}}

The '''divergence of a cross-product''', as we might expect, is simpler:
:{{big|&lt;math&gt;\begin{align}\operatorname{div}(\mathbf{a}\!\times\!\mathbf{b})
    &amp;= \mathbf{e}_i ~\!\!\cdot\part_i(\mathbf{a}\times\mathbf{b}) \\
    &amp;= \mathbf{e}_i ~\!\!\cdot(\part_i\mathbf{a}\times\mathbf{b} +
                               \mathbf{a}\times\part_i\mathbf{b}) \\
    &amp;= \mathbf{e}_i ~\!\!\cdot\part_i\mathbf{a}\times\mathbf{b} +
       \mathbf{e}_i ~\!\!\cdot\mathbf{a}\times\part_i\mathbf{b} \\
    &amp;= \mathbf{e}_i ~\!\!\cdot\part_i\mathbf{a}\times\mathbf{b} -
       \mathbf{e}_i ~\!\!\cdot\part_i\mathbf{b}\times\mathbf{a} \\
    &amp;= \mathbf{b}\cdot\mathbf{e}_i ~\!\!\times\part_i\mathbf{a} -
       \mathbf{a}\cdot\mathbf{e}_i ~\!\!\times\part_i\mathbf{b} \,;
  \end{align}&lt;/math&gt;}}
i.e.,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div}(\mathbf{a}\!\times\!\mathbf{b})
    \equiv \mathbf{b}\cdot\operatorname{curl}\mathbf{a}
         - \mathbf{a}\cdot\operatorname{curl}\mathbf{b} \,.
  &lt;/math&gt;}}|{{EquationRef|67d}}}}
In particular, in electromagnetics,&amp;#8201; {{math|div('''E'''&amp;#8239;&amp;times;&amp;#8239;'''H''') &amp;equiv; '''H&amp;#8239;&amp;sdot;'''&amp;#8201;curl&amp;#8201;'''E''' &amp;minus; '''E&amp;#8239;&amp;sdot;'''&amp;#8201;curl&amp;#8201;'''H'''}}&amp;#8239;;&amp;#8239; this is the identity on which [[w:Poynting's theorem|Poynting's theorem]] is based. But if&amp;#8202; {{math|'''b'''}} in ({{EquationNote|67d}}) is uniform, then ({{EquationNote|67d}}) reduces to ({{EquationNote|8c}}).

The '''gradient of a dot-product''', by comparison, is surprisingly messy:
:{{big|&lt;math&gt;\begin{align}\nabla\,\mathbf{a}{\cdot}\mathbf{b}
    &amp;= \mathbf{e}_i
       \part_i(\mathbf{a}\!\cdot\!\mathbf{b}) \\
    &amp;= \mathbf{e}_i
       (\mathbf{a}\!\cdot\!\part_i\mathbf{b} +
        \mathbf{b}\!\cdot\!\part_i\mathbf{a}) \\
    &amp;= \mathbf{a}\!\cdot\!\part_i\mathbf{b}
       \;\mathbf{e}_i +
       \mathbf{b}\!\cdot\!\part_i\mathbf{a}
       \;\mathbf{e}_i \,.
  \end{align}&lt;/math&gt;}}
Now the first term on the right can be recognized as&amp;#8201; {{math|'''a'''&amp;#8202;&amp;times;&amp;#8239;('''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8202;&amp;times;&amp;#8239;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''b''')&amp;#8201;+&amp;#8201;'''a&amp;sdot;&amp;#8202;e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8201;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''b'''&amp;#8202;}};&amp;#8201; that is,&amp;#8201; {{math|'''a'''&amp;#8202;&amp;times;&amp;#8239;('''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8202;&amp;times;&amp;#8239;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''b''')&amp;#8201;+&amp;#8201;''a&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''b'''&amp;#8202;}};&amp;#8201; that is,&amp;#8201; &lt;math&gt;\mathbf{a}\!\times\!\operatorname{curl}\mathbf{b}+\mathbf{a}~\!{\cdot}\nabla\,\mathbf{b}&lt;/math&gt;.&amp;#8201; Similarly, the second term is&amp;#8201; &lt;math&gt;\mathbf{b}\!\times\!\operatorname{curl}\mathbf{a}+\mathbf{b}~\!{\cdot}\nabla\,\mathbf{a}&lt;/math&gt;.&amp;#8201; Thus we have
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla\,\mathbf{a}{\cdot}\mathbf{b}
    \equiv \mathbf{a}\!\times\!\operatorname{curl}\mathbf{b}
         + \mathbf{b}\!\times\!\operatorname{curl}\mathbf{a}
	 + \mathbf{a}~\!{\cdot}\nabla\,\mathbf{b}
	 + \mathbf{b}~\!{\cdot}\nabla\,\mathbf{a} \,.
  &lt;/math&gt;}}|{{EquationRef|68}}}}
For ''uniform''&amp;#8202; {{math|'''b'''&amp;#8202;,}} the first and third terms on the right vanish, and we can solve for the first term on the right, obtaining
:&lt;math&gt;\mathbf{b}\times\operatorname{curl}\mathbf{a} ~\!=
    \nabla\,\mathbf{b{\cdot}a} - \mathbf{b}~\!{\cdot}\nabla\,\mathbf{a}
  \qquad&lt;/math&gt;[&amp;#8202;for uniform {{math|'''b'''}}] ,
so that we can now drop the question mark after ({{EquationNote|8q}}). If we write the curl operator as&amp;#8202; {{math|&amp;nabla;&amp;#8202;&amp;times;&amp;#8239;,}}&amp;#8239; the last equation [or ({{EquationNote|8q}})]&amp;#8202; ''looks like'' the expansion of a vector triple product; but the identity is valid only for uniform{{math| '''b'''}}.

The '''gradient of a product of scalars''', unlike that of a dot-product, is as simple as the product rule for ordinary differentiation:
:{{big|&lt;math&gt;\begin{align}\nabla(p\varphi)
    &amp;= \mathbf{e}_i \part_i(p\varphi) \\
    &amp;= \mathbf{e}_i(p~\!\part_i\varphi + \varphi~\!\part_i p) \\
    &amp;= p~\!\mathbf{e}_i\part_i\varphi + \varphi~\!\mathbf{e}_i\part_i p \,;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla(p\varphi) \equiv p\;\!\nabla\varphi + \varphi\;\!\nabla p \,.
  &lt;/math&gt;}}|{{EquationRef|69}}}}

The '''advection of a product''' is equally simple, ''regardless of the type of product'', except that the order of a cross-product matters. Let {{mvar|&amp;psi;}} and {{mvar|&amp;chi;}} be scalar or vector fields, and let {{math|''&amp;psi;''&amp;#8202;&amp;lowast;''&amp;chi;''}} denote any meaningful product of the two. Then, by ({{EquationNote|64}}),
:{{big|&lt;math&gt;\begin{align}
    \mathbf{q}\;\!{\cdot}\nabla\,(\psi~\!\! * \!\chi)
    &amp;= q_i \part_i (\psi~\!\! * \!\chi) \\
    &amp;= q_i (\psi * \part_i \chi + \part_i \psi * \chi) \\
    &amp;= \psi * q_i \part_i \chi + q_i \part_i \psi * \chi \,;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,(\psi~\!\! * \!\chi)
    \equiv \psi * (\mathbf{q}\;\!{\cdot}\nabla\chi) +
           (\mathbf{q}\;\!{\cdot}\nabla\psi) * \chi \,.
  &lt;/math&gt;}}|{{EquationRef|70}}}}
The {{math|'''q&amp;sdot;'''&amp;nabla;}} operator is a ''scalar'' operator in the sense that it maps the operand field to a field of the same order&amp;mdash;a scalar field to a scalar field, a vector field to a vector field, a matrix field to a matrix field, etc.&amp;mdash;&amp;#8202;''as if''&amp;#8202; it were multiplication by a scalar or differentiation w.r.t. a scalar; and indeed a differentiation w.r.t. path length appears in the coordinate-free definition ({{EquationNote|11}}) of the operator. Moreover, we did not need coordinates to obtain rule ({{EquationNote|70}}); as the reader may verify, the same rule can be obtained directly from the definition ({{EquationNote|11}}) in a similar manner. From these points of view, the simplicity of the rule is unsurprising.

The '''curl of the product of a scalar and a vector''' is
:{{big|&lt;math&gt;\begin{align}\operatorname{curl}p\mathbf{b}
    &amp;= \mathbf{e}_i ~\!\!\times \part_i(p\mathbf{b}) \\
    &amp;= \mathbf{e}_i ~\!\!\times(p~\!\part_i\mathbf{b}+\part_i p\;\mathbf{b}) \\
    &amp;= p~\!\mathbf{e}_i{\times}~\!\part_i\mathbf{b} +
       \mathbf{e}_i\part_i p \times\mathbf{b} \\
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}p\mathbf{b}
    \,\equiv\, p\operatorname{curl}\mathbf{b} ~\!+ \nabla p \times\mathbf{b} \,.
  &lt;/math&gt;}}|{{EquationRef|71c}}}}
For uniform {{math|'''b'''&amp;#8202;,}} this reduces to ({{EquationNote|8p}}), which was used to derive the Cartesian form of the curl ({{EquationNote|59c}}).

For the '''divergence of the product of a scalar and a vector''', we proceed likewise except that we use a dot instead of a cross. The result is
{{NumBlk|:|{{big|&lt;math&gt;
     \operatorname{div}p\mathbf{b}
     \,\equiv\, p\operatorname{div}\mathbf{b} ~\!+ \nabla p \cdot \mathbf{b} \,,
  &lt;/math&gt;}}|{{EquationRef|71d}}}}
which has the same form as ({{EquationNote|17}}), delivering the promised confirmation that ({{EquationNote|17}}) is an identity. For uniform {{math|'''b'''&amp;#8202;,}}&amp;#8201; ({{EquationNote|71d}}) reduces to ({{EquationNote|8g}}), which was used to derive the Cartesian form of the divergence ({{EquationNote|60d}}).

That exhausts the first-order product rules. For curiosity's sake, we shall also derive one second-order rule.

The '''Laplacian of the product of a scalar field and a generic field''', by ({{EquationNote|63L}}), is
:{{big|&lt;math&gt;\begin{align}\triangle(p\psi)
    &amp;= \part_i \part_i (p\psi) \\
    &amp;= \part_i (p~\!\part_i \psi + \psi~\!\part_i p) \\
    &amp;= p\,\part_i \part_i \psi + \part_i \psi\,\part_i p +
       \psi~\!\part_i \part_i p + \part_i p\,\part_i \psi \\
    &amp;= p\,\part_i \part_i \psi + 2\part_i p\,\part_i \psi +
       \psi~\!\part_i \part_i p \\
    &amp;= p~\!\triangle\psi + 2\part_i p\,\part_i \psi +
       \psi~\!\triangle p \,.
  \end{align}&lt;/math&gt;}}
In the middle term, by ({{EquationNote|58g}}), {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p}}&amp;#8202; is the {{mvar|i&amp;#8202;}}th component of{{math|&amp;#8202; &amp;nabla;''p''}}&amp;#8239; so that, by ({{EquationNote|64o}}),&amp;#8201; {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p&amp;#8239;&amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8239; is the {{math|'''q&amp;sdot;'''&amp;nabla;}} operator for&amp;#8239; {{math|'''q'''&amp;#8239;{{=}}&amp;#8239;&amp;nabla;''p''}}.&amp;#8201; So we have
{{NumBlk|:|{{big|&lt;math&gt;
     \triangle(p\psi)
     \equiv p~\!\triangle\psi + 2(\nabla p \cdot~\!\! \nabla)\psi +
              \psi~\!\triangle p \,.
  &lt;/math&gt;}}|{{EquationRef|72}}}}
The argument assumes a scalar{{mvar| p}} but is indifferent to whether {{mvar|&amp;psi;}} is a scalar or a vector or a higher-order tensor.

&lt;br /&gt;

Finally we turn to ''chain rules''&amp;#8202;&amp;mdash;&amp;#8202;especially the simple cases of the gradient, curl, divergence, advection, and Laplacian of a function of a scalar field{{mvar| u}}. As usual, let {{mvar|p}} denote a scalar field, {{math|'''q'''}} a vector field, and {{mvar|&amp;psi;}} a generic field.

'''Gradient&amp;#8202;&amp;#10744;&amp;#8202;curl&amp;#8202;&amp;#10744;&amp;#8202;divergence of a function of a scalar''':  By the general Cartesian formula ({{EquationNote|60s}}) and the chain rule for{{math| ''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;,}}
:{{big|&lt;math&gt;\begin{align}\nabla~\!\! * \big(\psi(u)\big)
    &amp;= \mathbf{e}_i ~\!\! * \part_i \big(\psi(u)\big) \\
    &amp;= \mathbf{e}_i ~\!\! * \psi'~\!\!(u) ~\!\part_i u \\
    &amp;= \mathbf{e}_i \part_i u * \psi'~\!\!(u) \,;
  \end{align}&lt;/math&gt;}}
i.e., by ({{EquationNote|58g}}),
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla~\!\! * \big(\psi(u)\big)
    \equiv \nabla u * \psi'~\!\!(u) \,.
  &lt;/math&gt;}}|{{EquationRef|73}}}}
In particular, if&amp;#8202; {{math|&amp;lowast;}} is a null,
{{NumBlk|:|&lt;math&gt;
    \nabla\big(p(u)\big) \equiv \nabla u \;p'~\!\!(u) \,;
  &lt;/math&gt;|{{EquationRef|73g}}}}
and if&amp;#8202; {{math|&amp;lowast;}} is a cross,
{{NumBlk|:|&lt;math&gt;
    \mathrm{curl}\big(\mathbf{q}(u)\big)
    \equiv \nabla u \times \mathbf{q}'~\!\!(u) \,;
  &lt;/math&gt;|{{EquationRef|73c}}}}
and if&amp;#8202; {{math|&amp;lowast;}} is a dot,
{{NumBlk|:|&lt;math&gt;
    \mathrm{div}\big(\mathbf{q}(u)\big)
    \equiv \nabla u \cdot \mathbf{q}'~\!\!(u) \,.
  &lt;/math&gt;|{{EquationRef|73d}}}}

'''Advection of a function of a scalar''':
:&lt;!-- SUBSCRIPTS ENLARGED FOR LEGIBILITY: --&gt;&lt;math&gt;\begin{align}
    \mathbf{q}\;\!{\cdot}\nabla\,\big(\psi(u)\big)
    &amp;= q_{\textstyle i} \part_{\textstyle i} \big(\psi(u)\big) \\
    &amp;= q_{\textstyle i} ~\!\psi'~\!\!(u) ~\!\part_{\textstyle i} u \\
    &amp;= q_{\textstyle i} \part_{\textstyle i} u \;\psi'~\!\!(u) \,;
  \end{align}&lt;/math&gt;
i.e.,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\big(\psi(u)\big)
    \equiv~\! \mathbf{q}\;\!{\cdot}\nabla u ~\psi'~\!\!(u) \,.
  &lt;/math&gt;}}|{{EquationRef|73q}}}}
This fits into the pattern set by ({{EquationNote|73}}) in that the gradient operator in ({{EquationNote|73g}}) is replaced by an advection operator.

Of the last four results, only ({{EquationNote|73c}}) is dependent on the order of the {{math|&amp;lowast;}} product; the others could equally well be written
{{NumBlk|:|&lt;math&gt;\begin{align}
    \nabla\big(p(u)\big) &amp;\equiv~\! p'~\!\!(u) ~\!\nabla u \\
    \mathrm{div}\big(\mathbf{q}(u)\big)
    &amp;\equiv~\! \mathbf{q}'~\!\!(u) \cdot~\!\! \nabla u \\
    \mathbf{q}\;\!{\cdot}\nabla\,\big(\psi(u)\big)
    &amp;\equiv~\! \psi'~\!\!(u)\;\mathbf{q}\;\!{\cdot}\nabla u ~.
  \end{align}&lt;/math&gt;|{{EquationRef|73z}}}}

The '''Laplacian of a function of a scalar''' departs from the above pattern.
:{{big|&lt;math&gt;\begin{align}\triangle\big(\psi(u)\big)
    &amp;= \part_i \part_i \big(\psi(u)\big) \\
    &amp;= \part_i\big(\psi'~\!\!(u) ~\!\part_i u\big) \\
    &amp;= \psi'~\!\!(u) ~\!\part_i\part_i u
       + \psi''~\!\!(u) ~\!\part_i u \,\part_i u \,,
  \end{align}&lt;/math&gt;}}
where the last line follows from the product rule for {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; and, in the second term, the chain rule for{{mvar| &amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; In that second term, the implicit sum&amp;#8202; {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;u&amp;#8239;&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;u}}&amp;#8239; can be recognized as&amp;#8202; {{math|{{abs|&amp;nabla;''u''}}&lt;sup&gt;2&lt;/sup&gt;}}&amp;#8202; by ({{EquationNote|58s}}). So we have
{{NumBlk|:|{{big|&lt;math&gt;
    \triangle\big(\psi(u)\big)
    \equiv \psi'~\!\!(u)~\!\triangle u + \psi''~\!\!(u)~\!\big|\nabla u\big|^2.
  &lt;/math&gt;}}|{{EquationRef|74}}}}

'''Multivariate chain rule''':  The foregoing chain rules involve ''one'' intermediate function of ''one'' scalar variable. It will be useful to have an elementary chain rule that can handle more than one of each. Let {{math|''p''('''r''')}} be a smooth scalar field, and let {{math|'''r'''}} in turn be a smooth function of several variables, one of which, say{{mvar| t&amp;#8202;}}, is allowed to vary while the others are held constant, so that {{math|'''r'''}} changes by {{math|''d'''''r'''}} when {{mvar|t}} changes by {{mvar|dt}}. Then dividing ({{EquationNote|26g}}) by {{mvar|dt}}&amp;#8202; gives
:&lt;math&gt;\part_t p = \nabla p \cdot \part_t \mathbf{r}&lt;/math&gt;
or, in indicial Cartesian coordinates with implicit summation,
:{{big|&lt;math&gt;\part_t p = \part_i p \,\part_t x_i&lt;/math&gt;}}
or, in traditional longhand,
:{{big|&lt;math&gt;\tfrac{\part}{\part t}~\!p(x,y,z)
    = \tfrac{\part p}{\part x}~\!\tfrac{\part x}{\part t}
    + \tfrac{\part p}{\part y}~\!\tfrac{\part y}{\part t}
    + \tfrac{\part p}{\part z}~\!\tfrac{\part z}{\part t} \,.
  &lt;/math&gt;}}
This is the desired multivariate chain rule for a scalar function of three intermediate real variables. The assumption that these variables are Cartesian coordinates is not a loss of generality, because any three real quantities can be suitably scaled and represented by perpendicular axes, so that any scalar function of them becomes a function of position, to which ({{EquationNote|26g}}) applies; and then the scaling can be reversed without changing the products in the last equation. Moreover, by the linearity of{{mvar| &amp;part;&lt;sub&gt;t&lt;/sub&gt;&amp;#8239;}}, the scalar field {{mvar|p}} may be replaced by any field expressible in terms of a uniform basis. For example, for a vector field{{math| '''q'''&amp;#8202;}},
:{{big|&lt;math&gt;\begin{align}\part_t \mathbf{q}
    &amp;= \part_t (q_j \mathbf{e}_j) \\
    &amp;= \mathbf{e}_j \part_t q_j \\
    &amp;= \mathbf{e}_j \part_i q_j \,\part_t x_i \\
    &amp;= \part_i (q_j \mathbf{e}_j) ~\!\part_t x_i
     = \part_i \mathbf{q} \,\part_t x_i \,,
  \end{align}&lt;/math&gt;}}
where the third line is obtained by applying the multivariate chain rule for a scalar field. Thus, for a generic field {{mvar|&amp;psi;&amp;#8202;}},
{{NumBlk|:|{{big|&lt;math&gt;
    \part_t \psi = \part_i \psi \,\part_t x_i
  \qquad&lt;/math&gt;}}[&amp;#8202;for generic {{mvar|&amp;psi;}} and {{mvar|x&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;}}].|{{EquationRef|75}}}}

'''Gradient&amp;#8202;&amp;#10744;&amp;#8202;curl&amp;#8202;&amp;#10744;&amp;#8202;divergence of a function of a scaled position vector''':  We end this subsection by deriving a lemma for use in the next subsection. If{{mvar| k}} is a uniform scalar multiplier and {{math|'''r'''}} is the position vector,
:{{big|&lt;math&gt;
    \nabla * \psi(k\mathbf{r})
    = \mathbf{e}_i * \part_i \psi(k\mathbf{r})
    = k\mathbf{e}_i * \part_{(kx_{\scriptstyle i})} \psi(k\mathbf{r}) \,,
  &lt;/math&gt;}}
where the third expression is obtained by from the second by multiplying each denominator (change in{{mvar| x&lt;sub&gt;i&lt;/sub&gt;}}) by{{mvar| k}}&amp;#8239; and compensating. But now we have
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla * \psi(k\mathbf{r}) = k\,(\nabla {*}~\! \psi)\Big|_{k\mathbf{r}} \,,
  &lt;/math&gt;}}|{{EquationRef|76}}}}
where the vertical bar and subscript indicate that the gradient, curl, or divergence is evaluated at{{math| ''k''&amp;#8202;'''r'''}}. We shall be interested in the curl (for which {{math|&amp;lowast;}} is a cross).

{{cob}}

=== Field with given curl ===

{{cot}}

Consider the vector field
{{NumBlk|:|&lt;math&gt;
    \mathbf{v}(\mathbf{r}) = \mathbf{q}(\mathbf{r})\times\mathbf{r} \,,
  &lt;/math&gt;|{{EquationRef|77}}}}
where {{math|'''q'''}} is a ''solenoidal''&amp;#8202; vector field and {{math|'''r''' }}is the position vector. By identity ({{EquationNote|67c}}),
:&lt;math&gt;\operatorname{curl}\mathbf{v}
  =   \mathbf{q}\operatorname{div}\mathbf{r}
    - \mathbf{r}\operatorname{div}\mathbf{q}
    + \mathbf{r}~\!{\cdot}\nabla\,\mathbf{q}
    - \mathbf{q}~\!{\cdot}\nabla\,\mathbf{r}
  &lt;/math&gt;
where, by hypothesis, {{math|div&amp;#8239;'''q'''}}&amp;#8202; is zero. Applying identities ({{EquationNote|62r}}) and ({{EquationNote|64r}}) then yields
:&lt;math&gt;\begin{align}
  \operatorname{curl}\mathbf{v}
  &amp;= 3\mathbf{q} + \mathbf{r}~\!{\cdot}\nabla\,\mathbf{q} - \mathbf{q} \\
  &amp;= 2\mathbf{q} + \mathbf{r}~\!{\cdot}\nabla\,\mathbf{q} \,.
  \end{align}&lt;/math&gt;

[[File:Vorticity_Figure_01_a-m.gif|thumb|Animation of a rigid-body-like velocity field, whose curl is twice the angular velocity.]]

In the special case in which {{math|'''q'''}} is the ''angular velocity'' {{math|'''&amp;omega;'''}} of a '''rigid body''' about an axis through the origin,&amp;#8201; {{math|'''v''' }}is the velocity field ({{math|'''&amp;omega;'''&amp;#8239;&amp;times;&amp;#8239;'''r'''}}) and {{math|'''&amp;omega;'''}} is uniform, so that the last result reduces to&amp;#8201; {{math|curl&amp;#8239;'''v'''&amp;#8201;{{=}}&amp;#8201;2'''&amp;omega;'''&amp;#8202;}}; that is, ''the vorticity is twice the angular velocity''. As the vorticity in this case is uniform and therefore independent of position relative to the axis, it does not change if the axis is shifted, provided that the angular velocity has the same magnitude and direction. And because a uniform velocity field has zero curl, the vorticity is also unchanged if a translational motion is superposed on the rotation. This is the most direct connection that we have seen between curl and rotation. But again I digress.

Returning to the more general case in which {{math|'''q''' }}is not necessarily uniform, but merely solenoidal,&lt;ref&gt;The following explanation takes some hints from Christopher Ford's note on &quot;Vector Potentials&quot; at [https://www.maths.tcd.ie/~houghton/231/Notes/ChrisFord/vp.pdf maths.tcd.ie/~houghton/231/Notes/ChrisFord/vp.pdf] (2006).&lt;/ref&gt; we have
:&lt;math&gt;\operatorname{curl}\mathbf{v}(\mathbf{r})
    = 2\mathbf{q}(\mathbf{r})
      + \mathbf{r}~\!{\cdot}\nabla\,\mathbf{q}(\mathbf{r}) \,,
  &lt;/math&gt;
to which we can apply our lemma ({{EquationNote|76}}) with a uniform real factor {{mvar|t&amp;#8202;}}, obtaining
:&lt;math&gt;
    \operatorname{curl}\mathbf{v}(t\mathbf{r})
    = 2t\mathbf{q}(t\mathbf{r})
       + t\mathbf{r}~\!{\cdot}\nabla\,\mathbf{q}(t\mathbf{r}) \,.
  &lt;/math&gt;
On the left we can recall ({{EquationNote|77}}); and on the right we can apply ({{EquationNote|11}}), noting that the magnitude of{{math| {{abs|'''r'''}}}} is{{mvar| r&amp;#8202;}}, which measures distance in the direction of{{math| '''r'''}}. Thus we obtain
:&lt;math&gt;
  \mathrm{curl}\big(\mathbf{q}(t\mathbf{r})\times t\mathbf{r}\big)
    = 2t\mathbf{q}(t\mathbf{r})
      + tr~\! \part_{\textstyle r} \mathbf{q}(t\mathbf{r}) \,.
  &lt;/math&gt;
Now if the direction of{{math| '''r'''}} is held constant,&amp;#8201; {{math|'''q'''(''t''&amp;#8202;'''r''')}} is a function of{{mvar| tr&amp;#8239;}}; and in general&amp;#8201; {{math|''r&amp;#8202;&amp;part;&lt;sub&gt;r&lt;/sub&gt;&amp;#8201;f''&amp;#8202;(''tr'')&amp;#8201;{{=}}&amp;#8201;''t&amp;#8202;&amp;part;&lt;sub&gt;t&lt;/sub&gt;&amp;#8201;f''&amp;#8202;(''tr'')}}.&amp;#8201; So we have
:&lt;math&gt;\begin{align}
  \mathrm{curl}\big(\mathbf{q}(t\mathbf{r})\times t\mathbf{r}\big)
    &amp;= 2t\mathbf{q}(t\mathbf{r}) + t^2 \part_t \mathbf{q}(t\mathbf{r}) \\
    &amp;= \part_t \big(t^2 \mathbf{q}(t\mathbf{r})\big) \,.
  \end{align}&lt;/math&gt;
Integrating w.r.t. {{mvar|t}}&amp;#8239; from 0 to 1 gives
:&lt;math&gt;
  \int_0^1\!\mathrm{curl}\big(\mathbf{q}(t\mathbf{r})\times t\mathbf{r}\big)\,dt
    = \big(t^2 \mathbf{q}(t\mathbf{r})\big)\Big|_0^1
    =~\! \mathbf{q}(\mathbf{r}) \,;
  &lt;/math&gt;
that is,
{{NumBlk|:|&lt;math&gt;\mathbf{q}(\mathbf{r}) \equiv
  \mathrm{curl}\int_0^1\!\mathbf{q}(t\mathbf{r})\!\times\!t\mathbf{r}\;dt\qquad
  &lt;/math&gt;[&amp;#8202;for solenoidal {{math|'''q'''&amp;#8202;}}].|{{EquationRef|78}}}}
Thus for any solenoidal vector field{{math| '''q'''}}&amp;#8239; we can construct a '''vector potential'''&amp;mdash;that is, a field whose curl is{{math| '''q'''&amp;#8202;}}; such a field is given by the integral on the right. This is the long-promised proof of the &quot;converse&quot; of identity ({{EquationNote|24d}}). Of course the vector potential is not unique, because any conservative field&amp;mdash;but ''only'' a conservative field&amp;mdash;can be added to it without changing its curl. Hence the existence of ''one'' vector potential implies the existence of infinitely many. The above integral gives us ''one''.

The proof of ({{EquationNote|78}}) assumes that {{math|'''q''' }}is solenoidal not only at position{{math| '''r'''&amp;#8202;,}} but also at{{math| ''t''&amp;#8202;'''r'''}}&amp;#8202; where&amp;#8239; {{math|0&amp;#8239;&amp;leq;&amp;#8239;''t''&amp;#8239;&amp;leq;&amp;#8202;1}}, i.e. at every point on the line-segment from the origin to{{math| '''r'''}}.&amp;#8201; A '''star-shaped''' region is one that contains a point{{mvar| O}}&amp;#8202; such that for every point{{mvar| P}} in the region, the line-segment {{mvar|OP}} is entirely contained in the region. We may choose any such {{mvar|O}}&amp;#8202; as the origin in the proof of ({{EquationNote|78}}). So the proof tells us that if a vector field is solenoidal within a star-shaped region, it has a vector potential in that region. As a special case, a vector field that is solenoidal everywhere has a vector potential everywhere.

{{cob}}

=== Notes on the curl of the curl ===

{{cot}}

Identity ({{EquationNote|65}}), namely
:&lt;math&gt;
    \operatorname{curl}\operatorname{curl}\mathbf{q}
    ~\!\equiv \nabla\operatorname{div}\mathbf{q} - \triangle\mathbf{q}
  &lt;/math&gt;
(&quot;curl curl is grad div minus del squared&quot;), has at least three implications worth noting here.

First, it can be rearranged as
{{NumBlk|:|&lt;math&gt;\triangle\mathbf{q}
    ~\!\equiv \nabla\operatorname{div}\mathbf{q}
              - \operatorname{curl}\operatorname{curl}\mathbf{q}
  &lt;/math&gt;|{{EquationRef|79}}}}
(&quot;del squared is grad div minus curl curl&quot;). This would serve as a coordinate-free definition of the Laplacian of a vector, if we did not already have one.&lt;ref&gt;''Cf''. [[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;#8202;71, and [[#moon-spencer-65|Moon &amp;amp; Spencer, 1965]], p.&amp;#8239;235; quoted in [[#tai-95|Tai, 1995]], pp.&amp;#8239;18,&amp;#8239;43.&lt;/ref&gt; But we do: we started with a coordinate-free definition ({{EquationNote|4L}}) for a generic field, established its unambiguity via ({{EquationNote|9L}}), and found its Cartesian form ({{EquationNote|63L}}), which we used in the derivation of ({{EquationNote|79}}). Wherever we start, we may properly assert by way of contrast that the Laplacian of a ''vector''&amp;#8202; is given by ({{EquationNote|79}}), whereas the Laplacian of a ''scalar''&amp;#8202; is given by the divergence of the gradient. But we should ''not'' conclude, as Moon &amp;amp; Spencer do, that representing the scalar and vector Laplacians by the same symbol is &quot;poor practice&amp;hellip; since the two are basically quite different&quot;,&lt;ref&gt;[[#moon-spencer-65|Moon &amp;amp; Spencer, 1965]], p.&amp;#8239;236.&lt;/ref&gt; because in fact the two have a common definition which is succinct, unambiguous, and coordinate-free: the Laplacian (of anything) is the closed-surface integral of the outward normal derivative, per unit volume.{{efn|Tai ([[#tai-95|1995]], pp.&amp;#8239;43–4) also disagrees with Moon &amp;amp; Spencer, but for a different reason: he regards the Laplacian as the divergence of the gradient even if the operand is a ''vector'' field. For better or worse, we do not consider the gradient of a vector in the present paper&amp;mdash;although the reader can probably work out how to modify ({{EquationNote|26g}}) if&amp;#8202; {{math|''d'''''r'''}} is written as a column vector and&amp;#8202; {{mvar|dp}}&amp;#8202; is ''replaced''&amp;#8202; by a column vector (compare the later footnote on ''dyadics'').}}

Second, by reason of identity ({{EquationNote|38}}) and the remarks thereunder, a given vector field{{math| '''v'''}} can be written
:&lt;math&gt;\mathbf{v}(\mathbf{r}) \,\equiv\, \triangle\bigg(\!{-}\!\iiint
    \frac{\,\mathbf{v}(\mathbf{r}')}{4\pi}\,
    \frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|}
  \,dV' \!\bigg) \,,&lt;/math&gt;
where the integral is over all space, or at least all of the space in which {{math|'''v'''}} may be non-zero. So, subject to the convergence of the integral, there exists a vector field{{math| '''q'''}} such that
:&lt;math&gt;\mathbf{v} = \triangle\mathbf{q} \,;&lt;/math&gt;
that is, by ({{EquationNote|79}}), there exists{{math| '''q'''}} such that
:&lt;math&gt;\mathbf{v}
    = \nabla\operatorname{div}\mathbf{q}
      - \operatorname{curl}\operatorname{curl}\mathbf{q} \,,
  &lt;/math&gt;
which implies the existence of a scalar field, say&lt;math&gt;\,\varphi~\!,\,&lt;/math&gt; and a vector field, say{{math| '''&amp;Psi;'''}}, such that
:&lt;math&gt;\mathbf{v}
    = -\nabla\varphi+\operatorname{curl}\boldsymbol{\Psi}
  &lt;/math&gt;
(namely&amp;#8201; &lt;math&gt;\varphi\!=\!-\!\operatorname{div}\mathbf{q}\,&lt;/math&gt; and&amp;#8201; {{math|'''&amp;Psi;'''&amp;#8201;{{=}}&amp;#8201;&amp;minus;&amp;#8202;curl&amp;#8239;'''q'''}}). In short, subject to the convergence of the said integral,
* ''a given vector field can be resolved into [minus] a gradient plus a curl''.
Such a resolution is called a '''Helmholtz decomposition''', and the proposition that it exists is the ''Helmholtz decomposition theorem''. Of course the gradient is irrotational and the curl is solenoidal so that, subject to the same convergence,
* ''a given vector field can be resolved into an irrotational field plus a solenoidal field''.
This is a second statement of the theorem, and follows from the first. And the first follows from the second because an irrotational field has a scalar potential by ({{EquationNote|29}}) and a solenoidal field has a vector potential by ({{EquationNote|78}}).

Third, if {{math|'''q'''}} is ''solenoidal'', the term&amp;#8201; {{math|&amp;nabla;&amp;#8201;div&amp;#8239;'''q'''}}&amp;#8239; in ({{EquationNote|65}}) or ({{EquationNote|79}}) vanishes. Hence ''for a solenoidal field, the curl of the curl is minus the Laplacian''. For example, in the ''dynamic'' case, in a ''vacuum'', the Maxwell&amp;ndash;Amp&amp;egrave;re law says that&amp;#8201; {{math|curl&amp;#8201;'''H''' {{=}} ''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8202;'''E&amp;#775;'''}}.&amp;#8201; Multiplying this by the physical constant {{math|''&amp;mu;''&lt;sub&gt;0&lt;/sub&gt;}} (called the '''vacuum permeability''' or simply the '''magnetic constant''') gives&amp;#8201; {{math|curl&amp;#8201;'''B''' {{=}} ''&amp;mu;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8202;''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8239;'''E&amp;#775;'''&amp;#8202;,}}&amp;#8239; whence
:&lt;math&gt;\operatorname{curl}\operatorname{curl}\mathbf{B}
    = \mu_0\epsilon_0\operatorname{curl}\mathbf{\dot{E}} \,.
  &lt;/math&gt;
But, by Gauss's law for magnetism, {{math|'''B'''}} is solenoidal so that, by ({{EquationNote|65}}), the left-hand side of the above is&amp;#8201; {{math|&amp;minus;&amp;#9651;'''B'''}}.&amp;#8201; And by '''Faraday's law''',&amp;#8201; &lt;math&gt;\operatorname{curl}\mathbf{E}=-\mathbf{\dot{B}}&lt;/math&gt;,&amp;#8201; so that&amp;#8201; &lt;math&gt;\operatorname{curl}\mathbf{\dot{E}}=-\mathbf{\ddot{B}}&lt;/math&gt;.&amp;#8201; Making these substitutions, we get&amp;#8201; &lt;math&gt;-\triangle\mathbf{B}=-\mu_0\epsilon_0\mathbf{\ddot{B}}~\!,\,&lt;/math&gt; i.e.
:&lt;math&gt;\mathbf{\ddot{B}}=\frac{1}{\mu_0\epsilon_0}~\!\triangle\mathbf{B} \,.&lt;/math&gt;
By comparison with ({{EquationNote|45}}), this is the wave equation with
:&lt;math&gt;c=\frac{1}{\sqrt{\mu_0\epsilon_0}} \,.&lt;/math&gt;
Thus the Maxwell&amp;ndash;Amp&amp;egrave;re law, Gauss's law for magnetism, and Faraday's law, with the aid of ({{EquationNote|65}}), predict the existence of '''electromagnetic waves''' together with their speed.

For these reasons, especially the last, one could hardly overstate the importance of identity ({{EquationNote|65}}).

{{cob}}

=== Digression: Proofs from formal products ===

{{cot}}

We have seen that Wilson ([[#wilson-1901|1901]], pp.&amp;#8239;150,&amp;#8239;152) interprets the divergence and curl as &quot;formal&quot; or &quot;symbolic&quot; scalar and vector products with the {{math|&amp;nabla; }}operator.&amp;#8201; {{nowrap|C.-T. Tai}}, in his [[#tai-95|1995 report]] (pp.&amp;#8239;26–9), alleges that this interpretation began with Wilson and not with Gibbs. Here I shall submit, on the contrary, that while the terminology may not be attributable to Gibbs, the concept certainly is.

Later in the same report, Tai confuses the picture by citing the first volume of [[w:Oliver Heaviside|Heaviside]]'s ''Electromagnetic Theory'' (1893), where Heaviside, although his notations for the scalar and vector products differ from those of Gibbs, nevertheless considers the {{math|&amp;nabla;}} operator as a factor in such products. Tai continues:
&lt;blockquote&gt;At the time of his writing he [Heaviside] was already aware of Gibbs' pamphlets on vector analysis but Wilson's book was not yet published. It seems, therefore, that Heaviside and Wilson independently introduced the misleading concept for the scalar and vector products between {{math|&amp;nabla;}} and a vector function. Both were, perhaps, induced by Gibbs' notations for the divergence and the curl. Heaviside did not even include the word 'formal' in his description of the products.&lt;ref&gt;[[#tai-95|Tai, 1995]], p.&amp;#8239;35.&lt;/ref&gt;
&lt;/blockquote&gt;
Whereas it was quite in character for Heaviside to treat an operator that way, the word &quot;independently&quot; would have surprised Wilson and is contradicted by Tai himself, who observes that Wilson's preface acknowledges Heaviside.&lt;ref&gt;[[#tai-95|Tai, 1995]], pp.&amp;#8239;25,&amp;#8239;29.&lt;/ref&gt; In Wilson's own words:
&lt;blockquote&gt;By far the greater part of the material used in the following pages has been taken from the course of lectures on Vector Analysis delivered annually at the University [Yale] by Professor Gibbs. Some use, however, has been made of the chapters on Vector Analysis in Mr. Oliver Heaviside's ''Electromagnetic Theory'' (Electrician Series, 1893) and in Professor F&amp;ouml;ppl's lectures on ''Die Maxwell'sche Theorie der Electricit&amp;auml;t'' (Teubner, 1894).&amp;#8239;....

Notwithstanding the efforts which have been made during more than half a century to introduce Quaternions into physics the fact remains that they have not found wide favor.{{efn|A ''quaternion''&amp;#8202; is a mathematical object invented by [[w:William Rowan Hamilton|William Rowan Hamilton]] in 1843, consisting of two parts which Hamilton later called the scalar part and the vector part. For most purposes the two parts were found to be more useful separately than together. By putting them together, however, Hamilton constructed a set which satisfied all the algebraic field axioms except commutativity of multiplication. This was, and is, considered a triumph.}} On the other hand there has been a growing tendency especially in the last decade toward the adoption of some form of Vector Analysis. The works of Heaviside and F&amp;ouml;ppl referred to before may be cited in evidence. As yet however no system of Vector Analysis which makes any claim to completeness has been published. In fact Heaviside says: &quot;I am in hopes that the chapter which I now finish may serve as a stopgap till regular vectorial treatises come to be written suitable for physicists, based upon the vectorial treatment of vectors&quot; (''Electromagnetic Theory'', Vol.&amp;#8239;{{serif|I}}., p.&amp;#8239;305). Elsewhere in the same chapter Heaviside has set forth the claims of vector analysis as against Quaternions, and others have expressed similar views.&lt;ref&gt;[[#wilson-1901|Wilson, 1901]], pp.&amp;#8239;ix,&amp;#8239;xi–xii.&lt;/ref&gt;
&lt;/blockquote&gt;

Most damaging to Tai's thesis, however, is Gibbs's original pamphlet, a copy of which Heaviside received from Gibbs himself in June 1888.&lt;ref&gt;[[#gibbs-1881-4|Gibbs, 1881–84]], privately printed version&amp;mdash;of which the scan in our bibliography is of the very copy that Gibbs sent to Heaviside, with annotations in Heaviside's hand. On the annotations see [[#rocci-20|Rocci, 2020]].&lt;/ref&gt; Sections 62 to 65 of the pamphlet appear under the heading
&lt;blockquote style=&quot;text-align: center&quot;&gt;{{math|&amp;nabla;,}} {{math|&amp;nabla;'''&amp;sdot;'''&amp;#8202;,}} ''and''&amp;#8202; {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8239; ''applied to Functions of Functions of Position''.
&lt;/blockquote&gt;In &amp;sect;&amp;#8239;62, Gibbs says that a constant scalar factor after such an operator may be placed before it (that is, taken outside the operator). {{nowrap|In &amp;sect;&amp;#8239;63}} he states our rule ({{EquationNote|73g}}) for the gradient of a function of a scalar field. His next section (in which I have bolded the vector field{{math| '''&amp;omega;'''}}) is worth quoting in full:
&lt;blockquote&gt;64.  If {{mvar|u}} or {{math|'''&amp;omega;'''}} is a function of several scalar or vector variables, which are themselves functions of the position of a single point, the value of&amp;#8202; {{math|&amp;nabla;''u''}} or {{math|&amp;nabla;'''&amp;sdot;&amp;#8239;&amp;omega;'''}}&amp;#8202; or {{math|&amp;nabla;&amp;#8202;&amp;times;&amp;#8202;'''&amp;omega;'''}}&amp;#8202; will be equal to the sum of the values obtained by making successively all but each one of these variables constant.
&lt;/blockquote&gt;
This proposition is a ''generalized product rule'' in the sense that the &quot;function of several scalar or vector variables&quot; may be, but is not restricted to, any sort of product of those variables. Gibbs continues:
&lt;blockquote&gt;65.  By the use of this principle, we easily derive the following identical equations:
&lt;/blockquote&gt;
Six &quot;equations&quot; follow. The first says that the gradient operation is distributive over addition, and the second says the same of the divergence and curl (on one line). The last four are our identities ({{EquationNote|69}}), ({{EquationNote|71d}}), ({{EquationNote|71c}}), and ({{EquationNote|67d}}), in that order (albeit with different symbols). Gibbs then remarks (with my italics):
&lt;blockquote&gt;The student will observe an analogy between these equations and the formul&amp;aelig; of ''multiplication''. (In the last four equations the analogy appears most distinctly when we regard all the factors but one as constant.) Some of the more curious features of this analogy are due to the fact that the {{math|&amp;nabla;}} contains implicitly the vectors {{math|'''i'''&amp;#8202;,}} {{math|'''j'''&amp;#8202;,}} and {{math|'''k'''&amp;#8202;,}} which are to be ''multiplied''&amp;#8202; into the following quantities.
&lt;/blockquote&gt;
Indeed, if the ''first''&amp;#8202; factor is constant, identities ({{EquationNote|69}}), ({{EquationNote|71d}}), ({{EquationNote|71c}}), and ({{EquationNote|67d}}) become
:&lt;math&gt;\begin{align}
    \nabla(p\varphi) &amp;= p\;\!\nabla\varphi \\
    \nabla\cdot p\mathbf{b} &amp;= p\,\nabla{\cdot}~\!\mathbf{b} \\
    \nabla\times p\mathbf{b} &amp;= p\,\nabla{\times}~\!\mathbf{b} \\
    \nabla\cdot(\mathbf{a}\!\times\!\mathbf{b})
    &amp;= -\mathbf{a}\cdot\nabla{\times}~\!\mathbf{b} \,,
  \end{align}&lt;/math&gt;
whereas if the ''second''&amp;#8202; factor is constant, they become respectively
:&lt;math&gt;\begin{align}
    \nabla(p\varphi) &amp;= \varphi\;\!\nabla p \\
    \nabla\cdot p\mathbf{b} &amp;= \nabla p \cdot \mathbf{b} \\
    \nabla\times p\mathbf{b} &amp;= \nabla p \times \mathbf{b} \\
    \nabla\cdot(\mathbf{a}\!\times\!\mathbf{b})
    &amp;= \nabla{\times}~\!\mathbf{a}\cdot\mathbf{b} \,.
  \end{align}&lt;/math&gt;
All eight equations look like rearrangements of ''products'' involving a vector{{math| &amp;nabla;}}.&amp;#8201; [Concerning the last ''three'' equations, we have made that observation before; see ({{EquationNote|15}}) above.]&amp;#8201; But only seven of the eight are explained by taking the constant outside the operator ({{nowrap|as in &amp;sect;&amp;#8239;62}}); the exception is the fourth, in which the minus sign is not explained by that step alone, but ''is'' explained by the change in the cyclic order of the formal triple product. And if we add the two right-hand sides corresponding to each of the four left-hand sides, we get the identities in which both factors are variable&amp;mdash;as claimed {{nowrap|in &amp;sect;&amp;#8239;64}}.

If &amp;sect;&amp;#8239;65 leaves any doubt that Gibbs approved of formal products with the symbolic vector{{math| &amp;nabla;}} (albeit without using those terms), this is dispelled {{nowrap|by &amp;sect;&amp;#8202;166}}, where he writes:
&lt;blockquote&gt;166.&amp;#8201; To the equations in No.&amp;#8239;65 may be added many others&amp;hellip;
&lt;/blockquote&gt;
followed by a list of seven identities terminated by &quot;etc.&quot; Six of the seven are beyond the scope of the present paper,{{efn|They involve ''dyadics'', i.e. 2nd-order tensors written in a vector-friendly notation. The fourth of the seven is
:{{math|&amp;nabla;('''&amp;tau;&amp;sdot;&amp;#8202;&amp;omega;''') {{=}} &amp;nabla;'''&amp;tau;&amp;#8239;&amp;sdot;&amp;#8201;&amp;omega;''' + &amp;nabla;'''&amp;omega;&amp;#8201;&amp;sdot;&amp;#8201;&amp;tau;'''&amp;#8239;,}}
which is our ({{EquationNote|68}}) expressed in terms of the dyadics {{math|&amp;nabla;'''&amp;tau;'''}} and{{math| &amp;nabla;'''&amp;omega;'''&amp;#8202;}}; the right-hand side is not to be confused with
:{{math|('''&amp;omega;&amp;#8202;&amp;sdot;'''&amp;nabla;)'''&amp;tau;''' + '''(&amp;tau;&amp;sdot;'''&amp;nabla;)'''&amp;omega;'''&amp;#8239;,}}
which would contradict our ({{EquationNote|68}}).}} while the third of the seven is our ({{EquationNote|67c}}). After that list comes the smoking gun ({{nowrap|&amp;sect;&amp;#8202;166, continued}}):
&lt;blockquote&gt;The principle in all these cases is that if we have one of the operators&amp;#8202; {{math|&amp;nabla;,}} {{math|&amp;nabla;'''&amp;sdot;'''&amp;#8202;,}} {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8239; prefixed to a ''product'' of any kind, and we make any transformation of the expression which would be allowable if the {{math|&amp;nabla;}} were a ''vector'', (viz: by changes in the order of the ''factors'', in the signs of ''multiplication'', in the parentheses written or implied, etc.,) by which changes the {{math|&amp;nabla;}} is brought into connection with one particular factor, the expression thus transformed will represent the part of the value of the original expression which results from the variation of that factor.
&lt;/blockquote&gt;
The italics are mine, but I have refrained from italicizing those instances of the word &quot;factor&quot; which are not applicable to{{math| &amp;nabla;}}. In particular, at the stage when &quot;the {{math|&amp;nabla;}} is brought into connection with one particular factor,&quot; the &quot;part of the value&amp;hellip; which results from the variation of that factor&quot; evidently means the term of the sum {{nowrap|in &amp;sect;&amp;#8239;64}}&amp;#8202;&amp;mdash;which, as we have noted, amounts to a generalized product rule. But, according to the stated &quot;principle', we reach that stage by treating{{math| &amp;nabla;}} as a factor. I rest my case.

&lt;br /&gt;

Wilson ([[#wilson-1901|1901]], p.&amp;#8239;157) gives a comprehensive list of sum and product rules for the gradient, divergence, and curl, and properly states (p.&amp;#8239;158) that the rules may be proven &quot;most naturally&quot; from Gibbs's definitions of the operators&amp;mdash;our equations ({{EquationNote|58g}}), ({{EquationNote|60d}}), and ({{EquationNote|59c}}). Understandably, Wilson uses a {{math|&amp;sum;}} sign rather than implicit summation. Less understandably, and less fortunately, he does not sum over a numerical index; e.g., he defines the curl operator as
:{{big|&lt;math&gt;\nabla\times
    \,=\, \textstyle\sum\,\mathbf{i}~\!\!\times\!\frac{\part}{\part x}
          \qquad\quad
  &lt;/math&gt;[sic]}}
and explains that &quot;The summation extends over {{math|''x'',&amp;#8239;''y'',&amp;#8239;''z''}}.&quot;&amp;#8201; With these definitions he proves our identities ({{EquationNote|71c}}) and ({{EquationNote|68}}) essentially as we have done, but inevitably with greater difficulty, which may explain why he then says &quot;The other formul&amp;aelig; are demonstrated in a similar manner&quot; before reverting to Gibbs's strategy of varying one factor at a time. He announces (p.&amp;#8239;159) that the variable held constant will be written as a subscript after the product, and he combines this notation with his {{math|&amp;sum;}} notation in a rigorous proof that varying one factor at a time is valid for our ({{EquationNote|68}}), i.e. the gradient of a dot-product. Noting that this result is analogous to
:&lt;math&gt;d(\mathbf{u}\cdot\mathbf{v})
    = \mathbf{u}\cdot d\mathbf{v} + d\mathbf{u}\cdot\mathbf{v} \,,
  &lt;/math&gt;
he then jumps to the conclusion that varying one factor at a time is valid for ''all''&amp;#8202; of his product rules&amp;mdash;notwithstanding that a small change in a vector is not related to its divergence or curl as a small change in a scalar is related to its gradient.

That ''per saltum''&amp;#8202; conclusion is his cue to go formal and symbolic. To obtain the curl of a cross-product [as in our ({{EquationNote|67c}})], he &quot;formally&quot; expands a vector triple product to obtain the curl when the first factor is constant, states the curl when the second factor is held constant, and adds the two partial curls ([[#wilson-1901|Wilson, 1901]], p.&amp;#8239;161). Next he gives various arrangements of our ({{EquationNote|8q}}), except that he presents the first vector not as strictly uniform, but as merely ''held'' constant for the gradient operation. He states in passing that a proof may be effected by &quot;expanding in terms of&amp;#8202; {{math|'''i'''&amp;#8202;,&amp;#8239;'''j''',&amp;#8239;'''k'''}}&quot;; but instead of such a proof, he offers a &quot;method of remembering the result&quot; by expanding the &quot;product&quot;&amp;#8239; {{math|'''u'''&amp;#8239;&amp;times;&amp;#8239;(&amp;nabla;&amp;#8239;&amp;times;&amp;#8239;'''v''')}}&amp;#8202; &quot;formally as if&amp;#8202; {{math|&amp;nabla;,&amp;#8239;'''u'''&amp;#8202;,&amp;#8202;'''v'''}}&amp;#8202; were all real vectors&quot; (pp.&amp;#8239;161–2). Concerning the curl of the gradient, and the divergence of the curl (pp.&amp;#8239;167,&amp;#8239;168), he recommends expanding in terms of&amp;#8202; {{math|'''i'''&amp;#8202;,&amp;#8239;'''j''',&amp;#8239;'''k'''&amp;#8202;,}}&amp;#8239; but does not elaborate. Concerning the curl of the curl, however, he shows what would happen if it were &quot;expanded formally according to the law of the triple vector product&quot; (p.&amp;#8239;169).

In defense of the &quot;formal product&quot; method, we should note that the operators {{math|''&amp;part;&lt;sub&gt;x&lt;/sub&gt;''&amp;#8202;,}} {{math|''&amp;part;&lt;sub&gt;y&lt;/sub&gt;''&amp;#8202;,}} and {{math|''&amp;part;&lt;sub&gt;z&lt;/sub&gt;''&amp;#8202;}} are ''linear'', so that they are distributive over addition and may be permuted with multiplication by a constant, as if the operators themselves were multipliers (like components of vectors). They may be similarly permuted with other like operators&amp;mdash;explaining why the formal-product method correctly deals with the curl of the gradient, the divergence of the curl, and the curl of the curl. But such an operator ''cannot'' be permuted with multiplication by a ''variable'', because then the product rule of differentiation applies, yielding an extra term. The formal-product system responds to this difficulty by generalizing the product rule as in &amp;sect;&amp;sect;&amp;#8239;64 &amp;amp;&amp;#8239;166 of Gibbs ([[#gibbs-1881-4|1881–84]]). As Borisenko &amp;amp; Tarapov put it ([[#borisenko-tarapov-68|1968]], p.&amp;#8239;169),
&lt;blockquote&gt;the operator {{math|&amp;nabla;}} acts on each factor separately with the other held fixed. Thus {{math|&amp;nabla;}} should be written after any factor regarded as a constant in a given term and before any factor regarded as variable.
&lt;/blockquote&gt;
In this they differ inconsequentially from Gibbs, who requires that the operator be &quot;brought into connection&quot; with the factor considered variable.

To illustrate, let us find the gradient of a dot-product, essentially in the manner of Borisenko &amp;amp; Tarapov ([[#borisenko-tarapov-68|1968]], p.&amp;#8239;180), quoted by Tai ([[#tai-95|1995]], p.&amp;#8239;46; the next five equation numbers are Tai's). In this case the generalized product rule gives
{{NumBlk|:|&lt;math&gt;\nabla(\mathbf{A ~\!\!\cdot B})
    = \nabla(\mathbf{A}_c {\cdot}~\!\mathbf{B}) +
      \nabla(\mathbf{A} ~\!\!\cdot \mathbf{B}_c) \,,
  &lt;/math&gt;|{{EquationRef|7.26}}}}
where the subscript {{mvar|c}} marks the factor held ''constant'' during the differentiation. In Wilson's notation, this equation would be written
:{{midsize|&lt;math&gt;\nabla(\mathbf{A ~\!\!\cdot B})
    = \nabla(\mathbf{A ~\!\!\cdot B})_{\mathbf{A}} +
      \nabla(\mathbf{A ~\!\!\cdot B})_{\mathbf{B}} \,,
  &lt;/math&gt;}}
where a trailing subscript indicates which factor is held constant.&amp;#8201; In the ''Feynman'' subscript notation, the subscript is attached to the {{math|&amp;nabla;}} operator and indicates which factor is allowed to ''vary'', so that the same equation would be written
:{{midsize|&lt;math&gt;\nabla(\mathbf{A ~\!\!\cdot B})
    = \nabla_{\mathbf{B}}(\mathbf{A ~\!\!\cdot B}) +
      \nabla_{\!\mathbf{A}}(\mathbf{A ~\!\!\cdot B}) \,.
  &lt;/math&gt;}}
But, as we are discussing Borisenko &amp;amp; Tarapov, we press on with ({{EquationNote|7.26}}).&amp;#8201; By the algebraic identity
{{NumBlk|:|&lt;math&gt;\mathbf{c}(\mathbf{a\cdot b})
    \,=\, (\mathbf{a\cdot c})\mathbf{b}
    \,-\, \mathbf{a}\times(\mathbf{b}\times\mathbf{c}) \,,
  &lt;/math&gt;|{{EquationRef|7.27}}}}
i.e.
:&lt;math&gt;\mathbf{c}(\mathbf{a\cdot b})
    \,=\, (\mathbf{a\cdot c})\mathbf{b}
    \,+\, \mathbf{a}\times(\mathbf{c}\times\mathbf{b}) \,,
  &lt;/math&gt;
we can say
{{NumBlk|:|&lt;math&gt;\nabla(\mathbf{A}_c {\cdot}~\!\mathbf{B})
    \,=\, (\mathbf{A}_c {\cdot}\nabla)\mathbf{B}
    \,+\, \mathbf{A}_c\times(\nabla\times\mathbf{B}) \,.
  &lt;/math&gt;|{{EquationRef|7.28}}}}
Similarly,&lt;ref&gt;In the next equation as printed in Borisenko &amp;amp; Tarapov ([[#borisenko-tarapov-68|1968]], p.&amp;#8239;180), the first cross should be &quot;&amp;equals;&quot;; Tai ([[#tai-95|1995]], p.&amp;#8239;46) corrects it.&lt;/ref&gt;
{{NumBlk|:|&lt;math&gt;\nabla(\mathbf{B}_c {\cdot}~\!\mathbf{A})
    \,=\, (\mathbf{B}_c {\cdot}\nabla)\mathbf{A}
    \,+\, \mathbf{B}_c\times(\nabla\times\mathbf{A}) \,.
  &lt;/math&gt;|{{EquationRef|7.29}}}}
Substituting ({{EquationNote|7.28}}) and ({{EquationNote|7.29}}) into ({{EquationNote|7.26}}), in which the order of the dot-products is immaterial, and dropping the {{mvar|c }}subscripts (because they are now outside the differentiations), we get the correct result
{{NumBlk|:|{{midsize|&lt;math&gt;\nabla(\mathbf{A{\cdot}B})
    =     (\mathbf{A}{\cdot}\nabla)\mathbf{B}
      +   (\mathbf{B}\;\!{\cdot}\nabla)\mathbf{A}
      +   \mathbf{A}{\times}(\nabla{\times}\mathbf{B})
      +   \mathbf{B}{\times}(\nabla{\times}\mathbf{A}) \,,
  &lt;/math&gt;}}|{{EquationRef|7.30}}}}
corresponding to our ({{EquationNote|68}}).

Tai ([[#tai-95|1995]], p.&amp;#8239;47) is unimpressed, asking why we cannot apply ({{EquationNote|7.27}}) directly to the left side of ({{EquationNote|7.26}}). The answer to that is obvious: on the left side, the {{math|&amp;nabla;}} operator is applied to a product of ''two variables'', and the variations of ''both'' must be taken into account. But there is a harder question which Tai does not ask: in ({{EquationNote|7.28}}), why can't we have {{math|&amp;nabla;'''&amp;sdot;A'''&lt;sub&gt;c&lt;/sub&gt;}} instead of{{math| '''A'''&lt;sub&gt;c&lt;/sub&gt;'''&amp;sdot;'''&amp;nabla;}}&amp;#8239;? (Or, in terms of Feynman subscripts, why can't we have {{math|&amp;nabla;'''&lt;sub&gt;B&lt;/sub&gt;&amp;#8202;&amp;sdot;&amp;#8202;A'''}} instead of{{math| '''A&amp;sdot;'''&amp;nabla;&lt;sub&gt;'''B'''&lt;/sub&gt;}}?) Because that would make the term vanish? Yes, it would; but, as there is only one variable factor on the left side, why do we need two terms on the right? Because the rule says {{math|&amp;nabla;}} should be written after the constant but before the variable? Yes, but that rule serves the purpose of varying ''each'' variable, whereas there is only one variable to vary on the left of ({{EquationNote|7.28}}). The same issue arises in ({{EquationNote|7.29}}). We cannot settle the question even by appealing to symmetry. Obviously the right side of ({{EquationNote|7.30}}), like the left, must be unchanged if we switch {{math|'''A'''}} and {{math|'''B'''}}; and indeed it is. But if the first term on the right of ({{EquationNote|7.28}}) and of ({{EquationNote|7.29}}) were to vanish, the necessary symmetry of ({{EquationNote|7.30}}) would be maintained. And unless I'm missing something, Tai's &quot;symbolic vector&quot; method does not circumvent the problem; Tai's &quot;Lemma 2&quot; ([[#tai-95|1995]], p.&amp;#8239;53) is the Gibbs&amp;#10744;Wilson method of &quot;varying one factor at a time&quot;, written with Feynman subscripts attached to the symbolic vector instead of the del operator.{{efn|I don't overlook the fact that Tai's symbolic vector, unlike the del operator, is subject to commutative and anticommutative laws. Neither do I see how it helps.}}

For another example of the same issue, consider the following two-liner offered by Panofsky &amp;amp; Phillips ([[#panofsky-phillips-62|1962]], pp.&amp;#8239;470–71) and rightly pilloried by Tai ([[#tai-95|1995]], pp.&amp;#8239;47–8):
:&lt;math&gt;\begin{align}
  &amp;   \nabla{\times}(\mathbf{A}{\times}\mathbf{B})
   =  (\nabla{\cdot}\;\!\mathbf{B})\mathbf{A}
    - (\nabla{\cdot}    \mathbf{A})\mathbf{B}               &amp;&amp;[\mathsf{sic}] \\
  &amp;=  (\nabla{\cdot}\;\!\mathbf{B}_c ~\!\!)\mathbf{A}
    + (\nabla{\cdot}\;\!\mathbf{B}~\!\!)\mathbf{A}_c ~\!\!
    - (\nabla\mathbf{\cdot A}_c ~\!\!)\mathbf{B}
    - (\nabla\mathbf{\cdot A}~\!\!)\mathbf{B}_c \!\!\!\!\!\!\!&amp;&amp;[\mathsf{sic}].
  \end{align}&lt;/math&gt;
If the first line were right, the authors would hardly bother to continue; but evidently it isn't, because it doesn't begin by &quot;varying one factor at a time&quot;. The second line does not follow from the first and includes divergences of constants, which ought to vanish but somehow apparently do not. Let's try again, this time sticking to the rules:
:&lt;math&gt;\begin{align}
  &amp; \nabla\!\times\!(\mathbf{A}\!\times\!\mathbf{B})
  \,=\,   \nabla\!\times\!(\mathbf{A}_c \!\times\!\mathbf{B})
    \,+\, \nabla\!\times\!(\mathbf{A}\!\times\!\mathbf{B}_c) \\
  &amp;~=\,   (\nabla{\cdot}\;\!\mathbf{B})\mathbf{A}_c
        - (\mathbf{A}_c {\cdot}\nabla)\mathbf{B}
    \,+\, (\mathbf{B}_c {\cdot}\nabla)\mathbf{A}
        - (\nabla{\cdot}    \mathbf{A})\mathbf{B}_c \\
  &amp;~=\,   \mathbf{A}(\nabla{\cdot}\;\!\mathbf{B})
        - \mathbf{B}(\nabla{\cdot}    \mathbf{A})
    \,+\, (\mathbf{B\;\!\cdot}\nabla)\mathbf{A}
        - (\mathbf{A    \cdot}\nabla)\mathbf{B} \,,
  \end{align}&lt;/math&gt;
in agreement with our ({{EquationNote|67c}}). Here the first line comes from the generalized product rule, and the third is obtained from the second by rearranging terms and dropping the (now redundant) subscripts. The interesting line is the second, which is obtained from the first by expanding the formal vector triple products. But again, why must we have {{math|'''A'''&lt;sub&gt;c&lt;/sub&gt;'''&amp;sdot;'''&amp;nabla;}} and {{math|'''B'''&lt;sub&gt;c&lt;/sub&gt;'''&amp;sdot;'''&amp;nabla;,}} instead of {{math|&amp;nabla;'''&amp;sdot;A'''&lt;sub&gt;c&lt;/sub&gt;}} and {{math|&amp;nabla;'''&amp;sdot;B'''&lt;sub&gt;c&lt;/sub&gt;&amp;#8202;,}} which would make the middle two terms vanish? Again symmetry does not give an answer. The right-hand side, like the left, must change sign if we switch {{math|'''A'''}} and {{math|'''B'''&amp;#8202;}}; but the disappearance of the {{math|'''A'''&lt;sub&gt;c&lt;/sub&gt;'''&amp;sdot;'''&amp;nabla;}} and {{math|'''B'''&lt;sub&gt;c&lt;/sub&gt;'''&amp;sdot;'''&amp;nabla;}} terms would maintain the required (anti)symmetry. Funnily enough, the result would then agree with the incorrect first line given by Panofsky &amp;amp; Phillips (above). But then how would we know that it is incorrect?

The foregoing examples show that &quot;formal product&quot; arguments can be tenuous, even on their own terms. Before these examples, we might have been troubled by the omission of a general proof of the &quot;generalized&quot; product rule. After them, we might wonder whether the rule is even well defined.

I submit, however, that none of this matters. I submit that the popularity of using &quot;formal products&quot; with the del operator, in derivations of vector-analytic identities, is a reaction to the failure of early writers to use indicial notation in the Cartesian definitions of differential operators.{{efn|Indicial notation is standard in higher-order tensor analysis, which however tends not to use unit vectors of coordinate systems, and therefore tends not to encourage the indexing of unit vectors in elementary vector analysis&amp;mdash;whereas in the present paper, I have unapologetically indexed the unit vectors.}} The ensuing proliferation of terms in coordinate-based derivations led authors to seek shortcuts through &quot;formal products&quot; when more rigorous but no-less convenient shortcuts could have been taken through indicial notation, especially in combination with implicit summation. Our derivation of the gradient of a dot-product ({{EquationNote|68}}) is shorter than that of Borisenko &amp;amp; Tarapov, and even uses the right-hand sides of their identities ({{EquationNote|7.28}}) and ({{EquationNote|7.29}}), but obtains them rigorously with no ambiguity and no {{mvar|c }}subscripts. Our derivation of the curl of a cross-product ({{EquationNote|67c}}) takes six lines with a single column of &quot;&amp;equals;&quot; signs. Our subsequent formal-product derivation (not to be confused with the attempt of Panofsky &amp;amp; Phillips) seems to take only three lines; but it is only through our earlier indicial derivation that we have any confidence in our result (not to be confused with the result of Panofsky &amp;amp; Phillips). Our other indicial derivations of identities are mostly shorter than the two just mentioned. Having amassed so comprehensive a collection of identities so rigorously with so little effort, I submit that the use of formal products, Wilson subscripts, {{mvar|c }}subscripts, and Feynman subscripts for this purpose is a historical aberration, to be deciphered in other people's writings but avoided in one's own.

That being said, it is one thing to conclude, as Tai duly does, that the del-cross and del-dot notations should not be interpreted as products in derivations and proofs, and another thing to allege, as Tai also does ([[#tai-95|1995]], p.&amp;#8239;22), that&amp;#8202; {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; and {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; are &quot;not compound operators&quot; but only &quot;assemblies&quot;, or in other words that &quot;{{math|&amp;#8202;&amp;nabla; }}is not a constituent of the divergence operator nor of the curl operator.&quot; Against the latter proposition, our equations ({{EquationNote|14}}), ({{EquationNote|61o}}), and ({{EquationNote|62o}}) have been ''derived'', not merely defined, and our derivation of ({{EquationNote|14}}) is as general as we could wish. Moreover, whereas ({{EquationNote|61o}}) and ({{EquationNote|62o}}) are for Cartesian coordinates, we shall see that they have counterparts in more general coordinates.

{{cob}}

== General coordinates ==

{{cot}}

From our initial definitions of the differential operators, we derived certain identities, from which we derived expressions for the operators in Cartesian coordinates, from which we derived a comprehensive collection of identities, two of which (the multivariate chain rule, and the curl of the product of a scalar and a vector) will now be useful for expressing the operators in other coordinate systems. Cartesian coordinates are traditionally called {{math|''x'',&amp;#8239;''y'',&amp;#8239;''z'',}}&amp;#8201; which we renamed {{mvar|x&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; where&amp;#8202; {{math|''i''&amp;#8201;{{=}}&amp;#8239;1,&amp;#8202;2,&amp;#8202;3&amp;#8202;,}}&amp;#8201; respectively. The best-known 3D ''non''&amp;#8202;-Cartesian coordinate systems are the cylindrical coordinates {{math|(''&amp;rho;'',&amp;#8239;''&amp;phi;'',&amp;#8239;''z'')}} and the spherical coordinates {{math|(''r'',&amp;#8239;''&amp;theta;'',&amp;#8239;''&amp;phi;'')}}; we have already seen {{mvar|r}}&amp;#8202; in the guise of the magnitude of the position vector{{math| '''r'''}}.&amp;#8201; But now we want our coordinate system to be as general as possible&amp;mdash;with the Cartesian, cylindrical, and spherical systems and many others, and even ''classes'' of systems, as special cases.

{{cob}}

=== Natural and dual basis vectors ===

{{cot}}

We shall call our general coordinates {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}}&amp;#8202; where&amp;#8202; {{math|''i''&amp;#8201;{{=}}&amp;#8239;1,&amp;#8202;2,&amp;#8202;3&amp;#8202;}};&amp;#8201; yes, for reasons which will emerge, we shall write the coordinate index as a {{nowrap|''super''&amp;#8202;script}}. But we shall write {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; for{{math| ''{{sfrac|&amp;part;|&amp;part;u&lt;sup&gt;i&lt;/sup&gt;&amp;#8202;}}''&amp;#8202;,}}&amp;#8202; relying on context to distinguish it from the special case{{mvar| {{sfrac|&amp;part;|&amp;part;x&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202;}}.&amp;#8201; By describing the {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}}&amp;#8202; as ''coordinates''&amp;#8239; we mean two things. First, for some domain of interest, the position vector is a smooth function
:&lt;math&gt;\mathbf{r} = \mathbf{r}(u^1,u^2,u^3) \,,&lt;/math&gt;
which possesses partial derivatives w.r.t. its arguments. Second, for every position vector in the resulting range, there is only one ordered triplet&amp;#8239; {{math|(''u&lt;sup&gt;i&lt;/sup&gt;''&amp;#8202;)&amp;#8201;{{=}}&amp;#8201;(''u''&amp;sup1;,&amp;#8239;''u''&amp;sup2;,&amp;#8239;''u''&amp;sup3;),}}&amp;#8239; so that we can think of each coordinate as
:{{big|&lt;math&gt;u^i = u^i(\mathbf{r}) \,;&lt;/math&gt;}}
&amp;mdash;that is, we can think of each {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}}&amp;#8202; as a scalar field, which possesses a gradient.{{efn|Hence we want each {{math|''u&lt;sup&gt;i&lt;/sup&gt;''('''r''')}} to be, as far as possible, a ''smooth'' function. This may require some tweaking of definitions. E.g., in cylindrical coordinates, the angular coordinate {{mvar|&amp;phi;}} must be confined to some 360&amp;deg; range in order to make it unique, and we don't want it jumping from the end of the range to the beginning within the region of interest.}} (I say &quot;think of&quot; because {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}}, being obviously dependent on a coordinate system, would not normally be considered a true scalar; but sometimes we need to treat the coordinate system itself as an object under study.)

These two properties of coordinates respectively suggest two simple ways of choosing basis vectors related to the coordinates: we shall define the '''natural basis''' vectors as
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_i := \part_i \mathbf{r} \,,
  \qquad&lt;/math&gt;}}|{{EquationRef|80a}}}}
and the '''dual basis''' vectors as
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^i := \nabla u^i .
  \qquad&lt;/math&gt;}}|{{EquationRef|80b}}}}
(We could ''normalize'' the natural basis vectors by dividing them by their magnitudes to obtain unit vectors; but, for the moment, we won't bother.) Just as we may think of each {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}} as a scalar field and inquire after its directional derivative or its gradient or its Laplacian, so we may think of each {{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt;}} or{{math| '''h'''&lt;sup&gt;''i''&lt;/sup&gt;}} as a vector field and inquire after its directional derivative or its curl or its divergence or its Laplacian. (That the curl of{{math|&amp;#8202; '''h'''&lt;sup&gt;''i''&lt;/sup&gt;}}&amp;#8202; is zero&amp;#8239; will be especially useful.)

In Cartesian coordinates,&amp;#8201; {{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt;}} and {{math|'''h'''&lt;sup&gt;''i''&lt;/sup&gt;}} are both equal to the unit vector{{math| '''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;}}; thus, in Cartesian coordinates, the natural basis vectors are their own duals.&amp;#8201; In ''general'' coordinates,&amp;#8201; {{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt;}} and {{math|'''h'''&lt;sup&gt;''i''&lt;/sup&gt;}} may differ in both direction and magnitude and are not generally unit vectors. Nevertheless, even in general coordinates, there is a simple relation between the natural and dual basis vectors. Consider the dot-product
:{{big|&lt;math&gt;
    \mathbf{h}_i \cdot \mathbf{h}^j
    = \part_i\mathbf{r} \cdot \nabla u^j \,.
  &lt;/math&gt;}}
If{{math|&amp;#8202; ''i&amp;#8239;&amp;ne;&amp;#8239;j''&amp;#8202;,}} then {{math|''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''r'''&amp;#8202;,}} being in a direction in which {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}} varies while each other {{mvar|u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;}} does not, is tangential to a surface of constant {{mvar|u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;}} and therefore normal to {{math|&amp;nabla;''u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;'',}} so that the dot-product is zero. But by ({{EquationNote|26g}}),
:{{big|&lt;math&gt;
    du^i = \nabla u^i \cdot d\mathbf{r} \,;
  &lt;/math&gt;}}
and if we vary {{math|'''r'''}} by varying {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}} while holding each other {{mvar|u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;}} constant, we can divide by {{mvar|du&lt;sup&gt;i&lt;/sup&gt;}} and obtain
{{NumBlk|:|{{big|&lt;math&gt;
    1 ~\!= \nabla u^i \cdot \part_i\mathbf{r}
      = \mathbf{h}^i \!\cdot \mathbf{h}_i
  \qquad&lt;/math&gt;}}[with no summation].|{{EquationRef|81i}}}}
Putting the two cases together, we have
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_i \cdot \mathbf{h}^j =~\! \delta_i^j
  &lt;/math&gt;}}|{{EquationRef|81}}}}
where the right-hand function, known as the '''Kronecker delta''' function, is defined by
{{NumBlk|:|{{big|&lt;math&gt;\delta_i^j = \delta_{ij} = \delta^{ij}
  =~&lt;/math&gt;}}&lt;math&gt;\begin{cases}
    0 &amp;\mathsf{if}~\, i \neq j \\
    1 &amp;\mathsf{if}~\, i = j \,.
  \end{cases}&lt;/math&gt;|{{EquationRef|82}}}}
Obviously the function is symmetric: the indices {{mvar|i }}and{{mvar| j}}&amp;#8202; can be interchanged. If two lists of vectors are related so that the dot-product of the {{mvar|i&amp;#8202;}}th vector in one list and the {{mvar|j&amp;#8202;}}th in the other is{{mvar| &amp;delta;&lt;sub&gt;ij&lt;/sub&gt;&amp;#8202;}}, the two lists are described as '''reciprocal'''. Thus the triplets {{math|('''h'''&lt;sub&gt;''i''&lt;/sub&gt;)}} and {{math|('''h'''&lt;sup&gt;''i''&lt;/sup&gt;)}} are '''reciprocal bases''': the dual basis is the reciprocal of the natural basis and vice versa. Hence, taking the natural basis as a reference, the dual basis is sometimes called &quot;the&quot; reciprocal basis.

In Cartesian coordinates, ({{EquationNote|81}}) becomes
:{{big|&lt;math&gt;
    \mathbf{e}_i ~\!\!\cdot \mathbf{e}_j =~\! \delta_{ij} \,.
  &lt;/math&gt;}}
So we have a relation for general coordinates ({{EquationNote|81}}) which is just as simple as its special case for Cartesian coordinates, ''provided that we use the natural basis for one factor and the dual basis for the other''. This will be a recurring pattern.

We have deduced the reciprocity relation ({{EquationNote|81}}) from prior definitions of the natural basis {{math|('''h'''&lt;sub&gt;''i''&lt;/sub&gt;)}} and the dual basis {{math|('''h'''&lt;sup&gt;''i''&lt;/sup&gt;)}}.&amp;#8201; This result has a partial converse, in that a reciprocity relation between bases is enough to define either basis in terms of the other&amp;mdash;as we shall see later. But first we proceed to components of vector fields.

{{cob}}

=== Contravariant and covariant components ===

{{cot}}

A '''coordinate grid''' is a set of intersecting curves such that on each curve, one coordinate varies while the others are constant. If we could embed such a grid in an elastic medium, and then stretch and rotate the medium, the natural basis vectors{{math| '''h'''&lt;sub&gt;''i''&lt;/sub&gt;}} given by ({{EquationNote|80a}}) would stretch and rotate ''with the medium'' and ''with the grid''. Accordingly, the ''natural'' basis is also called the '''covariant''' basis. But according to ({{EquationNote|81}}), the dot-product of a natural basis vector and a dual basis vector is '''invariant''' (independent of the coordinate system), so that the variation of one factor ''compensates''&amp;#8202; for the variation of the other. So, as the natural basis is &quot;covariant&quot; with the coordinate grid, we say that the dual basis is '''contravariant'''. Notice that the {{nowrap|''co''&amp;#8202;variant}} factor has a {{nowrap|''sub''&amp;#8202;script}} index (easily remembered because &quot;''co''&amp;#8202; rhymes with ''low''&amp;#8239;&quot;) whereas the {{nowrap|''contra''&amp;#8202;variant}} factor has a {{nowrap|''super''&amp;#8202;script}} index, and that one kind of variation must combine with the other in order to produce an {{nowrap|''in''&amp;#8202;variant}} result; these will be recurring patterns.

A vector field {{math|'''q'''}} may be expressed in components w.r.t. the natural (covariant) basis as
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q} = q^i \mathbf{h}_i
  \qquad&lt;/math&gt;}}|{{EquationRef|83a}}}}
with summation, or in components w.r.t. the dual (contravariant) basis as
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q} = q_i \mathbf{h}^i
  \qquad&lt;/math&gt;}}|{{EquationRef|83b}}}}
with summation. If{{math| '''q'''}} is to be invariant (a true vector, existing independently of the coordinate system), the components must be contravariant in the former case and covariant in the latter, and accordingly are written with superscripts and subscripts respectively. In Cartesian coordinates, the two bases are the same, so that the components w.r.t. the two bases are also the same; that's why, in the above section headed &quot;[[#Cartesian coordinates|Cartesian coordinates]]&quot;, we got away with writing component indices as subscripts. In ''general'' coordinates, however, the basis vectors have subscripts and the components have superscripts or vice versa, so that ''the index of implicit summation appears once as a superscript and once as a subscript''.

Taking dot-products of ({{EquationNote|83a}}) with{{math| '''h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;}}, applying ({{EquationNote|81}}), and noting that only one term on the right is non-zero, we obtain
{{NumBlk|:|{{big|&lt;math&gt;
     q^j =~\! \mathbf{q} \cdot \mathbf{h}^j .
  \qquad&lt;/math&gt;}}|{{EquationRef|83c}}}}
Similarly, taking dot-products of ({{EquationNote|83b}}) with{{math| '''h'''&lt;sub&gt;''j''&lt;/sub&gt;}} yields
{{NumBlk|:|{{big|&lt;math&gt;
     q_j =~\! \mathbf{q} \cdot \mathbf{h}_j \,.
  \qquad&lt;/math&gt;}}|{{EquationRef|83d}}}}
These results depend on the reciprocity relation ({{EquationNote|81}}) but not on the earlier definitions of the bases to which that relation applies. They say:
* to find the contravariant components of a vector, take its dot-products with the contravariant basis vectors, and
* to find the covariant components of a vector, take its dot-products with the covariant basis vectors;
''or'', in terms of the bases themselves:
* to find the components of a vector w.r.t. either basis, take dot-products of that vector with the ''other'' basis.

If a particular {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}} has a particular name, such as{{mvar| &amp;theta;}} or{{mvar| &amp;phi;}}, then, if we're not using indexed summation, we may find it convenient to write that name in place of the index{{mvar| i}}&amp;#8202; in the superscript or subscript.

At the present level of generality, the basis vectors {{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;,}} unlike their Cartesian counterparts {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;,}} are ''not''&amp;#8202; assumed to be uniform (i.e., '''homogeneous'''). One consequence of this general non-uniformity (inhomogeneity) is that, although we can say&amp;#8239; {{math|'''r'''&amp;#8201;{{=}}&amp;#8201;''x&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''e'''&lt;sub&gt;''i''&lt;/sub&gt;}}&amp;#8239; in Cartesian coordinates and&amp;#8239; {{math|'''q'''&amp;#8201;{{=}}&amp;#8201;''q&lt;sup&gt;i&lt;/sup&gt;''&amp;#8239;'''h'''&lt;sub&gt;''i''&lt;/sub&gt;}}&amp;#8239; in general coordinates, we ''cannot'' say
:{{big|&lt;math&gt;\mathbf{r} = u^i \mathbf{h}_i \qquad&lt;/math&gt;[''sic!''&amp;#8239;]}}
in general coordinates. For example, we have seen that in spherical coordinates the position vector {{math|'''r'''}} is simply &lt;math&gt;r\mathbf{\hat{r}}&lt;/math&gt;, i.e.{{math| ''r''&amp;#8202;'''h'''&lt;sub&gt;''r''&lt;/sub&gt;&amp;#8239;}};&amp;#8202; it is ''not''&amp;#8201; {{math|''r''&amp;#8202;'''h'''&lt;sub&gt;''r''&lt;/sub&gt;&amp;#8202;+&amp;#8202;''&amp;theta;''&amp;#8202;'''h'''&lt;sub&gt;''&amp;theta;''&lt;/sub&gt;&amp;#8202;+&amp;#8202;''&amp;phi;''&amp;#8202;'''h'''&lt;sub&gt;''&amp;phi;''&lt;/sub&gt;&amp;#8202;,}} because {{mvar|&amp;theta;}} and {{mvar|&amp;phi;}} are encoded in the direction of{{math|&amp;#8202; '''h'''&lt;sub&gt;''r''&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; Similarly, in cylindrical coordinates the position vector {{math|'''r'''}} is {{math|''&amp;rho;''&amp;#8202;'''h'''&lt;sub&gt;''&amp;rho;''&lt;/sub&gt;&amp;#8202;+&amp;#8239;''z''&amp;#8202;'''h'''&lt;sub&gt;''z''&lt;/sub&gt;&amp;#8239;}};&amp;#8202; it is ''not''&amp;#8201; {{math|''&amp;rho;''&amp;#8202;'''h'''&lt;sub&gt;''&amp;rho;''&lt;/sub&gt;&amp;#8202;+&amp;#8202;''&amp;phi;''&amp;#8202;'''h'''&lt;sub&gt;''&amp;phi;''&lt;/sub&gt;&amp;#8202;+&amp;#8239;''z''&amp;#8202;'''h'''&lt;sub&gt;''z''&lt;/sub&gt;&amp;#8202;,}} because {{mvar|&amp;phi;}} is encoded in the direction of{{math|&amp;#8202; '''h'''&lt;sub&gt;''&amp;rho;''&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; In both examples, encoding one coordinate in the direction of another coordinate's unit vector is circular in that the said direction depends on the position vector, which is the very thing that we want to represent.

A non-uniform basis is not a ''global''&amp;#8202; basis. It cannot give a uniform representation of a uniform vector field, because the standard of representation changes; it is like having a compass whose orientation varies from place to place and&amp;#10744;or a measuring stick whose length varies from place to place. But it can serve as a '''local basis'''&amp;#8202;&amp;mdash;as in ({{EquationNote|83a}}) and ({{EquationNote|83b}}), each of which expresses a vector field at a given location in terms of a basis at that location, notwithstanding that the basis may be different at other locations. And although a local basis (as we have just seen) cannot generally represent the position vector in a non-circular manner, it ''can''&amp;#8202; represent a ''change''&amp;#8202; in the position vector. By the generality of the multivariate chain rule ({{EquationNote|75}}),
:{{big|&lt;math&gt;
    \part_t \mathbf{r} = \part_i \mathbf{r} \,\part_t u^i .
  &lt;/math&gt;}}
Multiplying by {{mvar|dt&amp;#8239;}} we get
{{NumBlk|:|{{big|&lt;math&gt;
    d\mathbf{r} = \part_i \mathbf{r} \,du^i
  &lt;/math&gt;}}|{{EquationRef|84}}}}
or, substituting from ({{EquationNote|80a}}),
{{NumBlk|:|{{big|&lt;math&gt;
    d\mathbf{r} = \mathbf{h}_i ~\!du^i .
  \qquad&lt;/math&gt;}}|{{EquationRef|85}}}}
Thus the small changes in the coordinates{{mvar| u&lt;sup&gt;i&lt;/sup&gt;}}&amp;#8202; are the components of the true vector{{math| ''d'''''r'''}} w.r.t. the ''covariant''&amp;#8202; basis. That means the changes in the coordinates must be ''contravariant''. Here at last is the explanation why we write general coordinates with superscript indices. And again the point is moot for Cartesian coordinates, for which the covariant basis is also contravariant.

Since {{mvar|du&lt;sup&gt;i&lt;/sup&gt;}}&amp;#8202; is contravariant,&amp;#8201; {{math|''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''r'''}}&amp;#8202; in ({{EquationNote|84}}) must be covariant in order to yield the true vector{{math| ''d'''''r'''}}. This vindicates our decision to write {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}} with a subscript. Recall, however, that {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; means{{math| ''{{sfrac|&amp;part;|&amp;part;u&lt;sup&gt;i&lt;/sup&gt;&amp;#8202;}}''&amp;#8202;}}. Thus ''the derivative w.r.t. the contravariant quantity is covariant''&amp;#8202;&amp;mdash;wherefore it is said that ''a superscript in the denominator of a derivative counts as a subscript in the derivative as a whole''.

In ({{EquationNote|85}}), the general term&amp;#8239; {{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8201;''du&lt;sup&gt;i&lt;/sup&gt;''}} (not the sum) is the displacement of{{math|&amp;#8202; '''r'''}} due to the small change {{mvar|du&lt;sup&gt;i&lt;/sup&gt;}} in the coordinate {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}}. The three such displacements of{{math|&amp;#8202; '''r'''}} make concurrent edges of a parallelepiped whose signed volume is
:&lt;math&gt;
  dV =~\! \mathbf{h}_1~\!du^1 \cdot~\!\mathbf{h}_2~\!du^2
          ~\!\!\times\mathbf{h}_3~\!du^3 \,;
  &lt;/math&gt;
that is,
{{NumBlk|:|&lt;math&gt;dV = J \,du^1 du^2 du^3&lt;/math&gt;|{{EquationRef|86}}}}
where
:&lt;math&gt;J := \mathbf{h}_1 ~\!\!\cdot \mathbf{h}_2 \!\times~\!\!\mathbf{h}_3&lt;/math&gt;
or, to use a standard abbreviation for the scalar triple product,
{{NumBlk|:|&lt;math&gt;
    J := [~\!\mathbf{h}_1 \mathbf{h}_2 \mathbf{h}_3] \,.
  &lt;/math&gt;|{{EquationRef|87}}}}
{{mvar|J}}&amp;#8202; is called the '''Jacobian''' of the natural (covariant) basis. We describe the basis and the associated coordinate system as '''right-handed''' if this Jacobian is ''positive'', and '''left-handed''' if this Jacobian is ''negative''. Thus the handedness depends on the standard order in which we write the vectors; e.g., the standard Cartesian basis is right-handed because we write it as{{math| ('''i''',&amp;#8202;'''j''','''k''')}} but would be left-handed if we wrote it as{{math| ('''i''','''k''',&amp;#8202;'''j''')}}.

If the covariant basis is indeed a basis, its member vectors must not be coplanar; that is, {{mvar|J }}must not be zero. Hence, if the covariant basis is to be a local basis in some region of interest, {{mvar|J }}must not vanish anywhere in that region, and therefore must have the same sign throughout the region; that is, the handedness of the coordinate system must be the same throughout the region.

{{cob}}

=== Properties of reciprocal bases ===

{{cot}}

We have noted that formulae ({{EquationNote|83c}}) and ({{EquationNote|83d}}), for the components of a vector w.r.t. the covariant and contravariant bases, depend only on the reciprocity relation ({{EquationNote|81}}) between the bases. Now, retaining the designations &quot;covariant&quot; and &quot;contravariant&quot; for convenience, let us see what else we can deduce from that relation.

Most obviously, the reciprocity relation leads to a simple component-based expression for the dot-product of two vector fields, say {{math|'''v'''}} and{{math| '''q'''&amp;#8202;,}} provided that we use the contravariant components and covariant basis ({{EquationNote|83a}}) for one vector, and the covariant components and contravariant basis ({{EquationNote|83b}}) for the other:
:{{big|&lt;math&gt;\mathbf{v} \cdot \mathbf{q}
    =~\! v^i ~\!\mathbf{h}_i \cdot q_j \mathbf{h}^j
    =~\! v^i \,\mathbf{h}_i \!\cdot\! \mathbf{h}^j \,q_j
    =~\! v^i ~\!\delta_i^j ~\!q_j \,,
  &lt;/math&gt;}}
whence selecting the non-zero terms gives
{{NumBlk|:|{{big|&lt;math&gt;\mathbf{v} \cdot \mathbf{q}
    =~\! v^i q_i \,.
  &lt;/math&gt;}}|{{EquationRef|88a}}}}
And the two vectors, being general, can swap roles in ({{EquationNote|83a}}) and ({{EquationNote|83b}}):
{{NumBlk|:|{{big|&lt;math&gt;\mathbf{v} \cdot \mathbf{q}
    =~\! v_i q^i \,.
  &lt;/math&gt;}}|{{EquationRef|88b}}}}

The cross-product needs a bit more preparation. First we define the '''permutation symbol''' {{mvar|&amp;epsiv;&lt;sub&gt;ijk&lt;/sub&gt;}} or{{mvar| &amp;epsiv;&lt;sup&gt;ijk&lt;/sup&gt;}} (also called the '''[[w:Tullio Levi-Civita|Levi-Civita]]''' symbol) as having the value&amp;#8202; {{math|+1}} if {{math|(''i'',&amp;#8201;''j'',&amp;#8239;''k'')}} is a permutation of{{math| (1,&amp;#8202;2,&amp;#8202;3)}} in the same cyclic order,&amp;#8201; {{math|&amp;minus;1 }}if {{math|(''i'',&amp;#8201;''j'',&amp;#8239;''k'')}} is a permutation of{{math| (1,&amp;#8202;2,&amp;#8202;3)}} in the reverse cyclic order, and {{math|0}} if {{math|(''i'',&amp;#8201;''j'',&amp;#8239;''k'')}} is not a permutation, i.e. if there is at least one repeated index. To put it more formally,
{{NumBlk|:|{{big|&lt;math&gt;\epsilon_{ijk\!} = \epsilon^{ijk\!} =&lt;/math&gt;}}
  {{resize|&lt;math&gt;\begin{cases}
    +1 &amp;\mathsf{if}\,\,(i,j,k)\in\big\{(1,2,3),~\!(2,3,1),~\!(3,1,2)\big\}\\
    -1 &amp;\mathsf{if}\,\,(i,j,k)\in\big\{(3,2,1),~\!(1,3,2),~\!(2,1,3)\big\}\\
    \phantom{-}0 &amp;\mathsf{otherwise}.
  \end{cases}&lt;/math&gt;}}|{{EquationRef|89}}}}
Note that because switching any two indices changes the cyclic order, ''switching any two indices changes the sign of the permutation symbol''. Now by ({{EquationNote|81}}),&amp;#8201; {{math|'''h'''&lt;sup&gt;1&lt;/sup&gt; }}is perpendicular to both {{math|'''h'''&lt;sub&gt;2&lt;/sub&gt; }}and{{math| '''h'''&lt;sub&gt;3&lt;/sub&gt;}}. So we can say
:&lt;math&gt;\mathbf{h}_2 \!\times\!\mathbf{h}_3 =~\! \alpha_1 ~\!\mathbf{h}^1&lt;/math&gt;
where {{math|''&amp;alpha;''&lt;sub&gt;1&lt;/sub&gt; }}is a real variable to be determined. Taking dot-products with{{math| '''h'''&lt;sub&gt;1&lt;/sub&gt;}} and applying ({{EquationNote|81}}) and ({{EquationNote|87}}), we find that&amp;#8202; {{math|''&amp;alpha;''&lt;sub&gt;1&lt;/sub&gt;&amp;#8239;{{=}}&amp;#8239;''J''&amp;#8202;,}} so that
{{NumBlk|:|&lt;math&gt;
    \mathbf{h}_2 \!\times\!\mathbf{h}_3 =~\! J \mathbf{h}^1 .
  &lt;/math&gt;|{{EquationRef|90.1}}}}
By the generality of the vectors we can rotate the three indices, but the sign of the left-hand side changes if we swap the two indices on the left. All six cases are covered by
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_i \!\times\!\mathbf{h}_j
    =~\! J \epsilon_{ijk\,} \mathbf{h}^k .
  &lt;/math&gt;}}|{{EquationRef|90a}}}}
Here we want only one term; but we need not specify &quot;no sum&quot;, because for given {{mvar|i&amp;#8202; }}and{{mvar| j}}&amp;#8201; the permutation symbol leaves only one non-zero term in the sum over{{mvar| k}}. In words, this result says that the cross-product of two covariant basis vectors, with their indices in the standard cyclic order, is the Jacobian times the contravariant basis vector with the omitted index. Similarly, or rather reciprocally,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^i \!\times\!\mathbf{h}^j
    =~\! J' \epsilon^{ijk\,} \mathbf{h}_k \,,
  &lt;/math&gt;}}|{{EquationRef|90b}}}}
where {{mvar|J&amp;prime;}}&amp;#8202; is the Jacobian ''of the contravariant basis''.

Equations ({{EquationNote|90a}}) and ({{EquationNote|90b}}), which we have obtained from the reciprocity relation ({{EquationNote|81}}), can be solved for {{math|'''h'''&lt;sup&gt;''k''&lt;/sup&gt; }}and{{math| '''h'''&lt;sub&gt;''k''&lt;/sub&gt;}} respectively; but now we ''do'' suppress the implicit sum, because {{mvar|k}}&amp;#8202; is &quot;given&quot; instead of {{mvar|i&amp;#8202; }}and{{mvar| j&amp;#8239;}}:
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^k = \tfrac{\,1\,}{J}~\! \mathbf{h}_i \!\times\!\mathbf{h}_j \quad
  &lt;/math&gt;}} [distinct {{math|''i'',&amp;#8201;''j'',&amp;#8239;''k''}}  in cyclic order];
  |{{EquationRef|90c}}}}
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_k = \tfrac{1}{\,J'}~\! \mathbf{h}^i \!\times\!\mathbf{h}^j \quad
  &lt;/math&gt;}}[distinct {{math|''i'',&amp;#8201;''j'',&amp;#8239;''k''}}  in cyclic order].
  |{{EquationRef|90d}}}}
Thus ''a reciprocity relation between bases is enough to define either basis in terms of the other''&amp;mdash;as claimed above.{{efn|Our ({{EquationNote|90c}}) corresponds to [[#stratton-41|Stratton, 1941]], p.&amp;#8239;39, eqs.&amp;#8239;(9). And our ({{EquationNote|90d}}) corresponds to Stratton's subsequent eqs.&amp;#8239;(11) except that Stratton has, in our notation, {{mvar|J}} instead of{{mvar| J&amp;prime;}}; the error is noted by Tai ([[#tai-95|1995]], p.&amp;#8239;59). See also our ({{EquationNote|92}}).}} If it is not convenient to suppress an implicit sum, the last two results can instead be written
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^k =
      \tfrac{1}{2J}~\!\epsilon^{ijk\,}\mathbf{h}_i {\times}~\!\mathbf{h}_j
  &lt;/math&gt;}}|{{EquationRef|90e}}}}
and
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_k =
      \tfrac{1}{2J'}~\!\epsilon_{ijk\,}\mathbf{h}^i {\times}~\!\mathbf{h}^j \,,
  &lt;/math&gt;}}|{{EquationRef|90f}}}}
where the factor 2 in each denominator is needed because the right-hand side has two equal non-zero terms&amp;mdash;the sign of the permutation symbol compensating for the order of the cross-product.

Now we're ready to consider the cross-product of two vector fields. In terms of the covariant basis,
:{{big|&lt;math&gt;\begin{align}\mathbf{v} \!\times\! \mathbf{q}
     =~\! v^i \mathbf{h}_i ~\!\!\times q^j \mathbf{h}_j
    &amp;=~\! v^i \,\mathbf{h}_i {\times}~\! \mathbf{h}_j \,q^j \\
    &amp;=~\! v^i J \epsilon_{ijk~\!} \mathbf{h}^k \;\!q^j \,;
  \end{align}&lt;/math&gt;}}
i.e.,
{{NumBlk|:|{{big|&lt;math&gt;\mathbf{v} \!\times\! \mathbf{q}
    =~\! J \epsilon_{ijk\,} v^i q^j \mathbf{h}^k .
  &lt;/math&gt;}}|{{EquationRef|91a}}}}
On the right, the two components and the basis vector are contravariant, but invariance is achieved by multiplying by the covariant Jacobian (which has three covariant factors). Similarly,
{{NumBlk|:|{{big|&lt;math&gt;\mathbf{v} \!\times\! \mathbf{q}
    =~\! J' \epsilon^{ijk} v_i q_j \;\!\mathbf{h}_k .
  &lt;/math&gt;}}|{{EquationRef|91b}}}}
On the right of ({{EquationNote|91a}}) or ({{EquationNote|91b}}), the implicit triple summation has 27 terms, of which only six&amp;mdash;corresponding to the six possible permutations of the three possible indices&amp;mdash;can be non-zero. Thus the factor following the Jacobian can be recognized as the familiar determinant whose columns (or rows), in cyclic order, are the components of{{math| '''v'''&amp;#8202;,}} the components of{{math| '''q'''&amp;#8202;,}} and the three basis vectors. In Cartesian coordinates, in which the Jacobians are equal to{{math| 1}} and we don't need the co&amp;#10744;contra distinction, both equations reduce to
:{{big|&lt;math&gt;\mathbf{v} \!\times\! \mathbf{q}
    =~\! \epsilon_{ijk\,} v_i q_j \;\!\mathbf{e}_k
  &lt;/math&gt;}}
&amp;mdash;a familiar result written in a possibly unfamiliar way.

The Jacobian of the contravariant basis is
:&lt;math&gt;J' =~\! \mathbf{h}^1 \cdot \mathbf{h}^2 \!\times\!\mathbf{h}^3&lt;/math&gt;
or, if we substitute from ({{EquationNote|90c}}),
:&lt;math&gt;\begin{align}J'
  &amp;= \frac{(\mathbf{h}_2 \!\times\!\mathbf{h}_3)}{J} \cdot
     \frac{(\mathbf{h}_3 \!\times\!\mathbf{h}_1)}{J} \times
     \frac{(\mathbf{h}_1 \!\times\!\mathbf{h}_2)}{J} \\[.5ex]
  &amp;= \frac{(\mathbf{h}_2 \!\times\!\mathbf{h}_3) \cdot
           (\mathbf{h}_3 \!\times\!\mathbf{h}_1) \times
           (\mathbf{h}_1 \!\times\!\mathbf{h}_2)}
          {J^3} \,.
  \end{align}&lt;/math&gt;
In the numerator, the cross-product of cross-products can be read as a vector triple product in which the first factor is a cross-product. Expanding that triple product and noting that one term is a scalar triple product with a repeated factor, we get
:&lt;math&gt;
    J' = \frac{(\mathbf{h}_2 \!\times\!\mathbf{h}_3) \cdot J\mathbf{h}_1}{J^3}
       = \frac{\,J^2}{~J^3 ~\!} = \frac{\,1\,}{J} \,,
  &lt;/math&gt;
so that we may write
{{NumBlk|:|&lt;math&gt;J' =~\! J^{-1} &lt;/math&gt;|{{EquationRef|92}}}}
in ({{EquationNote|90b}}), ({{EquationNote|90d}}), ({{EquationNote|90f}}), and ({{EquationNote|91b}}). In words, ''the Jacobian of the reciprocal basis is the reciprocal of the Jacobian'' of the original basis. Therefore the two Jacobians have the same sign. Therefore ''a basis is right-handed if and only if its reciprocal is right-handed''. Thus the natural and dual bases of a coordinate system have the same handedness, and the handedness of either may be identified with the handedness of the coordinate system.

{{cob}}

=== The gradient, del, and advection operators ===

{{cot}}

Let {{mvar|p}}&amp;#8202; be a scalar field, and let{{mvar| s}}&amp;#8202; be arc length in the direction of the unit vector{{math| '''s&amp;#770;'''}}. By the multivariate chain rule ({{EquationNote|75}}),
:{{big|&lt;math&gt;\begin{align}\part_s p
    &amp;= \part_i p \;\part_s u^i \\
    &amp;= \part_i p \;\mathbf{\hat{s}} \cdot \nabla u^i \\
    &amp;= \part_i p \;\mathbf{\hat{s}} \cdot \mathbf{h}^i \\
    &amp;=\mathbf{\hat{s}} \cdot \mathbf{h}^i \part_i p \,.
  \end{align}&lt;/math&gt;}}
So&amp;#8202; {{math|'''h'''&lt;sup&gt;''i''&lt;/sup&gt;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p''}}&amp;#8239; is the vector whose (invariant) scalar component in the direction of any{{math| '''s&amp;#770;'''}} is the directional derivative of{{mvar| p}} in that direction; that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla p = \mathbf{h}^i \part_i p \,,
  &lt;/math&gt;}}|{{EquationRef|93g}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla =~\! \mathbf{h}^i \part_i \,.
  &lt;/math&gt;}}|{{EquationRef|93o}}}}
Apart from the need to pair a superscript with a subscript, these two results look as simple as their Cartesian special cases ({{EquationNote|58g}}) and ({{EquationNote|58o}}).

If {{mvar|&amp;psi;}}&amp;#8202; is a generic field and {{math|'''q'''}} is a general vector in the direction of the same{{math| '''s'''&amp;#8202;}} then by definition ({{EquationNote|11}}),
:{{big|&lt;math&gt;\begin{align}
    \mathbf{q}\;\!{\cdot}\nabla\,\psi
    &amp;= |\mathbf{q}| \,\part_s \psi \\
    &amp;= |\mathbf{q}| \,\part_i \psi \,\part_s u^i \\
    &amp;= \part_i \psi \;|\mathbf{q}| ~\!\part_s u^i \\
    &amp;= \part_i \psi \;\mathbf{q} \cdot \nabla u^i \\
    &amp;= \part_i \psi \;\mathbf{q} \cdot \mathbf{h}^i \\
    &amp;= \part_i \psi \;q^j \mathbf{h}_j \cdot \mathbf{h}^i \\
    &amp;= \part_i \psi \;q^j \epsilon_j^i \\
    &amp;= \part_i \psi \,q^i \,;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\psi = q^i \part_i \psi \,,
  &lt;/math&gt;}}|{{EquationRef|94}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla =~\! q^i \part_i \,.
  &lt;/math&gt;}}|{{EquationRef|94o}}}}
These results likewise look as simple as their Cartesian special cases ({{EquationNote|64}}) and ({{EquationNote|64o}}).&amp;#8201; And by ({{EquationNote|88a}}), the {{math|'''q&amp;sdot;'''&amp;nabla;}} operator again turns out to be the formal dot-product of&amp;#8202; {{math|'''q'''}} and{{math| &amp;nabla;}}.

{{cob}}

=== The curl and divergence operators ===

{{cot}}

To express the curl of a vector field{{math| '''q'''&amp;#8202;,}} we choose the contravariant basis ({{EquationNote|83b}}) and apply identity ({{EquationNote|71c}}):
:{{big|&lt;math&gt;\begin{align}\operatorname{curl}\mathbf{q}
    &amp;= \operatorname{curl} q_j \mathbf{h}^j \\
    &amp;= q_j \operatorname{curl}\mathbf{h}^j + \nabla q_j \times \mathbf{h}^j .
  \end{align}&lt;/math&gt;}}
On the right, the first term vanishes because {{math|'''h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;}}&amp;#8202; is {{math|&amp;nabla;''u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;''}} (and the curl of a gradient is zero). Substituting from ({{EquationNote|93o}}) in the second term, we obtain
:{{big|&lt;math&gt;\operatorname{curl}\mathbf{q}
    = \mathbf{h}^i \part_i q_j \times \mathbf{h}^j
    = \mathbf{h}^i {\times}~\! \mathbf{h}^j ~\!\part_i q_j
  &lt;/math&gt;}}
or, using ({{EquationNote|90b}}),
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{curl}\mathbf{q}
    = J' \epsilon^{ijk\,} \mathbf{h}_k \part_i q_j
  &lt;/math&gt;}}|{{EquationRef|95c}}}}
or, in a more familiar form,
:&lt;math&gt;
    \operatorname{curl}\mathbf{q} \,=\, J'\,
      \begin{vmatrix}
        \mathbf{h}_1 &amp; \part_1 &amp; q_1 \\
        \mathbf{h}_2 &amp; \part_2 &amp; q_2 \\
        \mathbf{h}_3 &amp; \part_3 &amp; q_3
      \end{vmatrix} \,.
  &lt;/math&gt;
Formula ({{EquationNote|95c}}) agrees with a result obtained by Tai with his &quot;symbolic vector&quot; method.&lt;ref&gt;[[#tai-95|Tai, 1995]], p.&amp;#8239;66, eq.&amp;#8239;(9.41).&lt;/ref&gt; It is also what we would get by naively using ({{EquationNote|91b}}) to evaluate {{math|&amp;nabla;&amp;#8202;&amp;times;&amp;#8202;'''q'''&amp;#8202;;}}&amp;#8201; it comes out so simply because each contravariant basis vector{{math| '''h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;}}&amp;#8202; is the actual gradient of{{mvar| u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;}} and not (e.g.) merely a unit vector in the same direction (remember that {{mvar|q&lt;sub&gt;j&lt;/sub&gt;}} is the component w.r.t.{{math| '''h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;&amp;#8202;,}}&amp;#8202; not{{math| '''h'''&lt;sub&gt;''j''&lt;/sub&gt;}}).

But ({{EquationNote|95c}}) does not end in a subexpression for the operand{{math| '''q'''}}&amp;#8202; and therefore does not directly yield an expression for the curl ''operator''. To find this operator and the divergence operator, we return to the original definitions ({{EquationNote|4g}}), ({{EquationNote|4c}}), and ({{EquationNote|4d}}), noting that they can be combined as
{{NumBlk|:|&lt;math&gt;\nabla ~\!\!* \psi
    \,=\, \tfrac{1}{dV}\!\iint_{\delta S} (\mathbf{\hat{n}}~\!dS*\psi) \,,
  &lt;/math&gt;|{{EquationRef|96}}}}
where {{math|&amp;lowast;}} may be a null for the gradient, a cross for the curl, or a dot for the divergence.{{efn|But not dot-del for the Laplacian, as in ({{EquationNote|19}}), because we want to use an elementary product rule inside the integral.}} Recalling that the value of this expression does not depend on the shape of{{mvar| dS&amp;#8202;}}, let{{mvar| dS&amp;#8202;}} be the parallelepiped defined by the six equicoordinate surfaces at {{mvar|u&lt;sup&gt;i&lt;/sup&gt; }}and{{mvar| u&lt;sup&gt;i&lt;/sup&gt;+du&lt;sup&gt;i&lt;/sup&gt;}}, so that {{mvar|dV}}&amp;#8202; is given by ({{EquationNote|86}}). Then the contribution to the integral from the face at{{math| ''u''&amp;sup1;+''du''&amp;sup1; &amp;#8202;}}is
:&lt;math&gt;\Big[
    \big(\mathbf{h}_2 du^2 \!\times\! \mathbf{h}_3 du^3\big) * \psi
  \Big]_{u^1 + du^1}&lt;/math&gt;
where the square brackets and subscripting mean &quot;evaluated at&quot;. This can be written
:&lt;math&gt;du^2 du^3 \Big[
    (\mathbf{h}_2 {\times}~\! \mathbf{h}_3) * \psi
  \Big]_{u^1 + du^1}&lt;/math&gt;
or, by ({{EquationNote|90.1}}),
:&lt;math&gt;du^2 du^3 \big[J \mathbf{h}^1 ~\!\!* \psi \big]_{u^1 + du^1} \,.&lt;/math&gt;
Similarly, the contribution from the face at{{math| ''u''&amp;sup1;}} (where {{math|'''h'''&lt;sub&gt;1&lt;/sub&gt;}} points inward instead of outward) is
:&lt;math&gt;-du^2 du^3 \big[J \mathbf{h}^1 ~\!\!* \psi \big]_{u^1} \,.&lt;/math&gt;
The sum of the contributions from the two opposite faces can then be written
:&lt;math&gt;du^1 du^2 du^3 ~\!\part_1 \big(J \mathbf{h}^1 ~\!\!* \psi\big) ~,&lt;/math&gt;
so that when we add in the contributions from the other two pairs of opposite faces, the entire integral becomes
:{{big|&lt;math&gt;
    du^1 du^2 du^3 ~\!\part_i \big(J \mathbf{h}^i ~\!\!* \psi\big)
  &lt;/math&gt;}}
(with implicit summation over{{mvar| i}}). Substituting this and ({{EquationNote|86}}) into ({{EquationNote|96}}), we get
{{NumBlk|:|{{big|&lt;math&gt;\nabla ~\!\!* \psi
    = \tfrac{\,1\,}{J} ~\!\part_i \big(J \mathbf{h}^i ~\!\!* \psi\big) \,.
  &lt;/math&gt;}}|{{EquationRef|97}}}}
Now applying the product rule gives
{{NumBlk|:|{{big|&lt;math&gt;\nabla ~\!\!* \psi
    = \mathbf{h}^i ~\!\!* \part_i \psi
      + \tfrac{\,1\,}{J} ~\!\part_i(J \mathbf{h}^i) * \psi \,.
  &lt;/math&gt;}}|{{EquationRef|98}}}}
Here the left-hand side is {{math|&amp;nabla;&amp;#8202;&amp;lowast;''&amp;psi;''}}&amp;#8239; according to our original volume-based definition ({{EquationNote|4g}}) of the {{math|&amp;nabla; }}operator&amp;mdash;which is known to yield the curl or the divergence if&amp;#8202; {{math|&amp;lowast;}} is a cross or a dot, respectively&amp;mdash;whereas the first term on the right is what we would get for {{math|&amp;nabla;&amp;#8202;&amp;lowast;''&amp;psi;''}}&amp;#8239; by using our latest definition ({{EquationNote|93o}}) of the {{math|&amp;nabla; }}operator and allowing{{mvar| &amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; to &quot;pass by&quot; the star in the Wilsonian manner. So, if we can show that the second term on the right is zero, we shall have established the precise sense in which the del-cross and del-dot notations are valid in general coordinates. In that second term, by ({{EquationNote|90e}}),
:{{big|&lt;math&gt;\begin{align}
    \part_i(J \mathbf{h}^i)
    &amp;= \part_i \big(
         \tfrac{\,1\,}{2}~\!\epsilon^{jki}\mathbf{h}_j {\times}~\!\mathbf{h}_k
       \big) \\
    &amp;= \tfrac{\,1\,}{2}~\!\epsilon^{jki}
       \part_i \big(\mathbf{h}_j {\times}~\!\mathbf{h}_k \big) \\[.5ex]
    &amp;= \tfrac{\,1\,}{2}~\!\epsilon^{jki}
       \big(\part_i \mathbf{h}_j \!\times~\!\!\mathbf{h}_k +
            \mathbf{h}_j \!\times~\!\!\part_i \mathbf{h}_k \big) \\[.5ex]
    &amp;= \tfrac{\,1\,}{2}~\!\epsilon^{jki}
       \big(\part_i \part_j \mathbf{r} ~\!\!\times\! \mathbf{h}_k +
            \mathbf{h}_j {\times}~\! \part_i \part_k \mathbf{r} \big) \,,
  \end{align}&lt;/math&gt;}}
where the last line follows by ({{EquationNote|80a}}). But the order of partial differentiation can be switched. So, in the sum over the permutations, for each term in{{math| ''&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;&amp;part;&lt;sub&gt;j&lt;/sub&gt;''&amp;#8239;'''r'''}}&amp;#8239; there is an equal term in{{math| ''&amp;part;&lt;sub&gt;j&lt;/sub&gt;&amp;#8202;&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''r'''}}&amp;#8239; to which the permutation symbol attaches the opposite sign, so that the terms in{{math| ''&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;&amp;part;&lt;sub&gt;j&lt;/sub&gt;''&amp;#8239;'''r'''}}&amp;#8202; cancel. Similarly the terms in{{math| ''&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;&amp;part;&lt;sub&gt;k&lt;/sub&gt;''&amp;#8239;'''r'''}}&amp;#8202; cancel. Thus, as anticipated, the second term in ({{EquationNote|98}}) is zero and we have
{{NumBlk|:|{{big|&lt;math&gt;\nabla ~\!\!* \psi
    = \mathbf{h}^i ~\!\!* \part_i \psi \,.
  &lt;/math&gt;}}|{{EquationRef|99}}}}

If&amp;#8202; {{math|&amp;lowast;}} is a null and {{mvar|&amp;psi;}} is a scalar field {{math|''p''&amp;#8202;,}} then ({{EquationNote|99}}) becomes ({{EquationNote|93g}}) and thus (fortunately!) confirms ({{EquationNote|93o}}) as the form of the del operator in general coordinates.

Now let {{mvar|&amp;psi;}} be a ''vector'' field {{math|'''q'''&amp;#8202;}}.&amp;#8201; If{{math|&amp;#8202; &amp;lowast;}} is a cross, then ({{EquationNote|99}}) becomes
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{curl}\mathbf{q}
    = \mathbf{h}^i ~\!\!\times \part_i \mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|100c}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{curl}
    = \mathbf{h}^i ~\!\!\times \part_i \,.
  &lt;/math&gt;}}|{{EquationRef|100o}}}}
If instead {{math|&amp;lowast;}} is a dot, ({{EquationNote|99}}) becomes
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{div}\mathbf{q}
    = \mathbf{h}^i ~\!\!\cdot \part_i \mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|101d}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{div}
    = \mathbf{h}^i ~\!\!\cdot \part_i \,.
  &lt;/math&gt;}}|{{EquationRef|101o}}}}
But if we take the {{math|&amp;nabla;}} operator as given by ({{EquationNote|93o}}) and try to construct the curl and divergence operators (in the ''same''&amp;#8202; coordinates) as {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8239; respectively, we get&amp;#8202; {{math|'''h'''&lt;sup&gt;''i''&lt;/sup&gt;&amp;#8239;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;&amp;times;}}&amp;#8239; and&amp;#8239; {{math|'''h'''&lt;sup&gt;''i''&lt;/sup&gt;&amp;#8239;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''&amp;sdot;'''}}&amp;#8201; respectively [compare ({{EquationNote|61o}}) and ({{EquationNote|62o}})]; and if we then let{{mvar| &amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; &quot;pass by&quot; the cross and the dot, we get ({{EquationNote|100o}}) and ({{EquationNote|101o}}), or ({{EquationNote|100c}}) and ({{EquationNote|101d}}) if we include the operand{{math| '''q'''&amp;#8202;}}. Thus ''the del-cross and del-dot notations work in general coordinates''.

Equations ({{EquationNote|100c}}) to ({{EquationNote|101o}}) are apparently due to Tai ([[#tai-95|1995]], eqs.&amp;#8239;9.39, 9.40, 9.34, &amp;amp; 9.35, and text on p.&amp;#8239;66), who derives them, along with the corresponding form of the del operator (his eq.&amp;#8239;9.33), from volume-based definitions expressed in his &quot;symbolic vector&quot; notation. But he does not point out that the curl and divergence operators are obtainable from that del operator, as del-cross and del-dot, via the same &quot;pass by&quot; step that he condemns in the Cartesian context. Speaking of which, we should note that our equations ({{EquationNote|100c}}) to ({{EquationNote|101o}}), apart from the need to pair a superscript with a subscript, are as simple as their Cartesian special cases ({{EquationNote|59c}}), ({{EquationNote|59o}}), ({{EquationNote|60d}}), and ({{EquationNote|60o}}).

In ({{EquationNote|100c}}) and ({{EquationNote|101d}}), it goes without saying that&amp;#8202; {{math|''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''q'''}}&amp;#8202; must be evaluated correctly&amp;mdash;in particular, that if the operand is expressed in terms of non-uniform basis vectors, the non-uniformity must be taken into account. Formula ({{EquationNote|95c}}), for the curl, does not suffer from this complication, because it is already expressed in components w.r.t. the contravariant basis (whose non-uniformity has already been taken into account). To obtain a similarly convenient formula for the divergence, we use components w.r.t. the {{nowrap|''co''&amp;#8202;variant}} basis (i.e., contravariant components): in ({{EquationNote|97}}), if{{math|&amp;#8202; &amp;lowast;}} is a dot and {{mvar|&amp;psi;}} is a vector field{{math| '''q'''&amp;#8202;,}} we have
:{{big|&lt;math&gt;\operatorname{div}\mathbf{q}
    = \tfrac{\,1\,}{J} ~\!\part_i \big(J \mathbf{h}^i {\cdot}\, \mathbf{q}\big)
  &lt;/math&gt;}}
or, by ({{EquationNote|83c}}),
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{div}\mathbf{q}
    = \tfrac{\,1\,}{J} ~\!\part_i \big(J q^i \big) \,.
  &lt;/math&gt;}}|{{EquationRef|102d}}}}
This too agrees with Tai ([[#tai-95|1995]], p.&amp;#8239;65, eq.&amp;#8239;9.37).

{{cob}}

=== The Laplacian ===

{{cot}}

For a scalar operand, applying ({{EquationNote|101o}}) and reversing the &quot;pass by&quot;, we find that the Laplacian operator is
:{{big|&lt;math&gt;\operatorname{div}\nabla
    = \mathbf{h}^i \!\cdot \part_i \nabla
    = \mathbf{h}^i \part_i ~\!\!\cdot \nabla
    = \nabla {\cdot} \nabla = \nabla^2 .
&lt;/math&gt;}}
And by the linearity of the Laplacian, the{{math| &amp;nabla;&lt;sup&gt;2&lt;/sup&gt;}} formulation remains valid if the operand is a fixed linear combination of scalars&amp;mdash;including a vector field, because that is expressible (even if not actually expressed) w.r.t. a uniform basis. (And if it is expressed in terms of a non-uniform basis, the non-uniformity must be taken into account in differentiations.)

In what follows, however, we shall find it convenient to take a different approach. If{{mvar| &amp;psi;}}&amp;#8202; is a scalar field, its gradient as given by ({{EquationNote|93g}}) is&amp;#8202; {{math| '''h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;''&amp;part;&lt;sub&gt;j&lt;/sub&gt;&amp;#8239;&amp;psi;''&amp;#8202;,}} of which the {{mvar|i&amp;#8202;}}th contravariant component is{{math| '''h'''&lt;sup&gt;''i''&lt;/sup&gt;'''&amp;sdot;&amp;#8239;h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;''&amp;part;&lt;sub&gt;j&lt;/sub&gt;&amp;#8239;&amp;psi;''&amp;#8202;,}} which takes the place of{{mvar| q&lt;sup&gt;i&lt;/sup&gt;}} in ({{EquationNote|102d}}), so that the divergence of the gradient of{{mvar| &amp;psi; &amp;#8202;}}is
{{NumBlk|:|{{big|&lt;math&gt;\triangle\psi
    = \tfrac{\,1\,}{J} ~\!\part_i
        \big(J \mathbf{h}^i {\cdot}~\! \mathbf{h}^j \part_j \psi \big) \,.
  &lt;/math&gt;}}|{{EquationRef|103L}}}}
This remains well-defined if{{mvar| &amp;psi;}}&amp;#8202; is a generic field (although we still need to deal with any non-uniformity of the basis in which{{mvar| &amp;psi;}}&amp;#8202; might be expressed).

{{cob}}

=== Affine coordinates ===

{{cot}}

If a basis is ''uniform'' (homogeneous), so is its Jacobian. Hence, by ({{EquationNote|90c}}) and ({{EquationNote|90d}}), the dual (contravariant) basis is uniform if and only if the natural (covariant) basis is uniform. A coordinate system in which these bases are uniform is described as '''affine'''. In affine coordinates,
* by ({{EquationNote|80a}}), {{math|''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''r''' }}is uniform, so that the curves on which only one coordinate varies are straight parallel lines; and
* by ({{EquationNote|80b}}), {{math|&amp;nabla;''u&lt;sup&gt;i&lt;/sup&gt;'' }}is uniform, so that the level surfaces of each coordinate (being perpendicular to {{math|&amp;nabla;''u&lt;sup&gt;i&lt;/sup&gt;''}}) are parallel planes.
Obviously Cartesian coordinates are affine; but one can also construct affine coordinate systems in which the three vectors of each basis are not mutually perpendicular and&amp;#10744;or the coordinates have different scales or different units.

We have noted above that the correct application of the del-cross, del-dot, and del-squared notations must allow for non-uniformity of the basis vectors. Obviously this issue does not arise in affine coordinates, including Cartesian coordinates. Hence, while these notations are not (as is sometimes alleged) invalid in other coordinate systems, it would be fair to say that they are safer and more convenient in affine coordinates, including Cartesian coordinates.

{{cob}}

=== Orthogonal coordinates ===

{{cot}}

We know, e.g. from ({{EquationNote|90a}}) and ({{EquationNote|90b}}), that if two bases are reciprocal, the cross-product of the {{mvar|i&amp;#8202;}}th and {{mvar|j&amp;#8202;}}th members of one basis is collinear with the {{mvar|k&amp;#8202;}}th member of the other, if {{math|''i'',&amp;#8201;''j'',&amp;#8239;''k''}}&amp;#8239; are distinct. But if the first basis is ''orthogonal'' (that is, if its three member vectors are mutually orthogonal), the same cross-product is also collinear with the {{mvar|k&amp;#8202;}}th member of the ''same'' basis, so that ''corresponding members of the two bases are collinear''. It follows that ''the natural basis of a coordinate system is orthogonal if and only if the dual basis is orthogonal''. And if the bases are orthogonal, the coordinate system itself is said to be '''orthogonal'''.

Cartesian coordinates are obviously both affine and orthogonal, and we have already implied that there is a class of coordinate systems that are affine but not orthogonal. The most widely-used class of non-Cartesian systems, however, contains the systems that are orthogonal but not affine; this class, of which the cylindrical and spherical systems are the best-known members, is the class of '''curvilinear orthogonal coordinates'''. But we shall drop the word ''curvilinear''&amp;#8202; in order to include Cartesian coordinates as a special case.

In orthogonal coordinates, expressing a member of one basis in terms of its reciprocal basis is especially simple because corresponding members of the two bases are collinear, wherefore we can say
:{{big|&lt;math&gt;
    \mathbf{h}^i =~\! \beta_i \mathbf{h}_i \,,
  &lt;/math&gt;}}
where {{mvar|&amp;beta;&lt;sub&gt;i&lt;/sub&gt;}} is a real variable to be determined (and the single index on the left-hand side means ''no summation''). Substituting this into ({{EquationNote|81i}}) gives
:{{big|&lt;math&gt;
    \beta ~\!= 1/h_i^{~2}
  &lt;/math&gt;}}
where
{{NumBlk|:|{{big|&lt;math&gt;
    h_i = \big|\mathbf{h}_i \big| \,,
  &lt;/math&gt;}}|{{EquationRef|104}}}}
so that
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^i =~\! \mathbf{h}_i \big/ h_i^{~2} \,.
  &lt;/math&gt;}}|{{EquationRef|105}}}}
And substituting that into ({{EquationNote|83c}}), and comparing the result with ({{EquationNote|83d}}), we get
{{NumBlk|:|{{big|&lt;math&gt;
    q^i =~\! q_i \big/ h_i^{~2} \,.
  &lt;/math&gt;}}|{{EquationRef|106}}}}
Comparing ({{EquationNote|104}}) with definition ({{EquationNote|80a}}), we see that {{mvar|h&lt;sub&gt;i&lt;/sub&gt;}} is the magnitude of{{math| ''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''r'''}}. Accordingly {{mvar|h&lt;sub&gt;i&lt;/sub&gt;}} is called the '''scale factor''' associated with the coordinate{{mvar| u&lt;sup&gt;i&lt;/sup&gt;&amp;#8202;}}; it is the factor by which we multiply a small change in{{mvar| u&lt;sup&gt;i&lt;/sup&gt;}} to obtain the magnitude of the consequent change in position.{{efn|Hsu ([[#hsu-84|1984]], p.&amp;#8239;171) implies that the scale factors are also called &quot;metric coefficients&quot;, and Tai ([[#tai-94|1994]], [[#tai-95|1995]]) prefers the latter term. This is loose terminology because, in general, the ''metric coefficient''&amp;#8202; is defined as
:{{math|''g&lt;sub&gt;ij&lt;/sub&gt;'' {{=}} '''h'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;'''&amp;sdot;&amp;#8201;h'''&lt;sub&gt;''j''&lt;/sub&gt;&amp;#8201;}}.
Hence, in the special case of orthogonal coordinates, we have {{math|''g&lt;sub&gt;ij&lt;/sub&gt;''&amp;#8201;{{=}}&amp;#8201;0}}&amp;#8202; for {{math|''i&amp;#8239;&amp;ne;&amp;#8239;j''&amp;#8202;,}}&amp;#8239; and&amp;#8202; {{math|''g&lt;sub&gt;ii&lt;/sub&gt;&amp;#8239;{{=}}&amp;#8201;h&lt;sub&gt;i&lt;/sub&gt;''&lt;sup&gt;2&lt;/sup&gt;}}&amp;#8202; [no sum]. Thus the scale factors are not special cases of the metric coefficients, but the ''square roots''&amp;#8202; of special cases of the metric coefficients (''cf''. [[#tai-95|Tai, 1995]], p.&amp;#8239;43, line 3).}}

If we now define
{{NumBlk|:|&lt;math&gt;\varsigma \,=~\! \begin{cases}
    +1 &amp;\mathsf{for~a~right{\operatorname{-}}handed~system} \\[.5ex]
    -1 &amp;\mathsf{for~a~left{\operatorname{-}}handed~system} ~,
  \end{cases}&lt;/math&gt;|{{EquationRef|107}}}}
then, due to the orthogonality,&amp;#8201; ({{EquationNote|87}}) and ({{EquationNote|92}}) are respectively reduced to
{{NumBlk|:|&lt;math&gt;
    J = \varsigma\, h_1 h_2 h_3
  &lt;/math&gt;|{{EquationRef|108}}}}
and
{{NumBlk|:|&lt;math&gt;
    J' = \frac{\,1\,}{J} = \frac{1}{\varsigma\, h_1 h_2 h_3}
       = \frac{\varsigma}{h_1 h_2 h_3}
  &lt;/math&gt;|{{EquationRef|109}}}}
&amp;mdash;although, for brevity, we shall sometimes leave things in terms of{{mvar| J}}.

At this point, we ''could''&amp;#8202; substitute ({{EquationNote|105}}) and ({{EquationNote|106}}) into earlier equations and obtain a suite of formulae for the differential operators in terms of the covariant basis and {{nowrap|''co''&amp;#8202;variant}} components! But we can avoid this confusing breach of convention by ''normalizing'' the basis vectors.

An '''orthonormal''' basis is one whose members are mutually orthogonal ''unit'' vectors. The assumption of unit vectors is introduced so late because it is more useful with orthogonality than without. If one basis consisted of unit vectors that were not all orthogonal, then the reciprocal basis vectors given by ({{EquationNote|90c}}) or ({{EquationNote|90d}}) would not all be unit vectors.{{efn|Outline of proof: If the reciprocal vectors were unit vectors, then the angles between the ''original''&amp;#8202; unit vectors would need to be equal, in order that their cross products have the same magnitude as their scalar triple product (Jacobian); and the latter condition requires the common angle to be 90&amp;deg;.}} But if the basis{{math| ('''h'''&lt;sub&gt;''i''&lt;/sub&gt;)}} consists of orthogonal unit vectors, equation ({{EquationNote|105}}) implies that the reciprocal basis consists of the ''same'' vectors; and the converse is also true, by the symmetry of the reciprocity relations. Thus ''an orthonormal basis is its own reciprocal''. Hence, if we choose an orthonormal basis, we do not need superscripts to distinguish the reciprocal basis from the original, or to distinguish components w.r.t. the latter basis from those w.r.t. the former.

An orthonormal basis is not generally covariant, because it doesn't stretch with the coordinate grid (although it does rotate with the grid). Neither is it generally contravariant, because its reciprocal (i.e. itelf) is not generally covariant. Hence, if a ''non''&amp;#8202;-orthonormal natural or dual basis of an orthogonal coordinate system is ''normalized'' (replaced by unit vectors in the same directions), the resulting orthonormal basis is not covariant or contravariant, and components with respect thereto are not contravariant or covariant, and the new basis vectors are not given in terms of the coordinates by ({{EquationNote|80a}}) or ({{EquationNote|80b}}); the basis is therefore described as a '''non-coordinate basis'''. By default, the indices of the orthonormal basis vectors and associated components are written as subscripts, but these are not indicative of covariance. The coordinates themselves remain contravariant (e.g., if the grid dilates, the same movement in space corresponds to ''smaller'' changes in the coordinates); but, for want of covariant basis vectors to pair them with, we tend to write the coordinates with subscripts when the basis is orthonormal.

Nevertheless, it is convenient to have one basis instead of two. Moreover, the components of a vector w.r.t. an orthonormal basis are '''physical components''': they have the same dimension (same units) as the represented vector, and they are the components that we would have in mind if we wanted to ''measure'' the &quot;components&quot; in the directions of the basis vectors. Hence an orthonormal basis is called a '''physical basis'''. Accordingly, it is indeed common practice to normalize the basis vectors of orthogonal coordinate systems. This together with the prevalence of such coordinate systems helps to account for the familiarity of subscripts as indices, and for the jarring unfamiliarity of superscript indices when general (possibly non-orthogonal) coordinates are encountered for the first time.

To normalize the covariant basis, let{{math| '''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;}} (as usual) be the unit vector in the direction of{{math|&amp;#8202; '''h'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8202;.}} Then, by ({{EquationNote|104}}),
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_i =~\! h_i \mathbf{\hat{h}}_i
  &lt;/math&gt;}}|{{EquationRef|110}}}}
(again with no summation, due to the single index on the left). Hence ({{EquationNote|105}}) becomes:
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^i =~\! \mathbf{\hat{h}}_i \big/ h_i \,.
  &lt;/math&gt;}}|{{EquationRef|111}}}}
A vector field {{math|'''q'''}} is expressed in components w.r.t. the basis{{math| ('''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;)}} as
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q} = \hat{q_i} \;\!\mathbf{\hat{h}}_i
  \qquad&lt;/math&gt;}}|{{EquationRef|112}}}}
(with summation), where
{{NumBlk|:|{{big|&lt;math&gt;
    \hat{q_i} =~\! \mathbf{q} \!\cdot\! \mathbf{\hat{h}}_i \,.
  \qquad&lt;/math&gt;}}|{{EquationRef|113}}}}
(Here the hat on{{math| ''q''&amp;#770;&lt;sub&gt;''i''&lt;/sub&gt;}} is needed to distinguish the coefficient of{{math| '''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;}} from the coefficient of{{math| '''h'''&lt;sup&gt;''i''&lt;/sup&gt;,}} and indicates that{{math| ''q''&amp;#770;&lt;sub&gt;''i''&lt;/sub&gt;}} is the coefficient of a unit vector&amp;mdash;''not'' that{{math| ''q''&amp;#770;&lt;sub&gt;''i''&lt;/sub&gt;}} has unit magnitude.) Taking{{mvar| q&lt;sub&gt;i&lt;/sub&gt;}} as given by ({{EquationNote|83d}}) and applying ({{EquationNote|110}}) and ({{EquationNote|113}}), we get
{{NumBlk|:|{{big|&lt;math&gt;
    q_i =~\! h_i \hat{q_i} \,,
  \qquad&lt;/math&gt;}}|{{EquationRef|114}}}}
whence ({{EquationNote|106}}) gives
{{NumBlk|:|{{big|&lt;math&gt;
    q^i =~\! \hat{q_i} \big/ h_i \,.
  \qquad&lt;/math&gt;}}|{{EquationRef|115}}}}

Equation ({{EquationNote|110}}) quantifies the non-covariance of the orthonormal basis; substituting ({{EquationNote|110}}) into ({{EquationNote|85}}), we find that the components of{{math| ''d'''''r'''}} with respect to{{math| '''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;}}&amp;#8202; are not simply{{math| ''du&lt;sup&gt;i&lt;/sup&gt;'',}} but{{math| ''h&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;du&lt;sup&gt;i&lt;/sup&gt;''}}&amp;#8202; [no sum]. So, as the coordinates{{mvar| u&lt;sup&gt;i&lt;/sup&gt;}} are still contravariant, the orthonormal basis vectors{{math| '''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;}} are not covariant unless the scale factors{{mvar| h&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; are equal to{{math| 1}}&amp;#8202;&amp;mdash;&amp;#8239;that is, unless the coordinates are Cartesian (except possibly for the handedness). And in Cartesian coordinates we can use subscripts throughout. This is another reason why, when using an orthonormal basis, we might as well write the coordinates as{{math| ''u&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;}}.

We can now re-express dot- and cross-products w.r.t. the orthonormal basis{{math| ('''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;)}}. If we apply ({{EquationNote|114}}) and ({{EquationNote|115}}) in ({{EquationNote|88a}}) or ({{EquationNote|88b}}), the scale factors cancel and we are left with
{{NumBlk|:|{{big|&lt;math&gt;\mathbf{v} \cdot \mathbf{q}
    =~\! \hat{v_i} \hat{q_i} \,,
  &lt;/math&gt;}}|{{EquationRef|116}}}}
as if the coordinates were Cartesian. And if we apply ({{EquationNote|108}}), ({{EquationNote|115}}), and ({{EquationNote|111}}) in ({{EquationNote|91a}}), the product of the scale factors cancels and we are left with
:{{big|&lt;math&gt;\mathbf{v} \!\times\! \mathbf{q} =~\!
    \varsigma~\!\epsilon_{ijk\,} \hat{v_i} \hat{q_j} \;\!\mathbf{\hat{h}}_k \,,
  &lt;/math&gt;}}
again as if the coordinates were Cartesian, except that the handedness symbol {{mvar|&amp;varsigma;}}&amp;#8202; gives a change of sign for left-handed coordinates. The last result is confirmed by applying ({{EquationNote|109}}), ({{EquationNote|114}}), and ({{EquationNote|110}}) in ({{EquationNote|91b}}). It can also be written
{{NumBlk|:|&lt;math&gt;\mathbf{v} \!\times\! \mathbf{q} \,=\,
    \varsigma\, \begin{vmatrix}
      \hat{v_1} &amp; \hat{q_1} &amp; \mathbf{\hat{h}}_1 \\
      \hat{v_2} &amp; \hat{q_2} &amp; \mathbf{\hat{h}}_2 \\
      \hat{v_3} &amp; \hat{q_3} &amp; \mathbf{\hat{h}}_3
    \end{vmatrix} \,.
  &lt;/math&gt;|{{EquationRef|117}}}}

We can similarly re-express the first-order differential operators. Applying ({{EquationNote|111}}) in ({{EquationNote|93o}}), ({{EquationNote|101o}}), and ({{EquationNote|100o}}) gives respectively
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla = \tfrac{1}{\,h_{\scriptstyle i}}\;\!\mathbf{\hat{h}}_i \part_i \,,
  &lt;/math&gt;}}|{{EquationRef|118}}}}
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div}
      = \tfrac{1}{\,h_{\scriptstyle i}}\;\!\mathbf{\hat{h}}_i ~\!\!\cdot \part_i \,,
  &lt;/math&gt;}}|{{EquationRef|119}}}}
and
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}
      = \tfrac{1}{\,h_{\scriptstyle i}}\;\!\mathbf{\hat{h}}_i ~\!\!\times \part_i \,.
  &lt;/math&gt;}}|{{EquationRef|120}}}}
And applying ({{EquationNote|115}}) in ({{EquationNote|94o}}) and ({{EquationNote|102d}}) gives
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla
      = \tfrac{1}{\,h_{\scriptstyle i}} \;\!\hat{q_i} \part_i
  &lt;/math&gt;}}|{{EquationRef|121}}}}
and
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{div}\mathbf{q}
    = \tfrac{\,1\,}{J} ~\!\part_i \Big(\tfrac{J}{\,h_{\scriptstyle i}} ~\!\hat{q_i} \Big) \,.
  &lt;/math&gt;}}|{{EquationRef|122}}}}
And applying ({{EquationNote|109}}), ({{EquationNote|110}}), and ({{EquationNote|114}}) in ({{EquationNote|95c}}) gives
:{{big|&lt;math&gt;\operatorname{curl}\mathbf{q}
    = \tfrac{\,1\,}{J} ~\!\epsilon_{ijk\,}
      h_k \mathbf{\hat{h}}_k ~\!\part_i \big(h_j \hat{q_j} \big)
  &lt;/math&gt;}}
or, in determinant form,
{{NumBlk|:|&lt;math&gt;
    \operatorname{curl}\mathbf{q} \,=\, \frac{\varsigma}{h_1 h_2 h_3}\,
      \begin{vmatrix}
        h_1 \mathbf{\hat{h}}_1 &amp; \part_1 &amp; h_1 \hat{q_1} \\
        h_2 \mathbf{\hat{h}}_2 &amp; \part_2 &amp; h_2 \hat{q_2} \\
        h_3 \mathbf{\hat{h}}_3 &amp; \part_3 &amp; h_3 \hat{q_3}
      \end{vmatrix} \,.
  &lt;/math&gt;|{{EquationRef|123}}}}

For the Laplacian, applying ({{EquationNote|111}}) twice in ({{EquationNote|103L}}) gives
:{{big|&lt;math&gt;\triangle\psi =
    \tfrac{\,1\,}{J} ~\!\part_i \Big(
      \tfrac{J}{h_i h_j} (\mathbf{\hat{h}}_i {\cdot}~\! \mathbf{\hat{h}}_j)
      ~\!\part_j \psi
    \Big) \,,
  &lt;/math&gt;}}
where the parenthesized dot-product is simply{{mvar| &amp;delta;&lt;sub&gt;ij&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; Selecting the non-zero terms, we are left with
{{NumBlk|:|{{big|&lt;math&gt;\triangle\psi =
    \tfrac{\,1\,}{J} ~\!\part_i \Big(
      \tfrac{J}{\,h_{\scriptstyle i}^{~2}} ~\!\part_i \psi
    \Big) \,.
  &lt;/math&gt;}}|{{EquationRef|124}}}}

Working entirely within the coordinates{{math| ''u&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;,}} we can use equations ({{EquationNote|118}}), ({{EquationNote|121}}) to ({{EquationNote|123}}), and ({{EquationNote|124}}) for scalar{{math| ''&amp;psi;''&amp;#8202;,}} provided that we know the scale factors in terms of{{mvar| u&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;}}.{{efn|In ({{EquationNote|124}}), if {{mvar|&amp;psi;}}&amp;#8202; is a ''vector'' field, we also need the derivatives of its basis vectors w.r.t.{{mvar| u&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;}} in terms of{{mvar| u&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;}}.}}&amp;#8201; And we can find the scale factors in terms of{{mvar| u&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; if we know the Cartesian coordinates{{mvar| x&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; in terms of{{mvar| u&lt;sub&gt;i&lt;/sub&gt;}}.&amp;#8201; For then the position vector can be written
:{{big|{{math|'''r''' {{=}} ''x&lt;sub&gt;j&lt;/sub&gt;''&amp;#8239;'''e'''&lt;sub&gt;''j''&lt;/sub&gt; ,}}}}
whence
:{{big|{{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt; {{=}} ''&amp;part;&lt;sub&gt;u&lt;sub&gt;i&lt;/sub&gt;&lt;/sub&gt;''&amp;#8239;'''r''' {{=}} ''&amp;part;&lt;sub&gt;u&lt;sub&gt;i&lt;/sub&gt;&lt;/sub&gt;&amp;#8201;x&lt;sub&gt;j&lt;/sub&gt;''&amp;#8201;'''e'''&lt;sub&gt;''j''&lt;/sub&gt; ,}}}}
so that the scale factors can be found from
:{{big|{{math|''h&lt;sub&gt;i&lt;/sub&gt;''&lt;sup&gt;2&lt;/sup&gt; {{=}} &amp;sum;&lt;sub&gt;&amp;#8202;''j'' &lt;/sub&gt;(''&amp;part;&lt;sub&gt;u&lt;sub&gt;i&lt;/sub&gt;&lt;/sub&gt;&amp;#8201;x&lt;sub&gt;j&lt;/sub&gt;'')&lt;sup&gt;2&lt;/sup&gt;.}}}}

In ({{EquationNote|121}}) to ({{EquationNote|123}}), the hat on{{math| ''q''&amp;#770;&lt;sub&gt;''i''&lt;/sub&gt;}} was needed because we treated the orthonormal basis as a special case, having used a hatless{{mvar| q&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; in less special cases; the hat would not have been needed if we had assumed an orthonormal basis at the outset. In more elementary introductions to curvilinear orthogonal coordinates, the basis vectors are indeed chosen as unit vectors and consequently as orthonormal vectors. Hence, if the coordinates are called&lt;math&gt;~u,v,w\,&lt;/math&gt; and the respective basis vectors are called&lt;math&gt;~\mathbf{e}_u , \mathbf{e}_v , \mathbf{e}_w&lt;/math&gt; (understood to be unit vectors), the components of the vector{{math| '''q'''}} w.r.t. that basis are called&lt;math&gt;~q_u , q_v , q_w ~\!,\,&lt;/math&gt; with no hats. In this notation, in which sums are written out longhand without numerical indices, it is convenient also to write out the Jacobian in full, in order to exploit cancellations of scale factors. If the Jacobian appears in both a numerator inside parentheses and a denominator outside, the handedness symbol{{mvar| &amp;varsigma;}}&amp;#8202; also cancels. Thus the equations numbered ({{EquationNote|116}}) to ({{EquationNote|124}}) can be rewritten as, respectively,
{{NumBlk||&lt;math&gt;\begin{align}
    \mathbf{f} \cdot \mathbf{q} ~\!&amp;=~\!
      f_u q_u + f_v q_v + f_w q_w \\[1ex]
    \mathbf{f} ~\!\!\times\! \mathbf{q} ~\!&amp;=~\!
      \varsigma\, \begin{vmatrix}
        f_u &amp; q_u &amp; \mathbf{e}_u \\
        f_v &amp; q_v &amp; \mathbf{e}_v \\
        f_w &amp; q_w &amp; \mathbf{e}_w
      \end{vmatrix} \\[1ex]
    \nabla &amp;=
      \tfrac{1}{~\!h_{\scriptstyle u}}\;\!\mathbf{e}_u \part_u +
      \tfrac{1}{~\!h_{\scriptstyle v}}\;\!\mathbf{e}_v \part_v +
      \tfrac{1}{~\!h_{\scriptstyle w}}\;\!\mathbf{e}_w \part_w \\[.5ex]
    \operatorname{div} &amp;=
      \tfrac{1}{~\!h_{\scriptstyle u}} \mathbf{e}_u ~\!\!\cdot \part_u +
      \tfrac{1}{~\!h_{\scriptstyle v}} \mathbf{e}_v ~\!\!\cdot \part_v +
      \tfrac{1}{~\!h_{\scriptstyle w}} \mathbf{e}_w ~\!\!\cdot \part_w \\[.5ex]
    \operatorname{curl} ~\!&amp;=
      \tfrac{1}{~\!h_{\scriptstyle u}} \mathbf{e}_u ~\!\!\times \part_u +
      \tfrac{1}{~\!h_{\scriptstyle v}} \mathbf{e}_v ~\!\!\times \part_v +
      \tfrac{1}{~\!h_{\scriptstyle w}} \mathbf{e}_w ~\!\!\times \part_w\\[.5ex]
    \mathbf{q}\;\!{\cdot}\nabla &amp;=
      \tfrac{1}{~\!h_{\scriptstyle u}} \;\!q_u \part_u +
      \tfrac{1}{~\!h_{\scriptstyle v}} \;\!q_v \part_v +
      \tfrac{1}{~\!h_{\scriptstyle w}} \;\!q_w \part_w \\[.5ex]
    \operatorname{div}\mathbf{q} ~\!&amp;= \tfrac{1}{h_u h_v h_w} \Big(
        \part_u (h_v h_w q_u) + \part_v (h_w h_u q_v) + \part_w (h_u h_v q_w)
      \Big) \\[.5ex]
    \operatorname{curl}\mathbf{q} ~\!&amp;=~\! \frac{\varsigma}{h_u h_v h_w}\,
      \begin{vmatrix}
        h_u \mathbf{e}_u &amp; \part_u &amp; h_u q_u \\
        h_v \mathbf{e}_v &amp; \part_v &amp; h_v q_v \\
        h_w \mathbf{e}_w &amp; \part_w &amp; h_w q_w
      \end{vmatrix} \\[.5ex]
    \triangle\psi ~\!&amp;=
      \tfrac{1}{h_{\scriptstyle u\;\!}h_{\scriptstyle v\;\!}h_{\scriptstyle w}\!}
      \bigg\{\!
        \tfrac{\part}{\part u\!}
          \Big(\!\tfrac{h_v h_w}{h_u\;\!}\tfrac{\part\psi}{\part u}\!\Big)
        \!+~\!\! \tfrac{\part}{\part v\!}
          \Big(\!\tfrac{h_w h_u}{\;\!h_v}\tfrac{\part\psi}{\part v}\!\Big)
        \!+~\!\! \tfrac{\part}{\part w\!}
          \Big(\!\tfrac{h_u h_v}{\;\!h_w}\tfrac{\part\psi}{\part w}\!\Big)
      \!\bigg\} .
  \end{align}&lt;/math&gt;|{{EquationRef|125}}}}
Only in the cross-product and the curl does the handedness factor{{mvar| &amp;varsigma;}}&amp;#8202; make any difference. If the system is right-handed&amp;mdash;as is also often assumed at the outset&amp;mdash;this factor is replaced by{{math| 1}}.

For some readers, equation group ({{EquationNote|125}}) will announce a return to familiar territory. For the writer, it offers a convenient place to stop.

{{cob}}

== Appendix: Mathematizing Huygens' principle ==

{{cot}}

If a wavelike disturbance originating ''outside''&amp;#8202; a region{{math| ''V'',}} bounded by a surface{{math| ''S''&amp;#8202;,}} enters the region, it must do so through the surface{{mvar| S}}.&amp;#8201; Unless we believe in &quot;action at a distance&quot;, we must conclude that ''the behavior of the wave function throughout the region is fully determined by its behavior on the bounding surface''. That reasoning, being qualitative, does not tell us precisely what aspects of the behavior at the boundary determine the behavior throughout the region, or how. In this appendix, we shall answer these questions using tools of vector analysis. The aim is to express the wave function in the region{{mvar| V}} as a surface integral, over the bounding surface{{math| ''S''&amp;#8202;,}} of an integrand related to the wave function incident at a general point on that surface.

'''[[w:Huygens' principle|Huygens' principle]]''' asserts not only that the behavior of the wave function throughout the region (containing no sources) is determined by the behavior at the boundary, but also that the behavior at the boundary is equivalent to a distribution of sources over the boundary, so that the wave function throughout the region is ''as if''&amp;#8202; the original sources outside the region ('''primary sources''') were ''replaced''&amp;#8202; by sources distributed over the boundary ('''secondary sources''').{{efn|Notice that the desired secondary sources are ''not''&amp;#8202; segments of the moving wavefronts, but segments of a stationary surface influenced by the passing waves. Compare Huygens' original statement: &quot;that ''each particle of matter''&amp;#8202; in which a wave spreads, ought not to communicate its motion only to the next particle which is in the straight line drawn from the luminous point, but that it also imparts some of it necessarily to all the others which touch it and which oppose themselves to its movement. So it arises that around each particle there is made a wave of which ''that particle''&amp;#8202; is the centre&quot; ([[#huygens-1690-thompson|Huygens, 1690, tr.&amp;#8239;Thompson]], p.&amp;#8239;19; my emphasis). Huygens chooses secondary sources on the same primary wavefront at the same time for the purpose of constructing the &quot;continuation&quot; of the wavefront (the same wavefront at a later time) in the same medium (''ibid.'', pp.&amp;#8239;19,&amp;#8239;50–51), but ''not''&amp;#8202; for the purpose of constructing a wavefront reflected or refracted at an interface between two media; for the latter purpose, he chooses secondary sources at various points on the reflecting or refracting surface, although the primary wavefront reaches those points at various times (''ibid.'', pp.&amp;#8239;23–4,&amp;#8239;35–7,&amp;#8239;etc.).}} We shall find that by appropriately arranging the integrand for the wave function inside the region, we can indeed recognize the distribution of boundary sources that would generate the wave function.

{{cob}}

=== Hints ===

{{cot}}

Let {{math|''&amp;psi;''('''r''',&amp;#8239;''t'')}} be the primary wave function, and let {{math|'''r&amp;prime;'''}} be the position of the observation point ('''field point''') at a distance{{mvar| s}}&amp;#8202; from position{{math| '''r'''}}. If the surface integrand is the wave function at{{math| '''r&amp;prime;'''}} due to a secondary source-strength density, it will not only be related to the primary wave function{{mvar| &amp;psi;}} at a general point{{math| '''r'''}} on{{mvar| S}}, but will also be delayed by the propagation time from {{math|'''r'''}} to{{math| '''r&amp;prime;'''}}, and attenuated in accordance with the propagation distance{{mvar| s}}. Hence the integrand (or at least the dominant term thereof) will be proportional to
{{NumBlk|:|{{big|{{math|{{sfrac|&amp;#8239;''s''&amp;#8239;}}&amp;#8202;''&amp;psi;''('''r''', ''t&amp;#8201;&amp;minus;&amp;#8201;s''&amp;#10744;''c'') .}}}}|{{EquationRef|126}}}}
But the necessary operations may cause{{mvar| &amp;psi;}} to be replaced by, e.g., one of its derivatives or a linear combination of its derivatives; such a replacement would be &quot;proportional&quot; to{{mvar| &amp;psi;}} in the requisite sense.

Of course we would like our distribution of secondary sources to be valid for an arbitrarily shaped boundary{{mvar| S}}. This preference will be easier to satisfy if the distribution of secondary sources, by itself, produces a zero wave function outside{{mvar| V}}&amp;#8202;&amp;mdash;in other words, ''no backward secondary waves''&amp;#8202;&amp;mdash;because in that case, even if {{mvar|S}}&amp;#8202; is concave outward, the wave function inside{{mvar| V}}&amp;#8239; will not be complicated by &quot;backward&quot; waves generated at one point on{{mvar| ''S''}}&amp;#8202; and entering{{mvar| V}}&amp;#8202; through another point on{{mvar| S}}.&amp;#8201; Accordingly, we would like our surface integral to be equal to the ''volume''&amp;#8202; integral over{{mvar| V}}&amp;#8239; of
{{NumBlk|:|{{big|{{math|''&amp;psi;''('''r''',&amp;#8239;''t'')&amp;#8239;''&amp;delta;''('''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;''') ,}}}}|{{EquationRef|127}}}}
because that volume integral will be {{math| ''&amp;psi;''('''r&amp;prime;''',&amp;#8239;''t'')}}&amp;#8202; if {{math|'''r&amp;prime;'''}} is inside{{mvar| V}} (where the secondary waves are forward), but zero if it is outside (where any secondary waves are backward).

Relating the volume integral of ({{EquationNote|127}}) to the surface integral of ({{EquationNote|126}}) would seem to require a surface-to-volume '''integral identity''' involving two different fields. Some promising identities are available; but, as we shall see, they tend to treat the two fields symmetrically, and they get simpler if the two fields have more properties in common. We might therefore seek fields with more in common than ({{EquationNote|126}}) and ({{EquationNote|127}}). In the first factor in ({{EquationNote|127}}),&amp;#8201; {{mvar|t}} can be replaced by&amp;#8202; {{math|''t&amp;#8202;&amp;minus;&amp;#8202;s''&amp;#10744;''c''}}&amp;#8202; because the second factor is zero for non-zero{{mvar| s}}. Thus the second factor in ({{EquationNote|127}}), by selecting the time, makes the primary wave function{{math| ''&amp;psi;''('''r''',&amp;#8239;''t'')}} equivalent to the second factor in ({{EquationNote|126}}). That primary wave function is of course a solution of the wave equation in{{mvar| V}}. So, if the factor
:{{big|{{math|''&amp;delta;''('''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;''')}}}}
in ({{EquationNote|127}}) can be replaced by solution of the wave equation with an equivalent &quot;selecting&quot; property, and especially if that solution includes the factor {{math|1&amp;#10744;''s''}} in ({{EquationNote|126}}), perhaps we can pick that solution and the primary wave function as the two fields to substitute into the integral identity. The &quot;solution&quot; that suggests itself is
{{NumBlk|:|{{big|{{math|{{sfrac|&amp;#8239;''s''&amp;#8239;}}&amp;#8202;''&amp;delta;''(''t&amp;#8201;+&amp;#8201;s''&amp;#10744;''c'') ,}}}}|{{EquationRef|128}}}}
where the defining properties of {{math|''&amp;delta;''(''t'')}} are that its integral over all ''time'' is zero and, ideally, that it is zero except at {{math|''t''&amp;#8239;{{=}}&amp;#8202;0}}; but, to ensure that {{math|''&amp;delta;''(''t'')}} is sufficiently differentiable for our purposes, we shall allow it to be a smooth function which is zero except within a negligibly short interval around{{math| ''t''&amp;#8239;{{=}}&amp;#8202;0}}. Solution ({{EquationNote|128}}) describes an ''incoming'' spherical wave converging on{{math| '''r&amp;prime;'''}} [recall the discussion of ({{EquationNote|54a}}) above], which is appropriate because, for an observer at{{math| '''r&amp;prime;'''}}, the secondary waves are incoming; to put it more precisely, the delta function selects the time{{math| ''t&amp;#8239;{{=}}&amp;#8202;&amp;minus;s''&amp;#10744;''c''}}, which is the time of emission of the secondary waves that affect the wave function at{{math| '''r&amp;prime;'''}} at{{math| ''t''&amp;#8239;{{=}}&amp;#8202;0}} (which is a general time, because the origin of{{mvar| t}} is arbitrary). Moreover, as{{math| ''t''&amp;rightarrow;&amp;#8202;0&lt;sup&gt;&amp;minus;&lt;/sup&gt;}}, the temporal delta function in ({{EquationNote|128}}) looks like the spatial delta function in ({{EquationNote|127}}).

So let us tentatively pick the primary wave function {{math| ''&amp;psi;''('''r''',&amp;#8239;''t'')}} and the auxiliary wave function ({{EquationNote|128}}) as the two fields to be related by the integral identity&amp;mdash;which we must now choose.

{{cob}}

=== Green's identities ===

{{cot}}

If&lt;math&gt;~u&lt;/math&gt; and&lt;math&gt;~v&lt;/math&gt; are scalar fields, then by identity ({{EquationNote|71d}}),
:&lt;math&gt;\mathrm{div}(u~\!\nabla v)
    \equiv u~\!\triangle v + \nabla u \cdot~\!\! \nabla v \,.
  &lt;/math&gt;
Integrating both sides over a volume{{math| ''V''}} enclosed by a surface{{math| ''S''&amp;#8202;,}} and applying the divergence theorem on the left, we get
:&lt;math&gt;\iint_S u~\!\nabla v \cdot \mathbf{\hat{n}}~\!dS \,\equiv
    \iiint_V \big(u~\!\triangle v + \nabla u \cdot~\!\! \nabla v \big)~\!dV \,,
  &lt;/math&gt;
where &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is the unit normal to{{mvar| S}}&amp;#8202; pointing out of{{mvar| V}}. This integral equation is called '''[[w:George Green (mathematician)|Green]]'s first identity'''. Switching the roles of&lt;math&gt;~u&lt;/math&gt; and&lt;math&gt;~v&lt;/math&gt; yields a second integral equation, which can be subtracted from the first to obtain
:&lt;math&gt;
    \iint_S \!\big(u~\!\nabla v - v~\!\nabla u \big) \cdot \mathbf{\hat{n}}~\!dS
    \,\equiv \iiint_V \!\big(u~\!\triangle v - v~\!\triangle u \big)~\!dV \,;
  &lt;/math&gt;
this is '''Green's second identity'''. If {{mvar|n}} is the normal distance from{{mvar| S}} (positive outside{{math| ''V'',}} negative inside), then, by relation ({{EquationNote|9g}}) between the gradient and the directional derivative, we can rewrite Green's second identity in the alternative form
{{NumBlk|:|&lt;math&gt;
    \iint_S \!\big(u~\!\part_n v - v~\!\part_n u\big)~\!dS
    \,\equiv \iiint_V \!\big(u~\!\triangle v - v~\!\triangle u\big)~\!dV \,,
  &lt;/math&gt;|{{EquationRef|129}}}}
which remains meaningful if one of the two operands is a ''generic'' field. And indeed, by the linearity of the various operators, the identity remains valid in that case (which is not always pointed out).

{{cob}}

=== Kirchhoff's integral theorem ===

{{cot}}

Now, as foreshadowed above,&lt;ref&gt;The following demonstration of the Kirchhoff integral theorem is indebted to Stratton ([[#stratton-41|1941]], pp.&amp;#8239;424–8), especially as regards the choice of the &quot;auxiliary&quot; wave function &lt;math&gt;v&lt;/math&gt; (my nomenclature) and the insight that the time origin is arbitrary (p.&amp;#8239;427). However, Stratton's treatment does not consider the case with {{math|'''r&amp;prime;'''}} outside{{mvar| V}}, handles the &quot;inside{{mvar|&amp;#8201;V&amp;#8239;}}&quot; case differently, and takes a less heuristic approach, without introductory &quot;hints&quot;.&lt;/ref&gt; let us see what happens if we put
{{NumBlk|:|&lt;math&gt;u = \psi(\mathbf{r},t)&lt;/math&gt;|{{EquationRef|130u}}}}
and
{{NumBlk|:|&lt;math&gt;v = \tfrac{\,1\,}{s}~\!\delta\big(t+s/c\big)&lt;/math&gt;|{{EquationRef|130v}}}}
in ({{EquationNote|129}}). As &lt;math&gt;u&lt;/math&gt; satisfies the wave equation in{{mvar| V}}, we have
{{NumBlk|:|&lt;math&gt;\triangle u
  = \tfrac{1}{c^2} \ddot{u} \,.&lt;/math&gt;|{{EquationRef|131u}}}}
With &lt;math&gt;v&lt;/math&gt;&amp;#8201; we need to be more careful, because &lt;math&gt;v&lt;/math&gt; is undefined at{{math| '''r&amp;prime;'''}}, which may be inside{{mvar| V}}.  By rule ({{EquationNote|54}}), the D'Alembertian of &lt;math&gt;v&lt;/math&gt; (with the {{math|&amp;#9744;}} operator written out in full) is
:&lt;math&gt;\triangle v - \tfrac{1}{c^2} \ddot{v}
  = -4\pi \delta(t)\,\delta(\mathbf{r}{-}\mathbf{r}') \,,&lt;/math&gt;
whence
{{NumBlk|:|&lt;math&gt;\triangle v = \tfrac{1}{c^2} \ddot{v}
  - 4\pi \delta(t)\,\delta(\mathbf{r}{-}\mathbf{r}') \,.
  &lt;/math&gt;|{{EquationRef|131v}}}}
Substituting ({{EquationNote|131u}}) and ({{EquationNote|131v}}) into ({{EquationNote|129}}) gives
:&lt;math&gt;\begin{align}
  &amp;\iint_S \big(u~\!\part_n v - v~\!\part_n u\big)~\!dS \\
  &amp;= \tfrac{1}{c^2}\!\iiint_V (u\ddot{v} - v\ddot{u})~\!dV
     - \!\iiint_V\!4\pi u~\!\delta(t)~\!\delta(\mathbf{r}{-}\mathbf{r}')~\!dV \\
  &amp;= \tfrac{1}{c^2}\!
     \iiint_V \tfrac{\part}{\part t} \big(u\dot{v} - v\dot{u}\big) ~\!dV
     - \!\iiint_V \!4\pi\psi(\mathbf{r},t)~\!\delta(t)
                  ~\!\delta(\mathbf{r}{-}\mathbf{r}') ~\!dV \,,
  \end{align}&lt;/math&gt;
using ({{EquationNote|130u}}) in the last term. In that term we may now set{{math| '''r'''}} to{{math| '''r&amp;prime;'''}} [because {{math|''&amp;delta;''('''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;''')}} is zero elsewhere] and then take the {{math|'''r'''}}-independent factor outside the volume integral, obtaining
:&lt;math&gt;\begin{align}
  \iint_S &amp;\big(u~\!\part_n v - v~\!\part_n u\big)~\!dS \\
  &amp;= \tfrac{1}{c^2}\!
     \iiint_V \tfrac{\part}{\part t} \big(u\dot{v} - v\dot{u}\big) ~\!dV
     - \,4\pi\psi(\mathbf{r}',t)~\!\delta(t)
       ~\!\operatorname{if}(\mathbf{r}'{\in}~\!V) \,,
  \end{align}&lt;/math&gt;
where {{math|if()}} is an ad-hoc function taking the value{{math|&amp;#8201;1}} if its argument is true ({{math|'''r&amp;prime;''' }}is in{{mvar| V&amp;#8202;}}), and{{math| 0}}&amp;#8202; if its argument is false. Then, to eliminate{{math| ''&amp;delta;''(''t'')}}, we integrate w.r.t.{{mvar| t}}&amp;#8202; over all time, obtaining
{{NumBlk|:|&lt;math&gt;\begin{align}
    \iint\limits_{S\;} &amp;\int_{-\infty}^{\infty}
      \!\!\big(u~\!\part_n v - v~\!\part_n u\big)
    dt\;dS \\
    &amp;= \tfrac{1}{c^2}\!\iiint_V\!(u\dot{v}-v\dot{u})\Big|_{-\infty}^{\infty} dV
       - \,4\pi\psi(\mathbf{r}',0)~\!\operatorname{if}(\mathbf{r}'{\in}~\!V)\,.
  \end{align}&lt;/math&gt;|{{EquationRef|132}}}}
In the remaining volume integral, substituting for &lt;math&gt;v&lt;/math&gt; from ({{EquationNote|130v}}), we have
{{NumBlk|:|&lt;math&gt;\begin{align}(u\dot{v}-v\dot{u})\Big|_{-\infty}^{\infty}
    &amp;= \Big(\tfrac{\,u\,}{s}~\!\delta'\!(t+s/c)
            - \tfrac{\,\dot{u}\,}{s}~\!\delta(t+s/c)\Big)
       \bigg|_{t\to-\infty}^{t\to\infty} \\
    &amp;= ~\!0
  \end{align}&lt;/math&gt;|{{EquationRef|133}}}}
because the expression in the big parentheses is zero except where {{math|''t&amp;#8201;&amp;#8776;&amp;#8201;&amp;minus;s''&amp;#10744;''c''}}, and{{mvar| s}} is finite in{{mvar| V}}.  So the volume integral in ({{EquationNote|132}}) vanishes, and what remains is
{{NumBlk|:|&lt;math&gt;
    \iint_{\!S} \!\textstyle\Big\{\!
      \int_{-\infty}^{\infty} \!u~\!\part_n v \,dt
      - \!\int_{-\infty}^{\infty} \!v~\!\part_n u \,dt
    \Big\} ~\!dS
    = - 4\pi\psi(\mathbf{r}'\!,0) ~\!\operatorname{if}(\mathbf{r}'{\in}~\!V)
  &lt;/math&gt;|{{EquationRef|134}}}}
&amp;mdash;which is what we wanted: a surface integral over{{mvar| S}}, equal (up to a scale factor) to the primary wave function at{{math| '''r&amp;prime;'''}} if {{math|'''r&amp;prime;'''}} is inside{{mvar| V}}, but zero if it is outside.

It remains to put the surface integral into a more convenient form, by substituting from ({{EquationNote|130u}}) and ({{EquationNote|130v}}) and simplifying. The second inner time-integral is
:&lt;math&gt;\begin{align}
    \int_{-\infty}^{\infty} \!v~\!\part_n u \,dt
    &amp;= \!\int_{-\infty}^{\infty}\!
         \tfrac{\,1\,}{s}~\!\delta\big(t+s/c\big)~\!\part_n\psi(\mathbf{r},t)
       \,dt \\[.5ex]
    &amp;= \!\int_{-\infty}^{\infty}\!
         \tfrac{\,1\,}{s}~\!\delta\big(t+s/c\big)~\!\part_n\psi(\mathbf{r},-s/c)
       \,dt \\
    &amp;= \tfrac{\,1\,}{s}~\!\part_n\psi(\mathbf{r},-s/c)\!\int_{-\infty}^{\infty}
         \!\delta\big(t+s/c\big)
       \,dt \,,
  \end{align}&lt;/math&gt;
i.e.
{{NumBlk|:|&lt;math&gt;\textstyle
    \int_{-\infty}^{\infty} \!v~\!\part_n u \,dt
    = \frac{\,1\,}{s}~\!\frac{\part\psi}{\part n}\big(\mathbf{r},-s/c\big) \,,
  &lt;/math&gt;|{{EquationRef|135}}}}
where the differentiation w.r.t.{{mvar| n}} does ''not'' account for the variation of{{mvar| s}} with{{mvar| n}}, because the {{mvar|s}}-dependence arises from selecting the time ''after'' the spatial differentiation. For the other time-integral in ({{EquationNote|134}}), however, it's the other way around: we differentiate a function of{{mvar| s}}, treating {{mvar|s}} as a function of{{mvar| n}} (and of two other coordinates which are also parameters of the surface{{mvar| S}}), using the chain rule and the product rule:
:&lt;math&gt;\begin{align}
    \int_{-\infty}^{\infty} \!u~\!\part_n v \,dt
    &amp;= \!\int_{-\infty}^{\infty}\!
         \psi(\mathbf{r},t)\,
         \part_n\!\Big(\!\tfrac{\,1\,}{s}~\!\delta(t+s/c)\Big)
       \,dt \\[.5ex]
    &amp;= \!\int_{-\infty}^{\infty}\!
         \psi(\mathbf{r},t)\,
         \part_s\!\Big(\!\tfrac{\,1\,}{s}~\!\delta(t+s/c)\Big)~\!
         \tfrac{\part s}{\part n}
       \,dt \\[.5ex]
    &amp;= \!\int_{-\infty}^{\infty}\!
         \psi(\mathbf{r},t)
         \Big(\tfrac{1}{cs}~\!\delta'\!(t+s/c)
              -\tfrac{1}{s^2}~\!\delta(t+s/c)\Big)
         \tfrac{\part s}{\part n}
       \,dt \\[.5ex]
    &amp;= \!\int_{-\infty}^{\infty}\!
         \tfrac{1}{cs}~\!\tfrac{\part s}{\part n}~\!\psi(\mathbf{r},t)
         ~\delta'\!(t+s/c)
       \,dt \\[.5ex]
    &amp;~~~~~- \int_{-\infty}^{\infty}\!
              \tfrac{1}{s^2}~\!\tfrac{\part s}{\part n}~\!\psi(\mathbf{r},t)
              \,\delta(t+s/c)
            \,dt \,.
  \end{align}&lt;/math&gt;
Expanding the first integral by parts, and processing the delta function in the second integral in the usual manner, we get
:&lt;math&gt;\begin{align}
    \int_{-\infty}^{\infty} \!u~\!\part_n v \,dt =\;
    &amp;\Big(\tfrac{1}{cs}~\!\tfrac{\part s}{\part n}~\!\psi(\mathbf{r},t)
         ~\delta(t+s/c)\Big)\bigg|_{-\infty}^{\infty} \\
    &amp; - \!\int_{-\infty}^{\infty}\!
          \tfrac{1}{cs}~\!\tfrac{\part s}{\part n}~\!\dot{\psi}(\mathbf{r},t)
          ~\delta(t+s/c)
        \,dt \\[.5ex]
    &amp; - \tfrac{1}{s^2}~\!\tfrac{\part s}{\part n}~\!\psi(\mathbf{r},-s/c) \,,
  \end{align}&lt;/math&gt;
in which we can now process the remaining delta functions to obtain
:&lt;math&gt;\textstyle\int_{-\infty}^{\infty} \!u~\!\part_n v \,dt \,=\,
    0 - \tfrac{1}{cs}~\!\tfrac{\part s}{\part n}~\!\dot{\psi}(\mathbf{r},-s/c)
      - \tfrac{1}{s^2}~\!\tfrac{\part s}{\part n}~\!\psi(\mathbf{r},-s/c) \,.
  &lt;/math&gt;
Substituting this and ({{EquationNote|135}}) into ({{EquationNote|134}}), renaming the (arbitrary) time origin as time{{mvar| t}}, and multiplying through by{{math| &amp;minus;1}}, we get the desired result:
{{NumBlk|:|&lt;math&gt;\begin{align}
    \iint_{S} \!\Big\{\!
    &amp; \tfrac{1}{cs} \tfrac{\part s}{\part n}
      ~\!\dot{\psi}\big(\mathbf{r},t{-}\tfrac{s}{c}\big)
      + \tfrac{1}{s^2} \tfrac{\part s}{\part n}
        ~\!\psi\big(\mathbf{r},t{-}\tfrac{s}{c}\big)
      + \tfrac{\,1\,}{s}~\!
        \tfrac{\part\psi}{\part n}\big(\mathbf{r},t{-}\tfrac{s}{c}\big)
    \Big\} ~\!dS \\
    &amp;=\, 4\pi\psi(\mathbf{r}',t) ~\!\operatorname{if}(\mathbf{r}'{\in}~\!V)\,.
  \end{align}&lt;/math&gt;|{{EquationRef|136}}}}
This is more usually written
:&lt;math&gt;
    \iint_{S} \!\Big\{\!
      \tfrac{1}{cs} \tfrac{\part s}{\part n} ~\!\big[\dot{\psi}\big]
      + \tfrac{1}{s^2} \tfrac{\part s}{\part n} ~\![\psi]
      + \tfrac{\,1\,}{s} \Big[\tfrac{\part\psi}{\part n}\Big]
    \Big\} ~\!dS
    \,=\, 4\pi\psi(\mathbf{r}',t) ~\!\operatorname{if}(\mathbf{r}'{\in}~\!V)\,,
  &lt;/math&gt;
where the square brackets indicate that the contents are to be ''delayed'' (or, in older literature, &quot;retarded&quot;) by the propagation time from {{math|'''r'''}} to{{math| '''r&amp;prime;'''}}&amp;mdash;that is, delayed by{{math| ''s''&amp;#10744;''c''}}&amp;#8202; relative to the default arguments{{math| ('''r''',&amp;#8239;''t'')}}.

It is common to write &lt;math&gt;-\tfrac{\part}{\part n}\big(\!\tfrac{\,1\,}{s}\!\big)&lt;/math&gt;&amp;#8202; instead of&amp;#8201; &lt;math&gt;\tfrac{1}{s^2}\tfrac{\part s}{\part n}&lt;/math&gt; (reversing the chain rule), so that the last result becomes
{{NumBlk|:|&lt;math&gt;
    \iint_{S} \!\Big\{\!
      \tfrac{1}{cs} \tfrac{\part s}{\part n} ~\!\big[\dot{\psi}\big]
      - [\psi]~\!\tfrac{\part}{\part n}\big(\!\tfrac{\,1\,}{s}\!\big)
      + \tfrac{\,1\,}{s} \Big[\tfrac{\part\psi}{\part n}\Big]
    \Big\} ~\!dS
    \,=\, 4\pi\psi(\mathbf{r}',t) ~\!\operatorname{if}(\mathbf{r}'{\in}~\!V)\,.
  &lt;/math&gt;|{{EquationRef|137}}}}

Although we derived ({{EquationNote|137}}) by supposing that {{mvar|V &amp;#8202;}}is a ''finite''&amp;#8202; region, we can extend the result to an infinite region by adding another sheet to the bounding surface{{mvar| S}}&amp;#8202; in such a way that (i) the region becomes finite, but (ii) the additional sheet makes no contribution to the surface integral. The simplest way to do this is to suppose that the additional sheet is at such a large distance that the disturbance has not reached it yet! Alternatively, we can consider how the wave function decays with distance.&lt;ref&gt;[[#baker-copson-39|Baker &amp;amp; Copson, 1939]], pp.&amp;#8239;37–8.&lt;/ref&gt; By such methods we can apply ({{EquationNote|137}}) not only to the region inside a closed surface, but also (e.g.) to the region outside a closed surface, or the region on one side of an infinite open surface.

Although we derived ({{EquationNote|137}}) by supposing, as usual in this paper, that&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; points out of{{mvar| V}}&amp;#8239; and that {{mvar|n &amp;#8202;}}is measured out of{{mvar| V}}, this has the arguably counterintuitive implication that&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is typically against the direction of propagation&amp;mdash;''directly'' against it in the simplest case, in which {{mvar|V}}&amp;#8202; is the exterior of a sphere with a monopole source at its center.

So, in the following formal statement of our result, let us drop the symbol {{mvar|V}}&amp;#8202; and define {{mvar|n}}&amp;#8202; as being measured out of the region containing the sources, and consequently ''into'' the region that satisfies the homogeneous wave equation, ''changing the signs''&amp;#8202; on the left side of ({{EquationNote|137}}).

'''[[w:Gustav Kirchhoff|Kirchhoff]]'s integral theorem''':  If
* the wave function {{mvar|&amp;psi;}}&amp;#8202; satisfies the wave equation (with speed{{mvar| c}}) in a region{{mvar| R}}&amp;#8202; bounded by a surface{{mvar| S}}&amp;#8202; (with all sources consequently on the other side of{{mvar| S&amp;#8202;}}), and
* {{mvar|s}}&amp;#8202; is the distance of the general point at position{{math| '''r'''}}&amp;#8202; from the observation point at position{{math| '''r&amp;prime;''',&amp;#8202;}} and
* quantities in square brackets are to be delayed by{{math| ''s''&amp;#10744;''c''&amp;#8202;,}} and
* {{mvar|n}}&amp;#8202; is the normal coordinate measured from the general point on{{mvar| S}}&amp;#8202; ''into''{{mvar| R}}&amp;#8202; [contrary to the usual direction for a named region, and contrary to the convention we have used above!],
then the expression
{{NumBlk|:|&lt;math&gt;
    \tfrac{1}{4\pi} \!\iint_{S} \!\Big\{\!
      [\psi]~\!\tfrac{\part}{\part n}\big(\!\tfrac{\,1\,}{s}\!\big)
      - \tfrac{1}{cs} \tfrac{\part s}{\part n} ~\!\big[\dot{\psi}\big]
      - \tfrac{\,1\,}{s} \Big[\tfrac{\part\psi}{\part n}\Big]
    \Big\} ~\!dS
  &lt;/math&gt;|{{EquationRef|138}}}}
is equal to the wave function at{{math| '''r&amp;prime;'''}}&amp;#8202; if{{math| '''r&amp;prime;'''}}&amp;#8202; is inside{{math| ''R''&amp;#8202;,}} but zero if it is outside.&lt;ref&gt;[[#born-wolf-02|Born &amp;amp; Wolf, 2002]], pp.&amp;#8239;420–21, eq.&amp;#8239;(13).&amp;#8201; ''Cf''. Baker &amp;amp; Copson ([[#baker-copson-39|1939]], p.&amp;#8239;37) and Miller ([[#miller-91|1991]], eq.&amp;#8239;2), who use {{mvar|r}}&amp;#8202; instead of{{mvar| s}}&amp;#8202; (among other notational differences). Baker &amp;amp; Copson, in their last equation on p.&amp;#8239;40, give the opposite sign because on this occasion they measure the normal coordinate&lt;math&gt;~\nu&lt;/math&gt; ''out'' of the region.&lt;/ref&gt;

The above derivation does not assume sinusoidal time-dependence at any stage. An alternative approach&lt;ref&gt;E.g., [[#baker-copson-39|Baker &amp;amp; Copson, 1939]], pp.&amp;#8239;36–7; [[#born-wolf-02|Born &amp;amp; Wolf, 2002]], pp.&amp;#8239;420–21.&lt;/ref&gt; is to derive the special case for sinusoidal time-dependence (due to [[w:Hermann von Helmholtz|Helmholtz]]) from Green's identities, and then generalize the time-dependence; this method has the advantage of being more readily applicable to ''dispersive''&amp;#8202; media (in which {{mvar|c &amp;#8202;}}is frequency-dependent), but the disadvantages of depending on complex numbers and on the premise that a general function of time can be expressed as a sum of sinusoids. Helmholtz's integrand is a sinusoidal version of our expression ({{EquationNote|139}}) below. That expression, and thence the Kirchhoff integral, can be obtained in a far more elementary manner, albeit with some loss of rigor, by ''assuming'' (instead of justifying) the form of the wave function due to a monopole source. From this, together with considerations of causality and superposition, we can work out the required distribution of secondary sources and then expresses the wave function as a surface integral.&lt;ref&gt;[[#putland-22|Putland, 2022&amp;ndash;]].&lt;/ref&gt; In the present paper, however, we argue in the other direction: from the integral to the secondary sources.

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=== Monopole and dipole secondary sources ===

{{cot}}

If the dependence on{{math| '''r'''}} is taken as implicit, the integrand inside the braces in ({{EquationNote|138}}) can be written out as
:&lt;math&gt;\begin{align}
    \psi&amp;\big(t\!-\!s/c\big)\,\part_n \tfrac{\,1\,}{s}
    - \tfrac{1}{cs}~\!\part_n s \,\psi'\big(t\!-\!s/c\big)
    - \tfrac{\,1\,}{s}~\!\tfrac{\part\psi}{\part n}\big(t\!-\!s/c\big) \\
    &amp;= \psi\big(t\!-\!s/c\big)\,\part_n \tfrac{\,1\,}{s}
       + \tfrac{\,1\,}{s}~\!\psi'\big(t\!-\!s/c\big)~\!
         \big({-}1/c\big)~\!\part_n s
       - \tfrac{\,1\,}{s}~\!\tfrac{\part\psi}{\part n}\big(t\!-\!s/c\big) \\
    &amp;= \psi\big(t\!-\!s/c\big)\,\part_n \tfrac{\,1\,}{s}
       + \tfrac{\,1\,}{s}~\!\part_n \psi\big(t\!-\!s/c\big)
       - \tfrac{\,1\,}{s}~\!\tfrac{\part\psi}{\part n}\big(t\!-\!s/c\big)
  \end{align}&lt;/math&gt;
or, recognizing the first two terms as the derivative of a product,
{{NumBlk|:|&lt;math&gt;
    \part_n \Big(\tfrac{\,1\,}{s}~\!\psi\big(t\!-\!s/c\big)\Big)
    - \tfrac{\,1\,}{s}~\!\tfrac{\part\psi}{\part n}\big(t\!-\!s/c\big) \,,
  &lt;/math&gt;|{{EquationRef|139}}}}
where {{mvar|&amp;part;&lt;sub&gt;n&lt;/sub&gt;}} accounts for the variation of {{mvar|s}}&amp;#8202; through{{mvar| n}}, but {{mvar|{{sfrac|&amp;part;&amp;psi;|&amp;part;n}}}} does not [see remarks after ({{EquationNote|135}}) above]. If{{mvar| h}} is a ''small''&amp;#8202; change in{{math| ''n''&amp;#8202;,}} from&amp;#8201; {{mvar|n&amp;#8201;{{=}}&amp;#8201;&amp;minus;h}}&amp;#8201; to&amp;#8201; {{math|''n''&amp;#8201;{{=}}&amp;#8201;0&amp;#8202;,}} the integrand can be written
{{NumBlk|:|&lt;math&gt;
    h~\!\part_n \bigg(\frac{\,1\,}{s}~\!\frac{\psi\big(t\!-\!s/c\big)}{h}\bigg)
    - \tfrac{\,1\,}{s}~\!\tfrac{\part\psi}{\part n}\big(t\!-\!s/c\big) \,.
  &lt;/math&gt;|{{EquationRef|140}}}}
The second term (including the minus sign) is recognizable as the contribution to the wave function from a monopole source with strength{{mvar| &amp;minus;{{sfrac|&amp;part;&amp;psi;|&amp;part;n}}&amp;#8202;}}.&lt;ref&gt;''Reminder&amp;#8202;:''&amp;#8201; There are rival definitions of the &quot;strength&quot; of a monopole source; see the text and footnote under equation ({{EquationNote|54}}) above.&lt;/ref&gt; Similarly, in the first term, the expression in the big parentheses is the contribution from a monopole source with strength{{math| ''&amp;psi;''&amp;#10744;''h''&amp;#8202;}}; and the operator {{mvar|h&amp;#8202;&amp;part;&lt;sub&gt;n&lt;/sub&gt;}} gives the change in that contribution due to{{mvar| n}}&amp;#8202; increasing from {{mvar|&amp;minus;h}}&amp;#8202; to{{math| 0&amp;#8202;,}}&amp;#8201; i.e. the change in that contribution due to moving the said monopole from&amp;#8201; {{mvar|n&amp;#8201;{{=}}&amp;#8201;&amp;minus;h}}&amp;#8201; to&amp;#8201; {{math|''n''&amp;#8201;{{=}}&amp;#8201;0&amp;#8202;,}}&amp;#8201; i.e. the whole contribution due to the combination of a monopole with strength{{math| &amp;minus;''&amp;psi;''&amp;#10744;''h''}}&amp;#8202; at&amp;#8201; {{mvar|n&amp;#8201;{{=}}&amp;#8201;&amp;minus;h}}&amp;#8201; and a monopole with strength{{math| ''&amp;psi;''&amp;#10744;''h''}}&amp;#8202; at&amp;#8201; {{math|''n''&amp;#8201;{{=}}&amp;#8201;0}}. This combination is called a '''dipole''' (or ''doublet'')&lt;ref&gt;The term ''doublet'', which seems to be older, is used by Baker &amp;amp; Copson ([[#baker-copson-39|1939]]), Born &amp;amp; Wolf ([[#born-wolf-02|2002, p.&amp;#8239;421]]), and Larmor ([[#larmor-1904|1904]]).&lt;/ref&gt; with strength{{mvar| &amp;psi;}}&amp;#8202; in the normal ({{mvar|n}}) direction.

According to ({{EquationNote|138}}), the expression ({{EquationNote|140}}) is to be scaled by {{math|{{sfrac|4''&amp;pi;''}}}}&amp;#8202; and integrated over the surface{{mvar| S}}. Thus the secondary source distribution can be described as a monopole distribution of strength density {{math|&amp;minus;{{sfrac|4''&amp;pi;''}}{{sfrac|''&amp;part;&amp;psi;''|''&amp;part;n''}}}}&amp;#8239; plus a normal dipole distribution of strength density{{math| {{sfrac|''&amp;psi;''|4''&amp;pi;''}}&amp;#8202;,}} where &quot;strength density&quot; means strength per unit area. This description is well known.&lt;ref&gt;E.g., [[#born-wolf-02|Born &amp;amp; Wolf, 2002]], p.&amp;#8239;421.&lt;/ref&gt;

The implication is not that the specified secondary sources really exist, or even that they ''could''&amp;#8202; exist, but only that the wave function in the region{{mvar| R}}&amp;#8202; is ''as if''&amp;#8202; it had been generated by the specified secondary sources (which would also give a null wave function outside the region). We should note, however, that a monopole contribution of the form ({{EquationNote|48}}) ''can''&amp;#8202; really exist, even for a vector wave function, notwithstanding that it requires not only the magnitude but also the direction of the vector to be independent of the direction of propagation. That requirement might seem to exclude electromagnetic waves, for which the electric and magnetic fields are transverse to the direction of propagation and therefore not independent of it. But it is possible to describe such waves in terms of an electric scalar potential and a magnetic vector potential, such that the contribution to the latter from a current element has the same direction as the current element for all directions of propagation.&lt;ref&gt;[[#stratton-41|Stratton, 1941]], pp.&amp;#8239;428–30.&lt;/ref&gt;

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=== Spatiotemporal-dipole secondary sources ===

{{cot}}

The &quot;dipole&quot; discussed so far is a ''spatial''&amp;#8202; dipole, in which the constituent monopoles differ only in sign and by a small spatial displacement. In the Helmholtz–Kirchhoff integrand ({{EquationNote|139}}), the second term (including the sign) represents a monopole strength density{{mvar| &amp;minus;{{sfrac|&amp;part;&amp;psi;|&amp;part;n}}}}&amp;#8202; and the first term represents a spatial dipole strength density{{mvar| &amp;psi;}}&amp;#8202; in the {{mvar|n }}direction; the dipole source per unit area of{{mvar| S}}&amp;#8202; comprises a monopole with strength{{math| &amp;minus;''&amp;psi;''&amp;#10744;''h''}}&amp;#8202; at&amp;#8201; {{mvar|n&amp;#8201;{{=}}&amp;#8201;&amp;minus;h}}&amp;#8202; (the ''inverted''&amp;#8202; monopole), and a monopole with strength{{math| ''&amp;psi;''&amp;#10744;''h''}}&amp;#8202; at&amp;#8201; {{math|''n''&amp;#8201;{{=}}&amp;#8201;0}} (the ''uninverted''&amp;#8202; monopole), where {{mvar|h}}&amp;#8202; is small (and the indicated strength densities are eventually to be divided by{{math| 4''&amp;pi;''}}).

If there is only a '''single monopole primary source''', this combination of a monopole and a spatial dipole is exactly equivalent to a modified dipole in which the inverted monopole has a certain fixed delay, and a certain fixed attenuation, relative to the uninverted monopole.&lt;ref&gt;The derivation of this &quot;generalized spatiotemporal dipole&quot; (GSTD) was first given in ver.&amp;#8239;0.3 of [[#putland-22|Putland, 2022&amp;ndash;]] (&amp;sect;&amp;#8239;3.7). It was included in earlier versions of the present paper, but is now more conveniently available in a much smaller document ([[#putland-25|Putland, 2025]]).&lt;/ref&gt;

If, in addition, the surface {{mvar|S}}&amp;#8202; coincides with a primary wavefront, the required &quot;fixed delay&quot; is simply{{math| ''h''&amp;#10744;''c''&amp;#8202;}}, i.e. the propagation time from the uninverted monopole to the inverted one. If the primary wavefronts are plane (for a general{{mvar| S&amp;#8202;}}), the inverted monopole should be unattenuated. If {{mvar|S}}&amp;#8202; coincides with a primary wavefront ''and''&amp;#8202; is plane (a large-{{mvar|r}} approximation), the modified dipole reduces to what D.A.B.&amp;#8239;Miller called a '''spatiotemporal dipole''',&lt;ref&gt;[[#miller-91|Miller, 1991]].&lt;/ref&gt; in which the only modification of the spatial dipole is the delay{{math| ''h''&amp;#10744;''c''}}.

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=== Application to diffraction by an aperture ===

{{cot}}

Suppose that the primary sources are partly obstructed by an opaque baffle with an aperture in it. What is the wave function that propagates beyond the baffle? Let us choose a surface{{mvar| S}}&amp;#8202; consisting of two segments, namely {{mvar|S&lt;sub&gt;a&lt;/sub&gt; }}spanning the aperture, and {{mvar|S&lt;sub&gt;b&lt;/sub&gt; }}on the side of the baffle facing away from the sources (the dark side or quiet side of the baffle). The obvious way to proceed is to suppose that the baffle simply eliminates the secondary sources on{{mvar| S&lt;sub&gt;b&lt;/sub&gt;}} while leaving the secondary sources on{{mvar| S&lt;sub&gt;a&lt;/sub&gt;}} unchanged (as if the baffle were not there). The result, as far as the wave function in{{mvar| R}} (beyond the baffle) is concerned, is simply that the integral is taken over {{mvar|S&lt;sub&gt;a&lt;/sub&gt; }}only.

Integrating over the aperture alone is indeed the standard answer, but there are various other ways of explaining it. Some explanations, including the famously inconsistent one offered by Kirchhoff himself, are discussed in [[#putland-22|Putland, 2022&amp;ndash;]] (&amp;sect;&amp;#8239;2.2 and Appendices A &amp;amp; B), and the references therein.

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== Acknowledgment ==

This learning resource uses images from ''Wikimedia Commons''.

== Notes ==

{{cot}}

{{notelist|30em}}

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== Citations ==

{{cot}}

{{reflist|19em}}

{{cob}}

== References ==

{{cot}}

&lt;div style=&quot;font-size: 111%&quot;&gt;

{{refbegin|indent=yes}}

*&lt;span id=&quot;axler-95&quot;&gt;S.J. Axler, 1995, &quot;Down with Determinants!&quot;&amp;#8201; ''American Mathematical Monthly'', vol.&amp;#8239;102, no.&amp;#8239;2 (Feb.&amp;#8239;1995), pp.&amp;#8239;139–54; [https://www.jstor.org/stable/2975348 jstor.org/stable/2975348].&amp;#8201; (Author's preprint, with different pagination: [https://www.researchgate.net/publication/265273063_Down_with_Determinants researchgate.net/publication/265273063_Down_with_Determinants].)&lt;/span&gt;
*&lt;span id=&quot;axler-23-&quot;&gt;S.J. Axler, 2023–, ''Linear Algebra Done Right'', 4th Ed., Springer; [https://linear.axler.net/ linear.axler.net] (open access).&lt;/span&gt;
*&lt;span id=&quot;baker-copson-39&quot;&gt;B.B. Baker and E.T. Copson, 1939, ''The Mathematical Theory of&amp;#8202; Huygens' Principle'', Oxford; 3rd Ed.&amp;#8201;(same pagination, with addenda), New York: Chelsea, 1987, [https://archive.org/details/mathematicaltheo0000bake archive.org/details/mathematicaltheo0000bake].&lt;/span&gt;
*&lt;span id=&quot;borisenko-tarapov-68&quot;&gt;A.I. Borisenko and I.E.&amp;#8239;Tarapov (tr.&amp;#8239;&amp;amp; ed. R.A.&amp;#8239;Silverman), 1968, ''Vector and Tensor Analysis with Applications'', Prentice-Hall; reprinted New York: Dover, 1979, [https://archive.org/details/vectortensoranal0000bori archive.org/details/vectortensoranal0000bori].&lt;!-- Typo on p.180: First cross in equation before (4.93) should be &quot;=&quot;. --&gt;&lt;/span&gt;
*&lt;span id=&quot;born-wolf-02&quot;&gt;M.&amp;#8201;Born and E.&amp;#8239;Wolf, 2002, ''Principles of Optics'', 7th Ed., Cambridge, 1999 (reprinted with corrections, 2002).&lt;/span&gt;
*&lt;span id=&quot;broyden-75&quot;&gt;C.G. Broyden, 1975, ''Basic Matrices'', London: Macmillan.&lt;/span&gt;
*&lt;span id=&quot;feynman-63&quot;&gt;R.P. Feynman, R.B. Leighton, &amp;amp; M.&amp;#8239;Sands, 1963 etc., ''The Feynman Lectures on Physics'', California Institute of Technology; [http://www.feynmanlectures.caltech.edu/ feynmanlectures.caltech.edu].&lt;/span&gt;
*&lt;span id=&quot;fletcher-74&quot;&gt;N.H. Fletcher, 1974, &quot;Adiabatic assumption for wave propagation&quot;, ''American Journal of Physics'', vol.&amp;#8239;42, no.&amp;#8239;6 (June 1974), pp.&amp;#8239;487–9; [https://doi.org/10.1119/1.1987757 doi.org/10.1119/1.1987757].&lt;/span&gt;
*&lt;span id=&quot;gibbs-1881-4&quot;&gt;J.W. Gibbs, 1881–84, &quot;Elements of Vector Analysis&quot;, privately printed New Haven: Tuttle, Morehouse &amp;amp; Taylor, 1881 (&amp;sect;&amp;sect;&amp;#8239;1–101), 1884 (&amp;sect;&amp;sect;&amp;#8239;102–189, etc.), [https://archive.org/details/elementsvectora00gibb archive.org/details/elementsvectora00gibb]; published in ''The Scientific Papers of J.&amp;#8239;Willard Gibbs'' (ed. H.A.&amp;#8239;Bumstead &amp;amp; R.G.&amp;#8239;Van Name), New York: Longmans, Green, &amp;amp; Co., 1906, vol.&amp;#8239;2, [https://archive.org/details/scientificpapers02gibbuoft archive.org/details/scientificpapers02gibbuoft], pp.&amp;#8239;17–90.&lt;/span&gt;
*&lt;span id=&quot;hsu-84&quot;&gt;H.P. Hsu, 1984, ''Applied Vector Analysis'', Harcourt Brace Jovanovich; [https://archive.org/details/appliedvectorana00hsuh archive.org/details/appliedvectorana00hsuh].&lt;/span&gt;
*&lt;span id=&quot;huygens-1690-thompson&quot;&gt;C. Huygens, 1690, tr. S.P.&amp;#8239;Thompson, ''Treatise on Light'', University of Chicago Press, 1912 / [https://gutenberg.org/files/14725/14725-h/14725-h.htm gutenberg.org/files/14725/14725-h/14725-h.htm], 2005. (See also &quot;Errata in various editions of Huygens' ''Treatise on Light''&amp;#8239;&quot;, ''www.grputland.com'' or ''grputland.blogspot.com'', June 2016.)&lt;/span&gt;
*&lt;span id=&quot;katz-79&quot;&gt;V.J. Katz, 1979, &quot;The history of Stokes' theorem&quot;, ''Mathematics Magazine'', vol.&amp;#8239;52, no.&amp;#8239;3 (May 1979), pp.&amp;#8239;146–56; [https://www.jstor.org/stable/2690275 jstor.org/stable/2690275].&lt;/span&gt;
*&lt;span id=&quot;kemin-et-al-00&quot;&gt;S. Kemin, X.&amp;#8239;Zhenting, T.&amp;#8239;Jinsheng, &amp;amp; H.&amp;#8239;Xuemei, 2000, &quot;The comprehension, some problems and suggestions to symbolic vector method and some defenses for Gibbs' symbol&quot;, ''Applied Mathematics and Mechanics'' (English Ed.), vol.&amp;#8239;21, no.&amp;#8239;5 (May 2000), pp.&amp;#8239;603–6; [https://doi.org/10.1007/BF02459044 doi.org/10.1007/BF02459044].&lt;/span&gt;
*&lt;span id=&quot;kemmer-77&quot;&gt;N. Kemmer, 1977, ''Vector Analysis: A physicist's guide to the mathematics of fields in three dimensions'', Cambridge; [https://archive.org/details/isbn_0521211581 archive.org/details/isbn_0521211581].&lt;/span&gt;
*&lt;span id=&quot;kreyszig-62-&quot;&gt;E. Kreyszig, 1962 etc., ''Advanced Engineering Mathematics'', New York: Wiley;&amp;#8201; 5th Ed., 1983;&amp;#8201; 6th Ed., 1988;&amp;#8201; 9th Ed., 2006;&amp;#8201; 10th Ed., 2011.&lt;/span&gt;
*&lt;span id=&quot;larmor-1904&quot;&gt;J. Larmor, 1904, &quot;On the mathematical expression of the principle of&amp;#8202; Huygens&quot; (read 8 Jan.&amp;#8239;1903), ''Proceedings of the London Mathematical Society'', Ser.&amp;#8239;2, vol.&amp;#8239;1 (1904), pp.&amp;#8239;1–13.&lt;!-- Listed as &quot;Issue 1&quot;; only issue for that volume. --&gt;&lt;/span&gt;
*&lt;span id=&quot;miller-91&quot;&gt;D.A.B. Miller, 1991, &quot;Huygens's wave propagation principle corrected&quot;, ''Optics Letters'', vol.&amp;#8239;16, no.&amp;#8239;18 (15 Sep.&amp;#8239;1991), pp.&amp;#8239;1370–72; [http://ee.stanford.edu/~dabm/146.pdf stanford.edu/~dabm/146.pdf].&lt;/span&gt;
*&lt;span id=&quot;moon-spencer-65&quot;&gt;P.H. Moon and D.E.&amp;#8239;Spencer, 1965, ''Vectors'', Princeton, NJ: Van Nostrand.&lt;/span&gt;
*&lt;span id=&quot;panofsky-phillips-62&quot;&gt;W.K.H. Panofsky and M.&amp;#8239;Phillips, 1962, ''Classical Electricity and Magnetism'', 2nd Ed., Addison-Wesley; reprinted Mineola, NY: Dover, 2005.&lt;/span&gt;
*&lt;span id=&quot;putland-22&quot;&gt;G.R. Putland, 2022&amp;ndash;, &quot;Consistent derivation of Kirchhoff's integral theorem and diffraction formula and the Maggi-Rubinowicz transformation using high-school math&quot; (working paper), [https://doi.org/10.5281/zenodo.7205781 doi.org/10.5281/zenodo.7205781] (Creative Commons).&lt;/span&gt;
*&lt;span id=&quot;putland-25&quot;&gt;G.R. Putland, 2025, &quot;Exact formulation of Huygens' principle in terms of generalized spatiotemporal-dipole secondary sources&quot;, [https://doi.org/10.48550/arXiv.2510.20825 doi.org/10.48550/arXiv.2510.20825] (Creative Commons).&lt;/span&gt;
*&lt;span id=&quot;rocci-20&quot;&gt;A. Rocci, 2020, &quot;Back to the roots of vector and tensor calculus: Heaviside versus Gibbs&quot; (online 10 Nov.&amp;#8239;2020), ''Archive for History of Exact Sciences'', vol.&amp;#8239;75, no.&amp;#8239;4 (July 2021), pp.&amp;#8239;369–413. (Author's preprint, with different pagination: [https://arxiv.org/abs/2010.09679 arxiv.org/abs/2010.09679].)&lt;/span&gt;
*&lt;span id=stratton-41&gt;J.A. Stratton, 1941, ''Electromagnetic Theory'', New York: McGraw-Hill; [https://archive.org/details/electromagnetict0000juli archive.org/details/electromagnetict0000juli].&lt;/span&gt;
*&lt;span id=&quot;tai-94&quot;&gt;C.-T. Tai, 1994, &quot;A survey of the improper use of &amp;nabla; in vector analysis&quot; (Technical Report RL&amp;#8239;909), Dept.&amp;#8201;of Electrical Engineering &amp;amp; Computer Science, University of Michigan; [https://deepblue.lib.umich.edu/handle/2027.42/7869 hdl.handle.net/2027.42/7869].&lt;/span&gt;
*&lt;span id=&quot;tai-95&quot;&gt;C.-T. Tai, 1995, &quot;A historical study of vector analysis&quot; (Technical Report RL&amp;#8239;915), Dept.&amp;#8201;of Electrical Engineering &amp;amp; Computer Science, University of Michigan; [https://deepblue.lib.umich.edu/handle/2027.42/7868 hdl.handle.net/2027.42/7868].&lt;/span&gt;
*&lt;span id=&quot;tai-fang-91&quot;&gt;C.-T. Tai and N.&amp;#8239;Fang, 1991, &quot;A systematic treatment of vector analysis&quot;, ''{{serif|IEEE}} Transactions on Education'', vol.&amp;#8239;34, no.&amp;#8239;2 (May 1991), pp.&amp;#8239;167–74; [https://doi.org/10.1109/13.81596 doi.org/10.1109/13.81596].&lt;/span&gt;
*&lt;span id=&quot;wilson-1901&quot;&gt;E.B.&amp;#8201;Wilson, 1901, ''Vector Analysis: A text-book for the use of students of mathematics and physics'' (&quot;Founded upon the lectures of J.&amp;#8239;Willard Gibbs&amp;hellip;&quot;), New York: Charles Scribner's Sons; 12th printing, Yale University Press, 1958, [https://archive.org/details/vectoranalysiste0000gibb archive.org/details/vectoranalysiste0000gibb].&lt;/span&gt;
*&lt;span id=&quot;wrede-spiegel-10&quot;&gt;R.C.&amp;#8201;Wrede and M.R.&amp;#8239;Spiegel, 2010, ''Advanced Calculus'', 3rd Ed., New York: McGraw-Hill (Schaum's Outlines); [https://archive.org/details/schaumsoutlinesa0000wred archive.org/details/schaumsoutlinesa0000wred].&lt;/span&gt;

{{refend}}

&lt;/div&gt;

{{cob}}

== Further reading ==

{{cot}}

M.J. Crowe, &quot;A History of Vector Analysis&quot; (address at the University of Louisville, Autumn term, 2002), [https://www.researchgate.net/publication/244957729_A_History_of_Vector_Analysis researchgate.net/publication/244957729_A_History_of_Vector_Analysis] (including much discussion of quaternions).

P. Lynch, &quot;Matthew O'Brien: An inventor of vector analysis&quot;, ''Bulletin of the Irish Mathematical Society'', No.&amp;#8239;74 (Winter 2014), pp.&amp;#8239;81–8; [https://doi.org/10.33232/BIMS.0074.81.88 doi.org/10.33232/BIMS.0074.81.88].

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[[Category:Mathematics]]
[[Category:Calculus]]
[[Category:Vectors]]
[[Category:Vector calculus]]
[[Category:Multivariable calculus]]
[[Category:Applied mathematics]]
[[Category:Mathematical physics]]
[[Category:Waves]]
[[Category:Coordinate systems]]</text>
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        <username>Gavin R Putland</username>
        <id>2838145</id>
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      <comment>Gavin R Putland moved page [[WikiJournal Preprints/Coordinates Last: Vector Analysis Done Fast]] to [[Coordinates Last: Vector Analysis Done Fast]]: Conversion from preprint to learning resource.</comment>
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      <text bytes="344303" sha1="9v5g8sjkrtrux8zloz23s8ww8tm7g9m" xml:space="preserve">{{Author|Gavin R Putland}}{{tertiary}}{{mathematics}}{{physics}}{{engineering}}{{testing}}

== Preface ==

This learning resource (which I call a &quot;paper&quot;, although it's a long one) is an attempt to reduce vector analysis from a second-year undergraduate subject to a ''first''-year undergraduate subject. Its strategy is to delay the use of coordinate systems until their use is required by upcoming topics&amp;mdash;and, behold, assisted by previous topics. It's about ''vector'' analysis as distinct from tensor analysis: it does not deal with dyadics or higher-order tensors, except by way of occasional hints; but, along its unusual path, it ''does'' treat some topics that one might not expect in a &quot;first&quot; course.

[[w:Sheldon Axler|Sheldon Axler]], in his essay &quot;Down with determinants!&quot; ([[#axler-95|1995]]) and his ensuing book ''Linear Algebra Done Right'' (4th Ed., [[#axler-23-|2023–]]), does not eliminate determinants, but introduces them as late as possible, and then exploits them for what he calls their &quot;main reasonable use in undergraduate mathematics&quot;, namely the change-of-variables formula for multiple integrals.&lt;ref&gt;[[#axler-95|Axler, 1995]], &amp;sect;9. The relegation of determinants was anticipated by C.G.&amp;#8239;Broyden ([[#broyden-75|1975]]). But Broyden's approach is less radical: he does not deal with abstract vector spaces or abstract linear transformations, and his eventual definition of the determinant, unlike Axler's, is traditional&amp;mdash;not a product of the preceding narrative.&lt;/ref&gt; Here I treat coordinates in vector analysis somewhat as Axler treats determinants in linear algebra: I introduce coordinate systems as late as possible, and then exploit them in unconventionally ''rigorous'' derivations of vector-analytic identities from (e.g.) vector-algebraic identities. But I contrast with Axler in at least two ways. First, I have no intention of expanding this &quot;paper&quot; into a book. Brevity is of the essence. Second, while one may well avoid determinants in ''numerical''&amp;#8202; linear algebra,&lt;ref&gt;[[#axler-95|Axler, 1995]], &amp;sect;1. But it is Broyden ([[#broyden-75|1975]]), not Axler, who discusses numerical methods at length.&lt;/ref&gt; one can hardly avoid coordinates in ''numerical'' vector analysis! So I cannot offer a coordinate-free path into computation. But I can prepare for computation by expressing the operators of vector analysis in general coordinates and orthogonal coordinates: indeed, readers who stay with me to the end will get a more general treatment of coordinates than is offered by a typical ''book''-length introduction to vector analysis. [''Continued&amp;#8239;&amp;hellip;'']

{{cot|&amp;hellip; Extended content (show or hide)}}

In the meantime, however, coordinates don't get in the way. Familiar coordinates may be mentioned in passing for purposes of illustration; but, until &quot;Cartesian coordinates&quot; are announced under their own heading, I work from ''conceptual'' definitions rather than coordinate-based definitions. This, I submit, keeps the exposition direct and accessible, and facilitates treating related concepts in parallel&amp;mdash;saving time and words, and highlighting similarities and differences.

Something else that doesn't get in the way is an exaggerated pretense of rigor. In the branch of pure mathematics known as ''analysis'', there is a thing called a ''limit'', whereby for every positive ''&amp;epsiv;''&amp;#8201; there exists a positive ''&amp;delta;'' such that if some increment is less than ''&amp;delta;'', some error is less than ''&amp;epsiv;''. In the branch of applied mathematics known as ''[[w:continuum mechanics|continuum mechanics]]'', there is a thing called reality, whereby if the increment is less than some positive ''&amp;delta;'', the assumption of a continuum becomes ridiculous, so that the error cannot be made less than an ''arbitrary &amp;epsiv;''. Yet vector &quot;analysis&quot; (or a superset thereof) is typically studied with the intention of applying it to some form of &quot;continuum&quot; mechanics&amp;mdash;such as the modeling of elasticity, plasticity, fluid flow, or (widening the net) electrodynamics of ordinary matter&amp;mdash;conveniently forgetting that, on a sufficiently small scale, matter is lumpy. (Even if we claim that &quot;particles&quot; of matter are wave functions and therefore continuous, these wave functions are still lumpy on a scale not normally contemplated by continuum mechanics.) One might therefore submit that to express the principles of vector analysis in the language of limits is to strain at a gnat and swallow a camel. Here I avoid that camel by referring to '''elements''' of length or area or volume, each of which is ''small'' enough to allow some quantity or quantities to be considered uniform within it, but, for the same reason, ''large'' enough to allow such local averaging of the said quantity or quantities as is necessary to tune out the lumpiness. We shall see bigger camels, where well-known authors define or misdefine a vector ''operator'' and then derive identities by treating it like an ordinary vector ''quantity''. These I also avoid.

A rough and ready premise is more rigorous than an absurd or meaningless one. Discarding the machinery of limits causes a small lapse in rigor where limits are applicable, but avoids a big lapse where they are not. Maintaining the distinction between operators and quantities cannot cause a loss of rigor, but avoids one wherever the alleged &quot;algebraic&quot; properties of operators get confusing. The resulting standard of rigor is economical but consistent.

This paper is a new arrangement of old knowledge. It does not pretend to offer any new mathematical results, and in that sense does not pretend to be [[original research]]. But, pursuant to its goals as a [[learning resource]], it ''does'' contain independent derivations and independent scholarship.

Much of that scholarship builds on the earlier scholarship of Professor Chen-To Tai, {{serif|FIEEE}}, who died in 2004, and who first came to my attention in 2018 through his invited paper &quot;On the presentation of Maxwell's theory&quot; [''Proc.&amp;#8239;{{serif|IEEE}}'', '''60'''(8):&amp;#8239;936–45, 1972]. In nearly every place where I mention him here, even if I do not accept his conclusion, I am entirely indebted to his works for drawing my attention to the issue raised. In particular, it was through Tai that I became aware of Gibbs's original definitions of the divergence and curl and their suitability for expression in indicial notation ([[#tai-95|Tai, 1995]], pp.&amp;#8239;17,&amp;#8239;21). And although he might not have been pleased, it was through Tai that I first knew with certainty that, if we allow for the variability of the basis vectors, the del-dot and del-cross notations are valid in general coordinates (''ibid.'', pp.&amp;#8239;64–5). Accordingly, this paper is dedicated to him.

{{right|&amp;mdash;&amp;#8201;[[w:User:Gavin R Putland|Gavin R.&amp;#8201;Putland]].}}

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== Overview (for instructors) ==

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The gradient, the curl, the divergence, and the Laplacian are initially defined, without coordinates, as closed-surface integrals per unit volume&amp;mdash;the definition of the Laplacian being indifferent to whether the operand is a scalar field or a vector field. Four integral theorems&amp;mdash;including the divergence theorem&amp;mdash;follow almost immediately, provided that the initial definitions are unambiguous. Their unambiguity, together with some examples of their usefulness, is established as follows, at a level suitable for beginners:
* The gradient is related to an acceleration through an equation of motion;
* The divergence is related to two time-derivatives of density (the partial derivative and the material derivative) through two forms of an equation of continuity;
* The component of the curl in a general direction is expressed as a divergence (now known to be unambiguous);
* The same is done for the general component of the gradient, yielding not only a second proof of unambiguity of the gradient, but also the relation between the gradient and the directional derivative; this together with the original definition of the Laplacian shows that the Laplacian of a ''scalar'' field is the divergence of the gradient and therefore unambiguous. The unambiguity of the Laplacian of a ''vector'' field then follows from a component argument (as for the curl) or a linearity argument.
The derivation of the relation between the gradient and the directional derivative yields a coordinate-free definition of the dot-del operator for a scalar right-hand operand. But, as the directional derivative is also defined for a non-scalar operand, the same relation offers a method of generalizing the dot-del operator, so that the definition of the Laplacian of a general field can be rewritten with that operator. The advection operator&amp;mdash;derived without coordinates, for both scalar and vector properties&amp;mdash;is likewise rewritten.

Meanwhile comparison between the definitions of the various operators leads to coordinate-free definitions of the del-cross, del-dot, and del-squared operators. These together with the dot-del operator allow the four integral theorems to be condensed into a single generalized volume-integral theorem.

If the volume of integration is reduced to a thin curved slab of uniform thickness, with an edge-face perpendicular to the broad faces, the four integral theorems are reduced to their two-dimensional forms, each of which relates an integral over a surface segment to an integral around its enclosing curve, provided that the original ''closed''-surface integral has no contribution from the broad faces of the slab. This proviso can be satisfied by construction in two of the four cases, yielding two general theorems, one of which is the Kelvin&amp;ndash;Stokes theorem. By applying these two theorems to a segment of a closed surface, and expanding the segment to cover the entire surface, it is shown that the gradient is irrotational and the curl is solenoidal.

The next part of the exposition is more conventional, but still coordinate-free. The gradient theorem is derived from the relation between the gradient and the directional derivative. An irrotational field is shown to have a scalar potential. The 1/''r''&amp;#8202; scalar field is shown to be the field whose negative gradient is the inverse-square vector field, whose divergence is a delta function, which is therefore also the negative Laplacian of the 1/''r''&amp;#8202; scalar field. These results enable the construction of a field with a given divergence or a given Laplacian. The wave equation is derived from small-amplitude sound waves in a non-viscous fluid, and shown to be satisfied by a spherical-wave field with a 1/''r''&amp;#8202; amplitude, whose D'Alembertian is a delta function, enabling the construction of a wave function with a given D'Alembertian. But further progress, including the construction of a field with a given ''curl'', seems to require the invocation of a coordinate system.

With the aid of identities already found, expressions are easily obtained for the gradient, curl, divergence, Laplacian, and advection operators in Cartesian coordinates&amp;mdash;with indicial notation and implicit summation, for brevity. While the resulting expressions for the curl and divergence may look unfamiliar, they match the initial definitions given by J.&amp;#8239;Willard Gibbs. The Cartesian expressions are found convenient for deriving further identities: a comprehensive collection (including a multivariate chain rule) is derived, leading to the construction of a field with a given curl in a star-shaped region and, as a by-product, a demonstration that the curl of the velocity field of a rigid body is twice the angular velocity. The curl-of-the-curl identity leads to a second definition of the Laplacian of a vector, the Helmholtz decomposition, and the prediction of electromagnetic waves.

The time-honored method of deriving vector-analytic identities&amp;mdash;treating the divergence and curl as &quot;formal products&quot; with the del operator, varying one field at a time, and adding the results&amp;mdash;is found to be less than rigorous, sometimes less than clear, and hard to justify in view of the ease with which the same thing can be done with Cartesian coordinates, indicial notation, and implicit summation.

The introduction of ''general'' coordinates proceeds through (non-normalized) natural and dual basis vectors, reciprocity, the Kronecker delta, covariance of the natural basis, contravariance of the dual basis, contravariant and covariant components, local bases, contravariance of coordinates, covariance of derivatives w.r.t. coordinates, the Jacobian, and handedness. Reciprocity leads to the dot-product of two vector fields and, via the permutation symbol, to the cross-products of the basis vectors, the definition of one basis in terms of the other, the cross-product of two vector fields, and reciprocity of the covariant and contravariant Jacobians. Thus the stage is set for expressing operators in general coordinates.

The multivariate chain rule leads to expressions for the directional derivative (in terms of the contravariant basis), hence the gradient (del) and advection operators. The identity for the curl of the product of a scalar and a vector leads to an expression for the curl in terms of covariant components. Expressions for the curl and divergence ''operators'' are obtained from the original volume-based definitions, and are found to agree with del-cross and del-dot respectively, with del expressed in the same general coordinates. The volume-based definition of the divergence leads, by a simpler path, to an expression in terms of contravariant components, which in turn yields an expression for the Laplacian.

Affine coordinates are briefly described before proceeding to orthogonal coordinates. In the latter, the Jacobian is simplified and we can choose an orthonormal basis, which is its own reciprocal, so that vectors can be specified in components w.r.t. a single basis. By expressing the old basis vectors and components in terms of the new ones, we can re-express dot-products, cross-products, and differential operators in terms of orthogonal coordinates with an orthonormal basis.

In an appendix, Huygens' principle is mathematized by deriving Green's identities and thence Kirchhoff's integral theorem (''without''&amp;#8202; assuming sinusoidal time-dependence), and then interpreting Kirchhoff's integrand as a distribution of secondary sources.

Some technicalities are relegated to the &quot;Notes&quot;, which are followed by the &quot;Citations&quot;, the &quot;References&quot; cited, and finally&amp;mdash;to compensate for the absence of a &quot;History&quot; section&amp;mdash;some suggested &quot;Further reading&quot;.

=== To-do list ===

Although this resource should be usable already, some improvements are envisaged, namely:
* More illustrations;
* A note on the metric tensor and its determinant.

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== Introduction ==

=== Scalars, vectors, tensors, and coordinates ===

{{cot}}

Elementary calculus concerns differentiation and integration with respect to a ''real'' variable. Vector analysis, or &quot;vector calculus&quot;, concerns what we might call differentiation and integration w.r.t. a ''vector'' variable&amp;mdash;usually the position vector. The function &quot;differentiated&quot; or &quot;integrated&quot; w.r.t. that vector may also be a vector.{{efn|Some authors treat &quot;vector analysis&quot; and &quot;vector calculus&quot; as synonymous. Others, apparently influenced by the difference between elementary &quot;calculus&quot; and real &quot;analysis&quot;, would say that &quot;vector analysis&quot; is more general, more theoretical, and more rigorous than &quot;vector calculus&quot;. That distinction might have surprised the inventors of &quot;vector analysis&quot;, as it was originally called; their motives were specific and practical, and their methods were ad-hoc.}}

But what exactly is a '''vector'''? Mathematicians define a &quot;vector&quot; as a member of a ''[[w:vector space|vector space]]'', which is a [[w:set (mathematics)|set]] whose members satisfy certain basic rules of algebra (called the ''vector-space axioms'') in relation to another set called a ''[[w:field (mathematics)|field]]'' (e.g., the real numbers), which has its own basic rules of algebra (the ''field axioms''), and whose members are called &quot;scalars&quot;. Physicists are more fussy. They typically want a &quot;vector&quot; to be not only a member of a vector space, but also a '''first-order tensor'''&amp;#8239;: a &quot;tensor&quot;, meaning that it exists independently of any coordinate system with which it might be specified; and &quot;first-order&quot; (or &quot;first-degree&quot;, or &quot;first-rank&quot;), meaning that it is specified by a ''one''-dimensional array of numbers. Similarly, a 2nd-order tensor is specified by a 2-dimensional array (a matrix), and a 3rd-order by a 3-dimensional array, and so on. Hence they want a &quot;scalar&quot;, which is specified by a single number (a zero-dimensional array), to be a ''zero-order tensor''. In &quot;vector analysis&quot;, we are greatly interested in applications to physical situations, and accordingly take the physicists' view on what constitutes a vector or a scalar.

So, for our purposes, defining a quantity by three components in (say) a Cartesian coordinate system is not enough to make it a vector, and defining a quantity as a real function of a list of coordinates is not enough to make it a scalar, because we still need to show that the quantity has an independent existence. One method of doing this (''not''&amp;#8202; the method we shall use here!) is to show that the coordinate representation behaves appropriately when the coordinate system is changed. Independent existence of a ''quantity'' means that its coordinate representation changes so as to compensate for the change in the coordinate system.&lt;ref&gt;E.g., Feynman ([[#feynman-63|1963]], vol.&amp;#8239;1, &amp;sect;&amp;#8202;11-5), having defined velocity from displacement in Cartesian coordinates, shows that velocity is a vector by showing that its coordinate representation contra-rotates (like that of displacement) if the coordinate system rotates.&lt;/ref&gt; But independent existence of an ''operator'' means that its expression in one coordinate system (with the operand[s] and the result ''in that system'') gives the same result as the corresponding expression in another coordinate system.&lt;ref&gt;E.g., Feynman ([[#feynman-63|1963]], vol.&amp;#8239;1, &amp;sect;&amp;#8202;11-7), having defined the magnitude and dot-product in Cartesian coordinates, proves that they are scalar functions by showing that the corresponding expressions in rotated (&quot;primed&quot;) coordinates give the same values as the original expressions (in &quot;unprimed&quot; coordinates). And Tai ([[#tai-95|1995]], pp.&amp;#8239;66–7), having found an expression for the &quot;gradient&quot; operator in a general coordinate system (the &quot;unprimed&quot; system), proves the &quot;invariance&quot; of the operator (its vector character in this case) by showing that the corresponding expression in any other general coordinate system (the &quot;primed&quot; system) has the same effect.&lt;/ref&gt;

Here we shall circumvent these complications by the most obvious route: by initially ''defining things without coordinates''. If, having defined something without coordinates, we then need to represent it ''with'' coordinates, we can choose the coordinate system for convenience rather than generality.

For example, without using coordinates, we can define displacements in three-dimensional space by their '''magnitudes''' and '''directions''' and show that they satisfy the vector-space rules, so that they are vectors in the mathematicians' sense, and therefore (because we have defined them without coordinates) in the physicists' sense. Then, by the same rules, we can show that the derivatives of these vectors w.r.t. time are vectors, and that products of these vectors with a scalar (such as mass) are vectors, with the result that not only displacement but also velocity, acceleration, momentum, and force are vectors. Having thus established that these things exist independently of any coordinate system, we can choose convenient coordinates.

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=== Prerequisites ===

{{cot}}

I assume that the reader is familiar with the algebra and geometry of vectors in 3D space, including the dot-product, the cross-product, and the scalar triple product, their geometric meanings, their expressions in Cartesian coordinates, and the identity
:{{big|{{math|'''a''' &amp;times; ('''b''' &amp;times; '''c''')  {{=}}  '''a&amp;sdot;&amp;#8202;c b''' &amp;minus; '''a&amp;sdot;&amp;#8202;b c''' ,}}}}
which we call the &quot;expansion&quot; of the vector triple product.&lt;ref&gt;There are many proofs and interpretations of this identity. My own effort, for what it's worth, is &quot;Trigonometric proof of vector triple product expansion&quot;, ''Mathematics Stack Exchange'', [https://math.stackexchange.com/a/4839213/307861 t.co/NM2v4DJJGo], 2024. The classic is [[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;sect;&amp;#8239;26–7.&lt;/ref&gt; I further assume that the reader can generalize the concept of a derivative, so as to differentiate a vector with respect to a scalar, e.g.
:&lt;math&gt;\mathbf{r}'(t) = \frac{d\mathbf{r}}{dt}
  =\, \lim_{h\to 0} \frac{\mathbf{r}(t+h) - \mathbf{r}(t)}{h} \,,&lt;/math&gt;
or so as to differentiate a function of several independent variables &quot;partially&quot; w.r.t. one of them while the others are held constant, e.g.
:&lt;math&gt;\tfrac{\part}{\part y} \psi\big(x,y,z\big)
  =\, \lim_{h\to 0} \frac{\psi(x,y{+}h~\!,z) - \psi(x,y,z)}{h} \,.&lt;/math&gt;

But, in view of the limited applicability of limits (see the [[#Preface|Preface]]), I also expect the reader to be tolerant of an argument like this: In a short time{{mvar| dt}}, let the vectors {{math|'''r'''}} and{{math| '''p'''}} change by {{math|''d'''''r'''}} and{{math| ''d'''''p'''}} respectively. Then
:&lt;math&gt;\begin{align}
  \tfrac{d}{dt}\big(\mathbf{r}\!\times\!\mathbf{p}\big)
  &amp;= \frac{(\mathbf{r}+d\mathbf{r})\times(\mathbf{p}+d\mathbf{p})
            \,-\, \mathbf{r}\times\mathbf{p}}{dt}\\[1ex]
  &amp;= \frac{\mathbf{r}\!\times\!d\mathbf{p}+d\mathbf{r}\!\times\!\mathbf{p}}{dt}
       ~~\quad [\mathsf{neglecting}~d\mathbf{r}\!\times\!d\mathbf{p}]\\[1ex]
  &amp;=\, \mathbf{r}\times\!\tfrac{d\mathbf{p}}{dt}
       + \tfrac{d\mathbf{r}}{dt}\!\times\mathbf{p}\\[1ex]
  &amp;=\, \mathbf{r}\times\mathbf{\dot{p}}\,+\,\mathbf{\dot{r}}\times\mathbf{p}\,,
\end{align}&lt;/math&gt;
where, as always, the orders of the cross-products matter.{{efn|If {{math|'''r'''}} is the position of a particle and {{math|'''p'''}} is its momentum, the last term vanishes. If the force is toward the origin, the previous term also vanishes, and we are left with ''conservation of angular momentum'' about the origin.}} Differentiation of a ''dot''-product behaves similarly, except that the orders ''don't'' matter; and if&amp;#8239;{{math| '''p'''&amp;#8201;{{=}}&amp;#8201;''m'''''v'''}}, where {{mvar|m}} is a scalar and {{math|'''v'''}} is a vector, then
:&lt;math&gt;~~\mathbf{\dot{p}} = m\mathbf{\dot{v}} + \dot{m}\mathbf{v} \,.&lt;/math&gt;

Or an argument like this:  If&lt;math&gt;~z\!=\!f(x,y)&lt;/math&gt;, then
:&lt;math&gt;\begin{align}
  \frac{\part^2 z}{\part x\,\part y}
  &amp;= \tfrac{\part}{\part x}\,\tfrac{\part}{\part y} f\big(x,y\big)\\
  &amp;= \frac{\part}{\part x}\,\frac{f(x,y{+}dy)-f(x,y)}{dy}\\[1ex]
  &amp;= \frac{\,\frac{f(x{+}dx~\!,\,y{+}dy)\,-\,f(x{+}dx~\!,\,y)}{dy}
           - \frac{f(x,\,y{+}dy)\,-\,f(x,y)}{dy}\,} {dx}\\[2ex]
  &amp;= \frac{\,\frac{f(x{+}dx~\!,\,y{+}dy)\,-\,f(x,\,y{+}dy)}{dx}
           - \frac{f(x{+}dx~\!,\,y)\,-\,f(x,y)}{dx}\,} {dy}\\[1ex]
  &amp;= \frac{\part}{\part y}\,\frac{f(x{+}dx~\!,~\!y)-f(x,y)}{dx}\\[1ex]
  &amp;= \tfrac{\part}{\part y}\,\tfrac{\part}{\part x} f\big(x,y\big)
   = \frac{\part^2 z}{\part y\,\part x} \,;
\end{align}&lt;/math&gt;
that is, we can switch the order of differentiation in a &quot;mixed&quot; partial derivative. If{{mvar| &amp;part;&lt;sub&gt;x&lt;/sub&gt;}} is an abbreviation for {{mvar|{{sfrac|&amp;part;|&amp;part;x}}&amp;#8202;}}, etc., this rule can be written in '''operational''' terms as
:{{big|{{mvar|&amp;part;&lt;sub&gt;x &lt;/sub&gt;&amp;part;&lt;sub&gt;y&lt;/sub&gt; {{=}} &amp;part;&lt;sub&gt;y &lt;/sub&gt;&amp;part;&lt;sub&gt;x &lt;/sub&gt;.}}}}
More generally, if {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}} is an abbreviation for {{mvar|{{sfrac|&amp;part;|&amp;part;x&lt;sub&gt;i&lt;/sub&gt;}}}}&amp;#8202; where&amp;#8202; {{math|''i''&amp;#8239;&amp;#8714;&amp;#8202;&amp;lcub;1,&amp;#8202;2,&amp;hellip;&amp;rcub;,}}&amp;#8202; the rule becomes
:{{big|{{mvar|&amp;part;&lt;sub&gt;i &lt;/sub&gt;&amp;part;&lt;sub&gt;j&lt;/sub&gt; {{=}} &amp;part;&lt;sub&gt;j &lt;/sub&gt;&amp;part;&lt;sub&gt;i &lt;/sub&gt;.}}}}

The above generalizations of differentiation, however, do not go beyond differentiation w.r.t. ''real'' variables, some of which are scalars, and some of which are coordinates. It is now time to consider various kinds of &quot;differentiation&quot; w.r.t. the position vector.

{{cob}}

== Closed-surface integrals per unit volume ==

{{cot}}

The term ''field'', mentioned above in the context of algebraic axioms, has another meaning, which will be its usual meaning from now on: if {{math|'''r'''}} is the position vector, a '''scalar field''' is a scalar-valued function of{{math| '''r''',}} and a '''vector field''' is a vector-valued function of{{math| '''r'''}}; both may also depend on time. These are the functions of which we want &quot;derivatives&quot; w.r.t. the vector{{math| '''r'''}}.

In this section I introduce four such derivatives&amp;mdash;the ''gradient'', the ''curl'', the ''divergence'', and the ''Laplacian''&amp;#8202;&amp;mdash;in a way that will seem unremarkable to those readers who aren't already familiar with them, but idiosyncratic to those who are. The gradient is commonly introduced in connection with a curve and its endpoints, the curl in connection with a surface segment and its enclosing curve, the divergence in connection with a volume and its enclosing surface, and the Laplacian as a composite of two of the above, initially applicable only to a scalar field. Here I introduce all four in connection with a volume and its enclosing surface, and I introduce the Laplacian as a concept in its own right, equally applicable to a scalar ''or vector''&amp;#8202; field; only later do I express the Laplacian in terms of other &quot;derivatives&quot;. My initial definitions of the gradient, the curl, and the Laplacian, although not novel, are usually thought to be more advanced than the common ones&amp;mdash;in spite of being conceptually simpler, and in spite of being obvious variations on the same theme.

{{cob}}

=== Instant integral theorems (with a caveat) ===

{{cot}}

Let {{mvar|V}} be a volume (3D region) enclosed by a surface {{mvar|S}} (a mathematical surface, ''not'' generally a physical barrier). Let &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; be the unit normal vector at a general point on {{mvar|S}}, pointing ''out'' of{{mvar| V}}. Let {{mvar|n}} be the distance from {{mvar|S}} in the direction of&lt;math&gt;~\mathbf{\hat{n}}&lt;/math&gt; (positive outside {{mvar|V}}, negative inside), and let {{mvar|&amp;part;&lt;sub&gt;n&lt;/sub&gt;}} be an abbreviation for{{mvar| {{sfrac|&amp;part;|&amp;part;n}}&amp;#8202;}}, where the derivative&amp;mdash;commonly called the '''normal derivative'''&amp;mdash;is tacitly assumed to exist.

In {{mvar|V}}, and on {{mvar|S}}, let {{mvar|p}} be a scalar field (e.g., pressure in a fluid, or temperature), and let {{math|'''q'''}} be a vector field (e.g., flow velocity, or heat-flow density), and let {{mvar|&amp;psi;}} be a generic field which may be a scalar or a vector. Let a general ''element'' (small segment) of the surface {{mvar|S}} have area {{mvar|dS}}, and let it be small enough to allow &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;, {{mvar|p}}, {{math|'''q'''}}, and {{mvar|&amp;part;&lt;sub&gt;n&lt;/sub&gt;&amp;#8202;&amp;psi;}} to be considered uniform over the element (making a tacit assumption of local continuity). Then, for every element, the following four products are well defined:
{{NumBlk|:|&lt;math&gt;\mathbf{\hat{n}}            ~\!p\,dS ~,\qquad
                 \mathbf{\hat{n}}\times\mathbf{q}\,dS ~,\qquad
                 \mathbf{\hat{n}         \cdot q}\,dS ~,\qquad
                 \part_n                     \psi\;dS \,.
          &lt;/math&gt;|{{EquationRef|1}}}}
If {{mvar|p}} is pressure in a non-viscous fluid, the first of these products is the force exerted by the fluid in {{mvar|V}}&amp;#8202; through the area {{mvar|dS}}. The second product does not have such an obvious physical interpretation; but if{{math| '''q'''}} is ''circulating'' clockwise about an axis directed through{{mvar| V}}, the cross-product will be exactly tangential to{{mvar| S}} and will tend to have a component in the direction of that axis. The third product is the ''flux'' of{{math| '''q'''}} through the surface element; if{{math| '''q'''}} is flow velocity, the third product is the volumetric flow rate (volume per unit time) ''out'' of{{mvar| V}}&amp;#8202; through{{mvar| dS&amp;#8202;}}; or if {{math|'''q'''}} is heat-flow density, the third product is the heat transfer rate (energy per unit time) ''out'' of{{mvar| V}}&amp;#8202; through{{mvar| dS}}. The fourth product, by analogy with the third, might be called the flux of the normal derivative of{{mvar| &amp;psi;}} through the surface element, but is equally well defined whether {{mvar|&amp;psi;}} is a scalar or a vector&amp;mdash;or, for that matter, a matrix, or a tensor of any order, or anything else that we can differentiate w.r.t.{{mvar| n}}.

If we add up each of the four products over all the elements of the surface {{mvar|S}}, we obtain, respectively, the four '''surface integrals'''
{{NumBlk|:|&lt;math&gt;\iint_S \!\mathbf{\hat{n}}            ~\!p\,dS \,,~
                 \iint_S \!\mathbf{\hat{n}}\times\mathbf{q}\,dS \,,~
                 \iint_S \!\mathbf{\hat{n}         \cdot q}\,dS \,,~
                 \iint_S \!\part_n                     \psi\;dS \,,
          &lt;/math&gt;|{{EquationRef|2}}}}
in which the double integral sign indicates that the range of integration is two-dimensional. The first surface integral takes a scalar field and yields a vector; the second takes a vector field and yields a vector; the third takes a vector field and yields a scalar; and the fourth takes (e.g.) a scalar field yielding a scalar, or a vector field yielding a vector. If{{mvar| p}} is pressure in a non-viscous fluid, the first integral is the force exerted by the fluid in {{mvar|V}}&amp;#8202; on the fluid outside {{mvar|V}}. The second integral may be called the ''skew'' surface integral of{{math| '''q'''}} over {{mvar|S&amp;#8202;}},&lt;ref&gt;[[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;#8239;56.&lt;/ref&gt; or, for the reason hinted above, the ''circulation'' of{{math| '''q'''}} over {{mvar|S}}.&amp;#8201; The third integral, commonly called the ''flux integral'' (or simply the surface integral) of{{math| '''q'''}} over {{mvar|S}}, is the total ''flux'' of{{math| '''q'''}} out of{{mvar| V}}. And the fourth integral is the surface integral of the outward normal derivative of{{mvar| &amp;psi;}}.

Let the volume {{mvar|V}}&amp;#8202; be divided into elements. Let a general volume element have the volume {{mvar|dV}} and be enclosed by the surface {{mvar|&amp;delta;S}}&amp;#8201;&amp;mdash;not to be confused with the area {{mvar|dS}} of a surface ''element'', which may be an element of{{mvar| S}} or of{{mvar| &amp;delta;S}}. Then consider what happens if, instead of evaluating each of the above surface integrals over {{mvar|S}}, we evaluate it over each {{mvar|&amp;delta;S}} and add up the results for all the volume elements. In the ''interior'' of{{mvar| V}}, each surface element of area {{mvar|dS}} is on the boundary between two volume elements, for which the unit normals &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; at {{mvar|dS}}, and the respective values of{{mvar| &amp;part;&lt;sub&gt;n&lt;/sub&gt;&amp;#8202;&amp;psi;}}, are equal and opposite. Hence when we add up the integrals over the surfaces {{mvar|&amp;delta;S}}, the contributions from the elements {{mvar|dS}} cancel in pairs, except on the original surface {{mvar|S}}, so that we are left with the original integral over {{mvar|S}}. So, for the four surface integrals in ({{EquationNote|2}}), we have respectively
{{NumBlk|:|&lt;math&gt;\begin{align}
    \iint_S \mathbf{\hat{n}}~\!p \,dS
    &amp; \,= \sum_V\iint_{\delta S} \mathbf{\hat{n}}~\!p \,dS \,, \\
    \iint_S \mathbf{\hat{n}}\times\mathbf{q} \,dS
    &amp; \,= \sum_V\iint_{\delta S} \mathbf{\hat{n}}\times\mathbf{q} \,dS \,, \\
    \iint_S \mathbf{\hat{n}\cdot q} \,dS
    &amp; \,= \sum_V\iint_{\delta S} \mathbf{\hat{n}\cdot q} \,dS \,, \\
    \iint_S \part_n \psi \;dS
    &amp; \,= \sum_V\iint_{\delta S} \part_n \psi \;dS \,.
  \end{align}&lt;/math&gt;|{{EquationRef|3}}}}

Now comes a big &quot;if&quot;:&amp;#8201; ''if''&amp;#8202; we define the '''gradient''' of{{mvar| p}} (pronounced &quot;grad {{mvar|p}}&quot;) inside {{mvar|dV}}&amp;#8202; as
{{NumBlk|:|&lt;math&gt;\nabla p
    \,:=\, \frac{1}{dV}\iint_{\delta S} \mathbf{\hat{n}}~\!p \,dS
  &lt;/math&gt;|{{EquationRef|4g}}}}
and the '''curl''' of {{math|'''q'''}} inside {{mvar|dV}}&amp;#8202; as
{{NumBlk|:|&lt;math&gt;\operatorname{curl}\mathbf{q}
    \,:=\, \frac{1}{dV}\iint_{\delta S} \mathbf{\hat{n}}\times\mathbf{q} \,dS
  &lt;/math&gt;|{{EquationRef|4c}}}}
and the '''divergence''' of {{math|'''q'''}} inside {{mvar|dV}}&amp;#8202; as
{{NumBlk|:|&lt;math&gt;\operatorname{div}\mathbf{q}
    \,:=\, \frac{1}{dV}\iint_{\delta S} \mathbf{\hat{n}\cdot q} \,dS
  &lt;/math&gt;|{{EquationRef|4d}}}}
and the '''Laplacian''' of {{mvar|&amp;psi;}} inside {{mvar|dV}}&amp;#8202; as
{{NumBlk|:|&lt;math&gt;\triangle\psi
    \,:=\, \frac{1}{dV}\iint_{\delta S} \part_n \psi \;dS
  &lt;/math&gt;|{{EquationRef|4L}}}}
(where the letters after the equation number stand for ''gradient'', ''curl'', ''divergence'', and ''Laplacian'', respectively), then equations ({{EquationNote|3}}) can be rewritten
:&lt;math&gt;\begin{align}
    \iint_S \mathbf{\hat{n}}~\!p \,dS
    &amp; \,= \sum_V \nabla p ~dV \,, \\
    \iint_S \mathbf{\hat{n}}\times\mathbf{q} \,dS
    &amp; \,= \sum_V \operatorname{curl}\mathbf{q} ~dV \,, \\
    \iint_S \mathbf{\hat{n}\cdot q} \,dS
    &amp; \,= \sum_V \operatorname{div}\mathbf{q} ~dV \,, \\
    \iint_S \part_n \psi \;dS
    &amp; \,= \sum_V \triangle\psi ~dV \,.
  \end{align}&lt;/math&gt;
(For the Laplacian operator, we have used the broad triangle symbol{{math| (&amp;#9651;)}} rather than the narrower Greek Delta{{math| (&amp;Delta;)}}; the latter would more readily be misinterpreted as &quot;change in&amp;hellip;&quot;)&amp;#8201; But because each term in each sum above has a factor {{mvar|dV}}, we call the sum an integral; and because the range of integration is three-dimensional, we use a triple integral sign. Thus we obtain the following four theorems relating integrals over an enclosing surface {{mvar|S}}&amp;#8202; to integrals over the enclosed volume {{mvar|V&amp;#8202;}}:
{{NumBlk|:|&lt;math&gt;~~~~~\!\iint_S \mathbf{\hat{n}}~\!p \,dS
    \,= \iiint_V \nabla p ~dV \,;
  &lt;/math&gt;|{{EquationRef|5g}}}}
{{NumBlk|:|&lt;math&gt;\iint_S \mathbf{\hat{n}}\times\mathbf{q} \,dS
    \,= \iiint_V \operatorname{curl}\mathbf{q} ~dV \,;
  &lt;/math&gt;|{{EquationRef|5c}}}}
{{NumBlk|:|&lt;math&gt;~~\!\iint_S \mathbf{\hat{n}\cdot q} \,dS
    \,= \iiint_V \operatorname{div}\mathbf{q} ~dV \,;
  &lt;/math&gt;|{{EquationRef|5d}}}}
{{NumBlk|:|&lt;math&gt;~~\iint_S \part_n \psi \;dS
    \,= \iiint_V \triangle\psi ~dV \,.
  &lt;/math&gt;|{{EquationRef|5L}}}}
Of the above four results, only the third ({{EquationNote|5d}}) seems to have a standard name; it is called the '''divergence theorem''' (or ''Gauss's theorem'' or, more properly, ''[[w:Mikhail Ostrogradsky|Ostrogradsky]]'s theorem''&lt;ref&gt;[[#katz-79|Katz, 1979]], pp.&amp;#8239;146–9.&lt;/ref&gt;), and is indeed the best known of the four&amp;mdash;although the other three, having been derived in parallel with it, may be said to be equally fundamental.

As each of the operators {{math|&amp;nabla;,}} {{math|curl,}} and {{math|div}} calls for an integration w.r.t. area and then a division by volume, the ''dimension'' (or unit of measurement) of the result is the dimension of the operand divided by the dimension of length, as if the operation were some sort of differentiation w.r.t. position. Moreover, in each of equations ({{EquationNote|5g}}) to ({{EquationNote|5d}}), there is a triple integral on the right but only a double integral on the left, so that each of the operators {{math|&amp;nabla;,}} {{math|curl,}} and {{math|div}} appears to compensate for a single integration. For these reasons, and for convenience, we shall describe them as '''differential operators'''. By comparison, the {{math|&amp;#9651; }}operator in ({{EquationNote|4L}}) or ({{EquationNote|5L}}) calls for a further differentiation w.r.t.{{mvar| n&amp;#8202;}}; we shall therefore describe {{math|&amp;#9651;}} as a ''2nd-order'' differential operator. (An additional reason for these descriptions will emerge later.) As promised, the four definitions ({{EquationNote|4g}}) to ({{EquationNote|4L}}) are &quot;obvious variations on the same theme&quot; (although the fourth is somewhat less obvious than the others).

But remember the &quot;if&quot;: Theorems ({{EquationNote|5g}}) to ({{EquationNote|5L}}) depend on definitions ({{EquationNote|4g}}) to ({{EquationNote|4L}}) and are therefore only as definite as those definitions! Equations ({{EquationNote|3}}), without assuming anything about the shapes and relative sizes of the closed surfaces {{mvar|&amp;delta;S}} (except, tacitly, that&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is piecewise well-defined), indicate that the surface integrals are ''additive with respect to volume''. But this additivity, by itself, does not guarantee that the surface integrals are shared among neighboring volume elements ''in proportion'' to their volumes, as envisaged by &quot;definitions&quot; ({{EquationNote|4g}}) to ({{EquationNote|4L}}). Each of these &quot;definitions&quot; is unambiguous if, and only if, the ratio of the surface integral to{{mvar| dV}}&amp;#8202; is insensitive to the shape and size of{{mvar| &amp;delta;S}}&amp;#8202; for a sufficiently small {{mvar|&amp;delta;S}}. Notice that the issue here is ''not'' whether the ratios specified in equations ({{EquationNote|4g}}) to ({{EquationNote|4L}}) are true vectors or scalars, independent of the coordinates; all of the operations needed in those equations have coordinate-free definitions. Rather, the issue is whether the resulting ratios are unambiguous ''notwithstanding the ambiguity of'' {{mvar|&amp;delta;S}}, provided only that {{mvar|&amp;delta;S}} is sufficiently small. That is the advertised &quot;caveat&quot;, which must now be addressed.

Our proofs of the unambiguity of the differential operators will rest on a few [[w:thought experiment|thought experiments]], each of which applies an operator to a physical field, say{{mvar| f}}, and obtains another physical field whose unambiguity is beyond dispute, provided only that it can be considered uniform over the (small) volume element. The conclusion of the thought experiment is then applicable to any operand field whose ''mathematical'' properties are consistent with the physical interpretation; the loss of generality, if any, is only what is incurred by that interpretation.

{{cob}}

=== Unambiguity of the gradient ===

{{cot}}

Suppose that a fluid with density {{mvar|&amp;rho;}} (a scalar field) flows with velocity{{math| '''v'''}} (a vector field) under the influence of the internal pressure {{mvar|p}} (a scalar field). Then the integral in ({{EquationNote|4g}}) is the force exerted by the pressure of the fluid inside {{mvar|&amp;delta;S}} on the fluid outside, so that ''minus'' the integral is the force exerted ''on'' the fluid inside{{mvar| &amp;delta;S}}&amp;#8202; by the pressure of the fluid outside. Dividing by {{mvar|dV}}, we find that {{math|&amp;minus;&amp;nabla;''p''}}, as defined by ({{EquationNote|4g}}), is the force per unit volume, due to the pressure outside the volume.&lt;ref&gt;In [[#feynman-63|Feynman, 1963]],&amp;#8201; {{math|&amp;minus;&amp;nabla;''p'' }}as the &quot;pressure force per unit volume&quot; eventually appears in the 3rd-last lecture of Volume 2 (&amp;sect;40-1).&lt;/ref&gt; If this is the ''only'' force per unit volume acting ''on'' the volume (e.g., because the fluid is non-viscous and in a weightless environment, and the volume element is not in contact with the container), then it is equal to the acceleration times the mass per unit volume; that is,
{{NumBlk|:|&lt;math&gt;
    \rho\,\frac{d\mathbf{v}}{dt} = -\nabla p \,.
  &lt;/math&gt;|{{EquationRef|6g}}}}
Now provided that the left-hand expression can be considered uniform inside the small {{mvar|&amp;delta;S}}, it is unambiguous, whence&amp;#8202; {{math|&amp;nabla;''p'' }}''is also unambiguous''. If there are additional forces on the fluid element, e.g. due to gravity and&amp;#10744;or viscosity, then {{math|&amp;minus;&amp;nabla;''p''}} is not the sole contribution to density-times-acceleration, but is still the contribution due to pressure, which is still unambiguous.

By showing the unambiguity of definition ({{EquationNote|4g}}), we have confirmed theorem ({{EquationNote|5g}}). In the process we have seen that the volume-based definition of the gradient is useful for the modeling of fluids, and intuitive in that it formalizes the common notion that a pressure &quot;gradient&quot; gives rise to a force.

{{cob}}

=== Unambiguity of the divergence ===

{{cot}}

In the aforesaid fluid, in a short time{{mvar| dt}}, the volume that flows out of fixed closed surface {{mvar|&amp;delta;S}}&amp;#8202; through a fixed surface element of area {{mvar|dS}}&amp;#8202; is &lt;math&gt;\mathbf{v}~\!dt\!\cdot\!\mathbf{\hat{n}}\,dS&lt;/math&gt;&amp;#8239; (i.e., the displacement normal to the surface element, times the area).&amp;#8201; Multiplying this by density and integrating over {{mvar|&amp;delta;S}}, we find that the mass flowing out of{{mvar| &amp;delta;S}}&amp;#8202; in time{{mvar| dt}} is&amp;#8201; &lt;math&gt;\textstyle\iint_{\delta S}\rho\mathbf{v}~\!dt\cdot\mathbf{\hat{n}}\,dS&lt;/math&gt;.  Dividing this by {{mvar|dV}}, and then by {{mvar|dt}}, we get the rate of reduction of density inside {{mvar|&amp;delta;S&amp;#8202;}}; that is,
:&lt;math&gt;\tfrac{1}{dV}\!\iint_{\delta S}\mathbf{\hat{n}}\cdot\rho\mathbf{v}\,dS
       \,= -\frac{\part\rho}{\part t} \,,&lt;/math&gt;
where the derivative w.r.t. time is evaluated at a fixed location (because {{mvar|&amp;delta;S}} is fixed), and is therefore written as a ''partial'' derivative (because other variables on which {{mvar|&amp;rho;}} might depend&amp;mdash;namely spatial coordinates&amp;mdash;are held constant). Provided that the right-hand side can be considered uniform inside {{mvar|&amp;delta;S}}, it is unambiguous, so that the left side is likewise unambiguous. But the left side is simply{{math| div&amp;#8239;''&amp;rho;'''''v'''}}&amp;#8201; as defined by ({{EquationNote|4d}}),{{efn|There is no need for parentheses around{{math| ''&amp;rho;'''''v'''&amp;#8202;,}} because {{math|div&amp;#8239;''&amp;rho;'''''v'''&amp;#8202;}} cannot mean {{math|(div&amp;#8239;''&amp;rho;'')'''v'''&amp;#8202;,}} because the divergence of a scalar field is not defined.}} which is therefore also unambiguous,&lt;ref&gt;A demonstration like the foregoing is outlined by Gibbs ([[#gibbs-1881-4|1881]], &amp;sect;&amp;#8239;55).&lt;/ref&gt; confirming theorem ({{EquationNote|5d}}). In short, the divergence operator is that which maps {{math|''&amp;rho;'''''v'''}} to the rate of reduction of density at a fixed point:
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\rho\mathbf{v} = -\frac{\part\rho}{\part t} \,.
  &lt;/math&gt;|{{EquationRef|7d}}}}
This result, which expresses ''conservation of mass'', is a form of the so-called '''equation of continuity'''.

The partial derivative {{mvar|{{sfrac|&amp;part;&amp;rho;|&amp;part;t}}&amp;#8202;}} in ({{EquationNote|7d}}) must be distinguished from the '''material derivative''' {{mvar|{{sfrac|d&amp;rho;|dt}}&amp;#8202;}}, which is evaluated at a point that moves ''with the fluid''.{{efn|The material derivative operator {{mvar|{{sfrac|d|dt}}}} is also called the ''substantive'' derivative, and is sometimes written {{mvar|{{sfrac|D|Dt}}}} if the result is meant to be understood as a field rather than simply a function of time ([[#kemmer-77|Kemmer, 1977]], pp.&amp;#8239;184–5).}} [Similarly, {{math|{{sfrac|''d''&amp;#8202;'''v'''|''dt''}}}} in ({{EquationNote|6g}}) is the ''material'' acceleration, because it is the acceleration of the mobile mass&amp;mdash;not of a fixed point!&amp;#8239;]&amp;#8239; To re-derive the equation of continuity in terms of the ''material'' derivative, the volume &lt;math&gt;\mathbf{v}~\!dt\!\cdot\!\mathbf{\hat{n}}\,dS~\!,&lt;/math&gt; which flows out through{{mvar| dS}} in time{{mvar| dt}} (as above), is integrated over {{mvar|&amp;delta;S}} to obtain the increase in volume of the mass ''initially'' contained in {{mvar|dV}}. Dividing this by the mass, {{mvar|&amp;rho;&amp;#8239;dV}}, gives the increase in ''[[w:specific volume|specific volume]]'' {{math|(1&amp;#10744;''&amp;rho;'')}} of that mass, and then dividing by {{mvar|dt}} gives the rate of change of specific volume; that is,
:&lt;math&gt;\tfrac{1}{\rho~\!dV}\!\iint_{\delta S}\mathbf{\hat{n}}\cdot\mathbf{v}\,dS
       \,= \tfrac{d}{dt}\big(\rho^{-1}\big)
       = -\rho^{-2\,}\tfrac{d\rho}{dt} \,.&lt;/math&gt;
Multiplying by {{math|''&amp;rho;''&amp;sup2;}} and comparing the left side with ({{EquationNote|4d}}), we obtain
{{NumBlk|:|&lt;math&gt;
    \rho\operatorname{div}\mathbf{v} = -\frac{d\rho}{dt} \,.
  &lt;/math&gt;|{{EquationRef|7d'}}}}
Whereas ({{EquationNote|7d}}) shows that {{math|div&amp;#8239;''&amp;rho;'''''v'''&amp;#8202;}} is unambiguous, ({{EquationNote|7d'}}) shows that {{math|div&amp;#8239;'''v'''&amp;#8202;}} is unambiguous (provided that other things are locally continuous). In accordance with the everyday meaning of &quot;divergence&quot;, ({{EquationNote|7d'}}) also shows that {{math|div&amp;#8239;'''v'''&amp;#8202;}} is positive if the fluid is expanding ({{mvar|&amp;rho; }}decreasing), negative if it is contracting ({{mvar|&amp;rho; }}increasing), and zero if it is incompressible. In the last case, the equation of continuity reduces to
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\mathbf{v} = 0
  \qquad&lt;/math&gt;[&amp;#8202;for an incompressible fluid&amp;#8202;].|{{EquationRef|7i}}}}

For incompressible flow, any tubular surface tangential to the flow velocity, and consequently with no flow in or out of the &quot;tube&quot;, has the same volumetric flow rate across all cross-sections of the &quot;tube&quot;, as if the surface were the wall of a pipe full of liquid (except that the surface is not necessarily stationary). Accordingly, ''a vector field with zero divergence is described as '''solenoidal''''' (from the Greek word for &quot;pipe&quot;). More generally, a solenoidal vector field has the property that for any tubular surface tangential to the field, the flux integrals across any two cross-sections of the &quot;tube&quot; are the same&amp;mdash;because otherwise there would be a net flux integral out of the closed surface comprising the two cross-sections and any segment of tube between them, in which case, by the divergence theorem ({{EquationNote|5d}}), the divergence would have to be non-zero somewhere inside, contrary to ({{EquationNote|7i}}).

{{cob}}

=== Unambiguity of the curl (and gradient) ===

{{cot}}

The unambiguity of the curl ({{EquationNote|4c}}) follows from the unambiguity of the divergence. Let {{math|'''b'''}} be any ''uniform'' vector (i.e., any vector that is independent of location&amp;mdash;e.g. a uniform vector field, possibly time-dependent). Taking dot-products of ({{EquationNote|4c}}) with{{math| '''b''',}} we get
:&lt;math&gt;\begin{align}
    \mathbf{b}\cdot\operatorname{curl}\mathbf{q}\,
    &amp;=\, \mathbf{b}\cdot\tfrac{1}{dV}\!\iint_{\delta S}
           \mathbf{\hat{n}}\times\mathbf{q}
         \,dS \\[.5ex]
    &amp;=\, \tfrac{1}{dV}\!\iint_{\delta S}
           \mathbf{b}\cdot\mathbf{\hat{n}}\!\times\!\mathbf{q}
         \,dS \\[1ex]
    &amp;=\, \tfrac{1}{dV}\!\iint_{\delta S}
           \mathbf{\hat{n}}\cdot\mathbf{q}\!\times\!\mathbf{b}
         \,dS \,;
  \end{align}&lt;/math&gt;
that is, by ({{EquationNote|4d}}),
{{NumBlk|:|&lt;math&gt;\operatorname{curl}\mathbf{q} \cdot \mathbf{b}
    = \operatorname{div}(\mathbf{q}\!\times\!\mathbf{b})
  \qquad&lt;/math&gt;[&amp;#8202;for uniform {{math|'''b'''}}].|{{EquationRef|8c}}}}
(The parentheses around&amp;#8202; {{math|'''q'''&amp;#8202;&amp;times;&amp;#8202;'''b'''}}&amp;#8202; on the right, although helpful because of the spacing, are not strictly necessary, because the alternative binding would be {{math|(div&amp;#8201;'''q''')}}, which is a scalar, whose cross-product with the vector {{math|'''b'''}} is not defined. And the left-hand expression does not need parentheses, because it can only mean the dot-product of a curl with the vector {{math|'''b'''}}; it cannot mean the curl of a dot-product, because the curl of a scalar field is not defined.) Equation ({{EquationNote|8c}}) is an identity for ''uniform''{{math| '''b'''}}. If we make {{math|'''b'''}} a ''unit'' vector in any fixed direction, the left-hand side of the identity is the (scalar) component of {{math|curl&amp;#8239;'''q'''}} in that direction, and the right-hand side is unambiguous. Thus ''the curl is unambiguous because its component in any direction is unambiguous''. This confirms theorem ({{EquationNote|5c}}).

Similarly, the unambiguity of the divergence implies the unambiguity of the gradient. Starting with ({{EquationNote|4g}}), taking dot-products with an arbitrary uniform vector {{math|'''b''',}} and proceeding as above, we obtain
{{NumBlk|:|&lt;math&gt;\nabla p \cdot \mathbf{b}
    = \operatorname{div} p\mathbf{b}
  \qquad&lt;/math&gt;[&amp;#8202;for uniform {{math|'''b'''}}].|{{EquationRef|8g}}}}
(The left-hand side does not need parentheses, because it can only mean the dot-product of a gradient with the vector {{math|'''b'''}}; it cannot mean the gradient of the dot-product of a scalar field with a vector field, because that dot-product would not be defined.) If we make {{math|'''b'''}} a ''unit'' vector, this result ({{EquationNote|8g}}) says that the (scalar) component of{{math| &amp;nabla;''p''}} in the direction of{{math|&amp;#8202; '''b'''}} is given by the right-hand side, which again is unambiguous. So here we have a second explanation of the unambiguity of the gradient: like the curl, it is unambiguous because its component in any direction is unambiguous.

We might well ask what happens if we take ''cross''-products with {{math|'''b'''}} on the left, instead of dot-products. If we start with ({{EquationNote|4g}}), the process is straightforward: in the end we can switch the order of the cross-product on the left, and change the sign on the right, obtaining
{{NumBlk|:|&lt;math&gt;\nabla p \times \mathbf{b}
    = \operatorname{curl} p\mathbf{b}
  \qquad&lt;/math&gt;[&amp;#8202;for uniform {{math|'''b'''}}].|{{EquationRef|8p}}}}
(Again no parentheses are needed.) If we start with ({{EquationNote|4c}}) instead, and take {{math|'''b'''}} inside the integral, we get a vector triple product to expand, which leads to
:&lt;math&gt;\mathbf{b} \times \operatorname{curl}\mathbf{q}
    = \tfrac{1}{dV}\!\iint_{\delta S}\!\mathbf{\hat{n}}\,\mathbf{b{\cdot}q}\,dS
    - \tfrac{1}{dV}\!\iint_{\delta S}\!\mathbf{b{\cdot}\hat{n}\,q}\,dS \,,
  &lt;/math&gt;
in which the first term on the right is simply&amp;#8202; {{math|&amp;nabla;&amp;#8239;'''b&amp;sdot;q'''}}&amp;#8201; (the gradient of the dot-product). The second term is more problematic. ''If''&amp;#8202; we had a scalar {{mvar|p}} instead of the vector {{math|'''q''',}} we could take {{math|'''b'''}} outside the second integral, so that the second term would be (minus) {{math|'''b&amp;#8239;&amp;sdot;'''&amp;#8239;&amp;nabla;''p''}}. This suggests that the actual second term should be (minus) {{math|'''b&amp;#8239;&amp;sdot;'''&amp;#8239;&amp;nabla;'''q'''}}.&amp;#8201; Shall we therefore adopt the second term (without the sign) as the ''definition'' of{{math|&amp;#8202; '''b&amp;sdot;'''&amp;nabla;&amp;#8239;'''q'''}} for a ''vector'' {{math|'''q'''}} (treating {{math|'''b&amp;sdot;'''&amp;nabla;}} as an operator), and write
{{NumBlk|:|&lt;math&gt;\mathbf{b} \times \operatorname{curl}\mathbf{q} ~\!=
    \nabla\,\mathbf{b{\cdot}q} - \mathbf{b}~\!{\cdot}\nabla\,\mathbf{q}
  \qquad&lt;/math&gt;[&amp;#8202;for uniform {{math|'''b'''}}] ?|{{EquationRef|8q}}}}
The proposal would be open to the objection that&amp;#8202; {{math|'''b&amp;sdot;'''&amp;nabla;&amp;#8239;'''q'''}}&amp;#8202; had been defined only for ''uniform''{{math| '''b'''&amp;#8202;,}} whereas&amp;#8202; {{math|'''b&amp;#8239;&amp;sdot;'''&amp;#8239;&amp;nabla;''p''&amp;#8202;}} (for scalar{{mvar| p}}) is defined whether {{math|'''b'''}} is uniform or not.&amp;#8201; So, for the moment, let us put ({{EquationNote|8q}}) aside and run with ({{EquationNote|8c}}), ({{EquationNote|8g}}), and ({{EquationNote|8p}}).

{{cob}}

=== Another meaning of the gradient ===

{{cot}}

Let {{math|'''s&amp;#770;'''}} be a unit vector in a given direction, and let {{mvar|s}} be a parameter measuring distance (arc length) along a path in that direction. By equation ({{EquationNote|8g}}) and definition ({{EquationNote|4d}}), we have
:&lt;math&gt;\nabla p \cdot \mathbf{\hat{s}} =
    \operatorname{div} p\mathbf{\hat{s}} =
    \tfrac{1}{dV}\!\iint_{\delta S}\mathbf{\hat{n}}\cdot p\mathbf{\hat{s}}\,dS\,,
  &lt;/math&gt;
where, by the unambiguity of the divergence, the shape of the closed surface {{mvar|&amp;delta;S}} enclosing {{mvar|dV}}&amp;#8202; can be chosen for convenience. So let {{mvar|&amp;delta;S}} be a right cylinder with cross-sectional area {{mvar|&amp;alpha;}}&amp;#8201; and perpendicular height {{mvar|ds&amp;#8202;,}} with the path passing perpendicularly through the end-faces at parameter-values {{mvar|s}} and {{mvar|s+ds&amp;#8202;,}} where the outward unit normal &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; consequently takes the values {{math|&amp;minus;'''s&amp;#770;'''}} and {{math|'''s&amp;#770;'''&amp;#8202;,}} respectively. And let the cross-sectional dimensions be small compared with {{mvar|ds}}&amp;#8202; so that the values of{{mvar| p}} at the end-faces, say {{mvar|p}} and {{mvar|p+dp}}, can be taken to be the same as where the end-faces cut the path. Then&amp;#8202; {{mvar|dV&amp;#8201;{{=}}&amp;#8201;&amp;alpha;&amp;#8239;ds&amp;#8202;}}, and the surface integral over{{mvar| &amp;delta;S}} includes only the contributions from the end-faces (because &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is perpendicular to {{math|'''s&amp;#770;'''}} elsewhere); those contributions are respectively&amp;#8201; &lt;math&gt;-\mathbf{\hat{s}}\!\cdot\!p\mathbf{\hat{s}}\,\alpha\,&lt;/math&gt; and&amp;#8201; &lt;math&gt;\mathbf{\hat{s}}\!\cdot\!(p\!+\!dp)\mathbf{\hat{s}}\,\alpha~\!,&lt;/math&gt;  i.e.&amp;#8239; &lt;math&gt;-p\alpha\,&lt;/math&gt; and &lt;math&gt;(p\!+\!dp)\alpha&lt;/math&gt;.  With these substitutions the above equation becomes
:&lt;math&gt;\begin{align}
    \nabla p \cdot \mathbf{\hat{s}}
    &amp;= \tfrac{1}{\alpha\,ds}\Big({-}p\alpha + (p\!+\!dp)\alpha\Big) \\[1ex]
    &amp;= \frac{\,p\!+\!dp ~-~ p\,}{ds} = \frac{\part p}{\part s} ~;
  \end{align}&lt;/math&gt;
that is,
{{NumBlk|:|&lt;math&gt;
    \nabla p \cdot \mathbf{\hat{s}} = \part_s p \,,
  &lt;/math&gt;|{{EquationRef|9g}}}}
where the right-hand side, commonly called the '''directional derivative''' of{{mvar| p}} in the {{math|'''s&amp;#770;'''}} direction,&lt;ref&gt;[[#wilson-1901|Wilson, 1901]], pp.&amp;#8239;147–8; [[#borisenko-tarapov-68|Borisenko &amp;amp; Tarapov, 1968]], pp.&amp;#8239;147–8 (again); [[#hsu-84|Hsu, 1984]], p.&amp;#8239;92; [[#kreyszig-62-|Kreyszig, 1988]], pp.&amp;#8239;485–6; [[#wrede-spiegel-10|Wrede &amp;amp; Spiegel, 2010]], p.&amp;#8239;198.&lt;/ref&gt; is the derivative of{{mvar| p}} w.r.t. distance in that direction. Although ({{EquationNote|9g}}) has been obtained by taking that direction as fixed, the equality is evidently maintained if {{mvar|s}} measures arc length along any path ''tangential''&amp;#8202; to{{math| '''s&amp;#770;'''}} at the point of interest.

Equation ({{EquationNote|9g}}) is an alternative definition of the gradient: it says that ''the gradient of&lt;math&gt;~p&lt;/math&gt; is the vector whose scalar component in any direction is the directional derivative of&lt;math&gt;~p&lt;/math&gt; in that direction''. For ''real&lt;math&gt;~p&lt;/math&gt;'', this component has its maximum, namely {{math|{{abs|&amp;nabla;''p''}}&amp;#8202;,}} in the direction of{{math| &amp;nabla;''p''&amp;#8202;}}; thus ''the gradient of&lt;math&gt;~p&lt;/math&gt; is the vector whose direction is that in which the derivative of&lt;math&gt;~p&lt;/math&gt; w.r.t. distance is a maximum, and whose magnitude is that maximum''. This is the usual conceptual definition of the gradient.&lt;ref&gt;Gibbs ([[#gibbs-1881-4|1881]], &amp;sect;&amp;#8239;50) ''introduces'' the gradient with this definition, except that he calls {{math|&amp;nabla;''u''}} simply the ''derivative'' of{{mvar| u}}, and {{mvar|u}} the ''primitive'' of{{math| &amp;nabla;''u''}}. Use of the term ''gradient'' as an alternative to ''derivative'' is reported by Wilson ([[#wilson-1901|1901]], p.&amp;#8239;138).&lt;/ref&gt; Sometimes it is convenient to work directly from this definition. For example, in Cartesian coordinates {{math|(''x'',&amp;#8239;''y'',&amp;#8239;''z''),}} if a scalar field is given by {{mvar|x&amp;#8202;,}} its gradient is obviously the unit vector in the direction of the {{mvar|x }}axis, usually called {{math|'''i'''&amp;#8202;}}; that is, {{math|&amp;nabla;''x''&amp;#8201;{{=}}&amp;#8201;'''i'''}}. Similarly, if&amp;#8201; &lt;math&gt;\mathbf{r}=r\mathbf{\hat{r}}&lt;/math&gt;&amp;#8201; is the position vector, then &lt;math&gt;\nabla r = \mathbf{\hat{r}}&lt;/math&gt;.

If&amp;#8202; {{math|'''s&amp;#770;'''}} is ''tangential''&amp;#8202; to a '''level surface''' of{{mvar| p}} (a surface of constant{{mvar| p}}), then {{mvar|&amp;part;&lt;sub&gt;s&lt;/sub&gt;&amp;#8201;p}}&amp;#8202; in that direction is zero, in which case ({{EquationNote|9g}}) says that {{math|&amp;nabla;''p''}} (if not zero) is orthogonal to{{math| '''s&amp;#770;'''}}.&amp;#8201; So&lt;math&gt;\,\nabla p&lt;/math&gt; ''is orthogonal to the surfaces of constant&lt;math&gt;~p\,&lt;/math&gt;'' (as we would expect, having just shown that the direction of{{math| &amp;nabla;''p''}} is that in which {{mvar|p}} varies most steeply). This result leads to a method of finding a vector normal to a curved surface at a given point: if the equation of the surface is&amp;#8239; {{math|''f''&amp;#8239;('''r''')&amp;#8201;{{=}}&amp;#8201;''C''&amp;#8202;,}}&amp;#8202; where {{math|'''r''' }}is the position vector and {{mvar|C&amp;#8202; }}is a constant (possibly zero), a suitable vector is {{math|&amp;nabla;''f''}}&amp;#8202; evaluated at the given point.

If {{mvar|p}} is ''uniform''&amp;#8202;&amp;mdash;that is, if it has no spatial variation&amp;mdash;then its derivative w.r.t. distance in every direction is zero; that is, the component of{{math| &amp;nabla;''p''}} in every direction is zero, so that {{math|&amp;nabla;''p''}} must be the zero vector. In short, ''the gradient of a uniform scalar field is zero''. Conversely, if {{mvar|p}} is ''not'' uniform, there must be some location and some direction in which its derivative w.r.t. distance, if defined at all, is non-zero, so that its gradient, if defined at all, is also non-zero. Thus ''a scalar field with zero gradient in some region is uniform in that region''.

{{cob}}

=== Unambiguity of the Laplacian ===

{{cot}}

Armed with our new definition of the gradient ({{EquationNote|9g}}), we can revisit our definition of the Laplacian ({{EquationNote|4L}}). If{{mvar| &amp;psi;}} is a ''scalar'' field, then, by ({{EquationNote|9g}}),  &lt;math&gt;\part_n \psi&lt;/math&gt; can be replaced by &lt;math&gt;\nabla\psi\cdot\mathbf{\hat{n}}\,&lt;/math&gt; in ({{EquationNote|4L}}), which then becomes
{{NumBlk|:|&lt;math&gt;\triangle\psi \,=\,
    \tfrac{1}{dV}\!\iint_{\delta S} \mathbf{\hat{n}}\cdot\nabla\psi \;dS \,;
  &lt;/math&gt;|{{EquationRef|9L}}}}
that is, by definition ({{EquationNote|4d}}),
{{NumBlk|:|&lt;math&gt;\triangle\psi
    \,=\, \operatorname{div}\nabla\psi
  \qquad&lt;/math&gt;[&amp;#8202;for scalar {{mvar|&amp;psi;}}].|{{EquationRef|9L'}}}}
So ''the Laplacian of a scalar field is the divergence of the gradient''. This is the usual ''introductory'' definition of the Laplacian&amp;mdash;and on its face is applicable only in the case of a scalar field. The unambiguity of the Laplacian, in this case, follows from the unambiguity of the divergence and the gradient.

If, on the contrary, {{mvar|&amp;psi;}} in definition ({{EquationNote|4L}}) is a ''vector'' field, then we can again take dot-products with a uniform vector {{math|'''b''',}} obtaining
:&lt;math&gt;(\triangle\psi)\cdot\mathbf{b} \,=\,
    \tfrac{1}{dV}\!\iint_{\delta S} \part_n(\psi\!\cdot\!\mathbf{b}) \,dS \,.
  &lt;/math&gt;
If we make {{math|'''b'''}} a ''unit'' vector, this says that ''the scalar component of the Laplacian of a vector field, in any direction, is the Laplacian of the scalar component of that vector field in that direction''. As we have just established that the latter is unambiguous, so is the former.

But the unambiguity of the Laplacian can be generalized further. If
:{{big|{{math|''&amp;psi;'' {{=}} &amp;sum;&lt;sub&gt;''i'' &lt;/sub&gt;''&amp;alpha;&lt;sub&gt;i &lt;/sub&gt;&amp;phi;&lt;sub&gt;i&lt;/sub&gt;''}}}}
where each {{mvar|&amp;phi;&lt;sub&gt;i&lt;/sub&gt;}} is a scalar field, and each {{mvar|&amp;alpha;&lt;sub&gt;i&lt;/sub&gt;}} is a constant, and the counter {{mvar|i}} ranges from (say) 1 to{{mvar| k&amp;#8202;}}, then it is clear from ({{EquationNote|4L}}) that
{{NumBlk|:|{{big|{{math|&amp;#9651;{{big|(}}&amp;sum;&lt;sub&gt;''i'' &lt;/sub&gt;''&amp;alpha;&lt;sub&gt;i &lt;/sub&gt;&amp;phi;&lt;sub&gt;i&lt;/sub&gt;''{{big|)}} {{=}} &amp;sum;&lt;sub&gt;''i'' &lt;/sub&gt;{{big|(}}''&amp;alpha;&lt;sub&gt;i &lt;/sub&gt;''&amp;#9651;''&amp;phi;&lt;sub&gt;i&lt;/sub&gt;''{{big|)}}}} .}}|{{EquationRef|10}}}}
In words, this says that ''the Laplacian of a linear combination of fields is the same linear combination of the Laplacians of the same fields''&amp;mdash;or, more concisely, that ''the Laplacian is '''[[w:linearity|linear]]'''''. I say &quot;it is clear&quot; because the Laplacian as defined by ({{EquationNote|4L}}) is itself a linear combination, so that ({{EquationNote|10}}) merely asserts that we can regroup the terms of a nested linear combination; the gradient, curl, and divergence as defined by ({{EquationNote|4g}}) to ({{EquationNote|4d}}) are likewise linear. It follows from ({{EquationNote|10}}) that ''the Laplacian of a linear combination of fields is unambiguous if the Laplacians of the separate fields are unambiguous''. Now we have supposed that the fields {{mvar|&amp;phi;&lt;sub&gt;i&lt;/sub&gt;}} are scalar and that the coefficients {{mvar|&amp;alpha;&lt;sub&gt;i&lt;/sub&gt;}} are constants. But the same logic applies if the &quot;constants&quot; are uniform basis vectors (e.g.,{{math| '''i''',&amp;#8202;'''j''','''k'''}}), so that the &quot;linear combination&quot; can represent any vector field, whence the Laplacian of any vector field is unambiguous. And the same logic applies if the &quot;constants&quot; are chosen as a &quot;basis&quot; for a space of tensors of any order, so that the Laplacian of any tensor field of that order is unambiguous, and so on. In short, ''the Laplacian of any field that we can express with a uniform basis is unambiguous''.

{{cob}}

=== The dot-del, del-cross, and del-dot operators ===

{{cot}}

The gradient operator {{math|&amp;nabla;}} is also called {{mvar|'''del'''}}.{{efn|Or ''nabla'', because it allegedly looks like the ancient Phoenician harp that the Greeks called by that name.}} If it simply denotes the gradient, we tend to pronounce it &quot;grad&quot; in order to emphasize the result. But it can also appear in combination with other operators to give other results, and in those contexts we tend to pronounce it &quot;del&quot;.

One such combination is &quot;dot del&quot;&amp;mdash;&amp;#8202;as in &quot;{{math|&amp;#8202;'''b&amp;sdot;'''&amp;nabla;&amp;#8202;}}&quot;, which we proposed for ({{EquationNote|8q}}), but did not quite manage to define satisfactorily for a vector operand. With our new definition of the gradient ({{EquationNote|9g}}), we can now make a second attempt. A general vector field {{math|'''q'''}} can be written &lt;math&gt;|\mathbf{q}|\,\mathbf{\hat{q}}~\!,&lt;/math&gt; so that
:&lt;math&gt;\mathbf{q}\cdot\nabla\psi
    \,=\, |\mathbf{q}| \,\mathbf{\hat{q}}\cdot\nabla\psi \,.
  &lt;/math&gt;
If {{mvar|&amp;psi;}} is a ''scalar'' field, we can apply ({{EquationNote|9g}}) to the right-hand side, obtaining
:{{big|&lt;math&gt;\mathbf{q}\cdot\nabla\psi
    ~\!=~\! |\mathbf{q}| \,\part_{s_q} \psi \,,
  &lt;/math&gt;}}
where {{mvar|s&lt;sub&gt;q&lt;/sub&gt;}} is distance in the direction of{{math| '''q'''}}. For ''scalar'' {{mvar|&amp;psi;}}, this result is an identity between previously defined quantities. For ''non-scalar'' {{mvar|&amp;psi;}}, we have not yet defined the left-hand side, but the right-hand side is still well-defined and self-explanatory (provided that we can differentiate {{mvar|&amp;psi;}} w.r.t.{{mvar| s&lt;sub&gt;q&lt;/sub&gt;}}). So we are free to adopt
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\psi \,:=\, |\mathbf{q}| \,\part_{s_q} \psi
  &lt;/math&gt;}}|{{EquationRef|11}}}}
(where {{mvar|s&lt;sub&gt;q&lt;/sub&gt;}} is distance in the direction of{{math| '''q'''}}) as the general definition of the ''operator'' {{math|'''q&amp;sdot;'''&amp;nabla;&amp;#8202;,}} and to interpret it as defining both a ''unary'' operator&amp;#8202; {{math|'''q&amp;sdot;'''&amp;nabla;}} which operates on a generic field, and a ''binary'' operator&amp;#8202; {{math|'''&amp;sdot;'''&amp;nabla;}} which takes a (possibly uniform) vector field on the left and a generic field on the right.

For any vector field {{math|'''q'''&amp;#8202;,}} it follows from ({{EquationNote|11}}) that ''if&lt;math&gt;~\psi&lt;/math&gt; is a uniform field, then&lt;math&gt;\,\,\mathbf{q}\;\!{\cdot}\nabla~\!\psi=0&lt;/math&gt;''.

For the special case in which {{math|'''q'''}} is a unit vector {{math|'''s&amp;#770;'''&amp;#8202;,}} with {{mvar|s}} measuring distance in the direction of{{math|&amp;#8202; '''s&amp;#770;'''&amp;#8202;,}} definition ({{EquationNote|11}}) reduces to
{{NumBlk|:|&lt;math&gt;\mathbf{\hat{s}}{\cdot}\nabla\,\psi = \part_s \psi \,,
  &lt;/math&gt;|{{EquationRef|12}}}}
which agrees with ({{EquationNote|9g}}) but now holds for a ''generic'' field {{mvar|&amp;psi;}} [whereas ({{EquationNote|9g}}) was for a ''scalar'' field, and was derived as a ''theorem'' based on earlier definitions]. So{{math| '''s&amp;#770;&amp;sdot;'''&amp;nabla;&amp;#8202;,}} with a unit vector {{math|'''s'''&amp;#8202;,}} is the '''directional-derivative operator''' on a generic field; and by ({{EquationNote|11}}),&amp;#8201; {{math|'''q&amp;sdot;'''&amp;nabla;}} is a '''scaled directional derivative''' operator on a generic field.

In particular, if&amp;#8202; {{math|'''s&amp;#770;'''}} is &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;,&amp;#8201; we have
:&lt;math&gt;\part_n \psi \,=\, \mathbf{\hat{n}}\;\!{\cdot}\nabla\,\psi \,,&lt;/math&gt;
which we may substitute into the original definition of the Laplacian ({{EquationNote|4L}}) to obtain
{{NumBlk|:|&lt;math&gt;\triangle\psi \,=\,
    \tfrac{1}{dV}\!\iint_{\delta S} \mathbf{\hat{n}}{\cdot}\nabla\,\psi \;dS \,,
  &lt;/math&gt;|{{EquationRef|13L}}}}
which is just ({{EquationNote|9L}}) again, except that it now holds for for a ''generic'' field.

If our general definition of the gradient ({{EquationNote|4g}}) is also taken as the general definition of the {{math|&amp;nabla;}} operator,&lt;ref&gt;''Cf''. [[#borisenko-tarapov-68|Borisenko &amp;amp; Tarapov, 1968]], p.&amp;#8239;157, eq.&amp;#8239;(4.43), quoted in [[#tai-95|Tai, 1995]], p.&amp;#8239;33, eq.&amp;#8239;(4.19).&lt;/ref&gt; then, comparing ({{EquationNote|4g}}) with ({{EquationNote|4c}}), ({{EquationNote|4d}}), and ({{EquationNote|13L}}), we see that
:&lt;math&gt;\begin{align}
    \operatorname{curl}\mathbf{q} ~\!&amp;= \nabla(\times\mathbf{q}) \\
    \operatorname{div}\mathbf{q}  ~\!&amp;= \nabla(\cdot\,\mathbf{q}) \\
    \triangle\psi                 ~\!&amp;= \nabla(\cdot\nabla\,\psi) \,,
  \end{align}&lt;/math&gt;
where the parentheses may seem to be required on account of the closing {{mvar|dS}}&amp;#8202; in ({{EquationNote|4g}}).&lt;ref&gt;The first two cases may be compared with Javid &amp;amp; Brown, 1963, cited in [[#tai-94|Tai, 1994]], p.&amp;#8239;15.&lt;/ref&gt; But if we write the factor {{mvar|dS}} ''before'' the integrand, the del operator in ({{EquationNote|4g}}) becomes
:&lt;math&gt;\nabla = \tfrac{1}{dV}\!\iint_{\delta S} dS\,\mathbf{\hat{n}} &lt;/math&gt;
&amp;mdash;''if''&amp;#8202; we insist that it is to be read as a operator looking for an operand, and not as a self-contained expression. Then, if we similarly bring forward the {{mvar|dS}} in ({{EquationNote|4c}}), ({{EquationNote|4d}}), and ({{EquationNote|13L}}), the respective operators become&lt;ref&gt;The first two cases may be compared with Neff, 1991, cited in [[#tai-94|Tai, 1994]], p.&amp;#8239;16.&lt;/ref&gt;
{{NumBlk|:|&lt;math&gt;\begin{align}
    \operatorname{curl} &amp;= \nabla\times \\
    \operatorname{div}  &amp;= \nabla~\!\boldsymbol{\cdot} \\
    \triangle           &amp;= \nabla\boldsymbol{\cdot}\nabla
  \end{align}&lt;/math&gt;|{{EquationRef|14}}}}
(pronounced &quot;del cross&quot;, &quot;del dot&quot;, and &quot;del dot del&quot;), of which the last is usually abbreviated as{{math| &amp;nabla;&lt;sup&gt;2&lt;/sup&gt;}}&amp;#8201; (&quot;del squared&quot;).&lt;ref&gt;But Gibbs ([[#gibbs-1881-4|1881]]) and Wilson ([[#wilson-1901|1901]]) were content to leave it as {{math|&amp;nabla;'''&amp;sdot;'''&amp;nabla;}}.  And they did not call it the ''Laplacian''; they used that term with a different meaning, which has apparently fallen out of fashion.&lt;/ref&gt; These notations are ubiquitous.

Another way to obtain the {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; operators (but ''not''{{math|&amp;#8202; &amp;nabla;&lt;sup&gt;2&lt;/sup&gt;}}), again inspired by ({{EquationNote|4g}}), is to define
{{NumBlk|:|&lt;math&gt;T(\nabla)
    \,:=\, \tfrac{1}{dV}\!\iint_{\delta S} T(\mathbf{\hat{n}}) \,dS \,,
  &lt;/math&gt;|{{EquationRef|14s}}}}
where {{mvar|T}}&amp;#8202; is any well-defined function that takes a vector argument. Setting{{math|&amp;#8202; ''T''&amp;#8202;(&amp;nabla;)}} to {{math|&amp;nabla;''p''&amp;#8202;,}} {{math|&amp;nabla;&amp;#8202;&amp;times;&amp;#8202;'''q'''&amp;#8202;,}} and{{math| &amp;nabla;'''&amp;sdot;&amp;#8239;q'''}}&amp;#8239; in ({{EquationNote|14s}}), we obtain respectively {{math|&amp;nabla;''p''&amp;#8202;,}}&amp;#8202; {{math|curl&amp;#8201;'''q'''&amp;#8202;,}} and&amp;#8202; {{math|div&amp;#8239;'''q'''}}&amp;#8239; as given by ({{EquationNote|4g}}) to ({{EquationNote|4d}}). But this approach has undesirable side-effects&amp;mdash;for example, that {{math|&amp;nabla;''p''}}&amp;#8202; becomes synonymous with{{math| ''p''&amp;nabla;}}.&amp;#8201; Accordingly, Chen-To Tai,&lt;ref&gt;[[#tai-fang-91|Tai &amp;amp; Fang, 1991]], pp.&amp;#8239;168–9.&lt;/ref&gt; on the left of ({{EquationNote|14s}}), replaces{{math| &amp;nabla;}} with his original symbol &lt;math&gt;\nabla\!\!\!\!^{\textstyle_-}~\!\!,\,&lt;/math&gt; which he calls the &quot;symbolic operator&quot; or the &quot;{{nowrap|''S''&amp;#8202;-operator}}&quot; or, later, the &quot;symbolic vector&quot; or the &quot;dummy vector&quot;. Tai in his later works (e.g.,&amp;#8239;[[#tai-94|1994]],&amp;#8239;[[#tai-95|1995]]) does not tolerate cross- or dot-products involving the del operator, but ''does'' tolerate such products involving his symbolic vector ([[#tai-95|1995]], pp.&amp;#8239;50–52).

There is a misconception that the operational equivalences in ({{EquationNote|14}}) apply ''only'' in Cartesian coordinates.&lt;ref&gt;Durney &amp;amp; Johnson, in ''Introduction to Modern Electromagnetics'' (1969, p.&amp;#8239;45, cited in [[#tai-94|Tai, 1994]], p.&amp;#8239;12), make the absurd statement that &quot;a{{math| &amp;nabla;}} operator cannot be defined in the other coordinate systems&amp;hellip;&quot; In the context, they apparently meant to say that&amp;#8202; {{math|div&amp;#8202;'''A'''}} isn't&amp;#8202; {{math|&amp;nabla;'''&amp;sdot;A'''}}&amp;#8202; in other coordinate systems. Robert S. Elliott, in ''Electromagnetics'' (1966, p.&amp;#8239;606, cited in [[#tai-94|Tai, 1994]], p.&amp;#8239;13), says that &quot;only in Cartesian coordinates&amp;hellip; do the gradient and divergence operators turn out to be identical.&quot; Apparently he meant to say that only in Cartesian coordinates do the two operators differ by a dot. But what these authors apparently meant to say is still wrong, as shown with counterexamples by Kemmer (next citation).&lt;/ref&gt; Tai does not accept them even in that case. But, because these equivalences have been derived from ''coordinate-free'' definitions of the operators, they must remain valid in any coordinate system ''provided that they are expressed correctly''&amp;mdash;without (e.g.) inadvertently taking dependent variables inside or outside differentiations.&lt;ref&gt;The perception that they are restricted to Cartesian coordinates arises partly from failure to allow for the variability of the basis vectors in curvilinear coordinate systems; ''cf''. [[#kemmer-77|Kemmer, 1977]], pp.&amp;#8239;163–5, 172–3 (Exs.&amp;#8239;2,&amp;#8239;3,&amp;#8239;5), 230–33 (sol'ns). From the del operator and the derivatives of the basis vectors w.r.t. the coordinates, Kemmer finds the curl and divergence in cylindrical coordinates, notes that we can do the same &quot;with a little greater effort&quot; in spherical coordinates (p.&amp;#8239;230), and finds the Laplacian of a scalar in both coordinate systems (p.&amp;#8239;231). He further reports that the method works for the Laplacian of a vector in cylindrical and spherical coordinates and is relatively convenient for the former (p.&amp;#8239;232), for which &quot;differentiation of the unit vectors is very simple&quot; (p.&amp;#8239;165).&lt;/ref&gt; That does ''not'' mean that they are always convenient, or easily verified, or conducive to the avoidance of error. But they sometimes make useful mnemonics; e.g., they let us rewrite identities ({{EquationNote|8c}}), ({{EquationNote|8g}}), and ({{EquationNote|8p}}) as
{{NumBlk|:|&lt;math&gt;\left.\begin{align}
    \nabla\!\times\!\mathbf{q}\cdot\mathbf{b}
    &amp;\,=\, \nabla\cdot\mathbf{q}\!\times\!\mathbf{b}\\[.5ex]
    \nabla p \cdot \mathbf{b}
    &amp;\,=\, \nabla\cdot\;\! p\mathbf{b}\\[.5ex]
    \nabla p \times \mathbf{b}
    &amp;\,=\, \nabla \times p\mathbf{b}
  \end{align}~\right\}\quad&lt;/math&gt;for uniform {{math|'''b'''}}.
    |{{EquationRef|15}}}}
These would be basic ''algebraic'' vector identities if&amp;#8202; {{math|&amp;nabla;}} were an ordinary vector, and one could try to derive them from the &quot;algebraic&quot; behavior of{{math| &amp;nabla;}}; but they're not, because it isn't, so we didn't&amp;#8239;!  Moreover, these simple &quot;algebraic&quot; rules are for a uniform {{math|'''b''',}} and do not of themselves tell us what to do if&amp;#8202; {{math|'''b'''}} is spatially variable; for example, ({{EquationNote|8g}}) is not applicable to ({{EquationNote|7d}}).

{{cob}}

=== The advection operator ===

{{cot}}

Variation or transportation of a property of a medium due to motion with the medium is called '''advection''' (which, according to its Latin roots, means &quot;carrying to&quot;). Suppose that a medium (possibly a fluid) moves with a velocity field {{math|'''v'''}} in some inertial reference frame. Let {{mvar|&amp;psi;}} be a field (possibly a scalar field or a vector field) expressing some property of the medium (e.g., density, or acceleration, or stress,{{efn|Stress is a second-order tensor, and the origin of the term &quot;tensor&quot;; but, for present purposes, it's just another possible example of a field called{{mvar| &amp;psi;}}.}}&amp;hellip; or even {{math|'''v''' }}itself). We have seen that the time-derivative of{{mvar| &amp;psi;}} may be specified in two different ways: as the ''partial'' derivative {{mvar|{{sfrac|&amp;part;&amp;psi;|&amp;part;t}}&amp;#8202;,}} evaluated at a fixed point (in the chosen reference frame), or as the ''material'' derivative {{mvar|{{sfrac|d&amp;psi;|dt}}&amp;#8202;}}, evaluated at a point moving at velocity {{math|'''v'''}} (i.e., ''with the medium''). The difference&amp;#8202; {{mvar|{{sfrac|d&amp;psi;|dt}}&amp;#8202;&amp;minus;&amp;#8202;{{sfrac|&amp;part;&amp;psi;|&amp;part;t}}&amp;#8202;}} is due to motion with the medium. To find another expression for this difference, let {{mvar|s}} be a parameter measuring distance along the path traveled by a particle of the medium. Then, for points along the path, the surface-plot of the small change in {{mvar|&amp;psi;}} (or any component thereof) as a function of small changes in {{mvar|t}} and {{mvar|s&amp;#8202;}} (plotted on perpendicular axes) can be taken as a plane through the origin, so that
:{{big|&lt;math&gt;d\psi
    = \tfrac{\part\psi}{\part t}~\!dt + \tfrac{\part\psi}{\part s}~\!ds \;;
  &lt;/math&gt;}}
that is, the change in {{mvar|&amp;psi;}} is the sum of the changes due to the change in {{mvar|t}} and the change in {{mvar|s&amp;#8202;}}. Dividing by {{mvar|dt}} gives
:{{big|&lt;math&gt;\begin{align}\tfrac{d\psi}{dt}
  &amp;= \tfrac{\part\psi}{\part t}+\tfrac{\part\psi}{\part s}~\!\tfrac{ds}{dt}\\[1ex]
  &amp;= \tfrac{\part\psi}{\part t}+\tfrac{\part\psi}{\part s}~\!|\mathbf{v}| \,;
  \end{align}&lt;/math&gt;}}
i.e.,
:{{big|&lt;math&gt;\tfrac{d\psi}{dt}
    = \tfrac{\part\psi}{\part t} + |\mathbf{v}|\,\part_s \psi
  &lt;/math&gt;}}
(and the first term on the right could have been written {{mvar|&amp;part;&lt;sub&gt;t&lt;/sub&gt;&amp;#8239;&amp;psi;}}). So the second term on the right is the contribution to the material derivative due to motion with the medium; it is called the '''advective term''', and is non-zero wherever a particle of the medium moves along a path on which {{mvar|&amp;psi;}} varies with location&amp;mdash;even if {{mvar|&amp;psi;}} at ''each'' location is constant over time.&amp;#8201; So the operator&amp;#8202; {{math|{{abs|'''v'''}}&amp;#8201;''&amp;part;&lt;sub&gt;s&lt;/sub&gt;''&amp;#8202;,}} where {{mvar|s}} measures distance along the path, is the ''advection operator''&amp;#8239;: it maps a property of a medium to the advective term in the time-derivative of that property. If{{mvar| &amp;psi;}} is {{math|'''v''' }}itself, the above result becomes
:{{big|&lt;math&gt;\tfrac{d\mathbf{v}}{dt}
    = \tfrac{\part\mathbf{v}}{\part t} + |\mathbf{v}|\,\part_s \mathbf{v} \,,
  &lt;/math&gt;}}
where the left-hand side (the ''material'' acceleration) is as given by Newton's second law, and the first term on the right (which we might call the &quot;partial&quot; acceleration) is the time-derivative of velocity in the chosen reference frame, and the second term on the right (the ''advective'' term) is the correction that must be added to the &quot;partial&quot; acceleration in order to obtain the material acceleration. This term is non-zero wherever velocity is non-zero and varies along a path, even if the velocity at each point on the path is constant over time (as when water speeds up while flowing at a constant volumetric rate into a nozzle). Paradoxically, while the material acceleration and the &quot;partial&quot; acceleration are apparently linear (first-degree) in {{math|'''v''',}} their difference (the advective term) is not. Thus the distinction between {{mvar|{{sfrac|&amp;part;&amp;psi;|&amp;part;t}}}} and {{mvar|{{sfrac|d&amp;psi;|dt}}}}&amp;#8202; has the far-reaching implication that ''fluid dynamics is non-linear''.

Applying ({{EquationNote|11}}) to the last two equations, we obtain respectively
{{NumBlk|:|{{big|&lt;math&gt;\tfrac{d\psi}{dt}
    = \tfrac{\part\psi}{\part t} + \mathbf{v}{\cdot}\nabla\,\psi
  &lt;/math&gt;}}|{{EquationRef|16}}}}
and
{{NumBlk|:|{{big|&lt;math&gt;\tfrac{d\mathbf{v}}{dt}
    = \tfrac{\part\mathbf{v}}{\part t} + \mathbf{v}{\cdot}\nabla\,\mathbf{v} \,,
  &lt;/math&gt;}}|{{EquationRef|16v}}}}
where, in each case, the second term on the right is the advective term. So ''the '''advection operator''' can also be written''&amp;#8239;{{math| '''v&amp;sdot;'''&amp;nabla;&amp;#8202;}}.

When the generic {{mvar|&amp;psi;&amp;#8202;}} in ({{EquationNote|16}}) is replaced by the density {{mvar|&amp;rho;&amp;#8202;}}, we get a relation between {{mvar|{{sfrac|&amp;part;&amp;rho;|&amp;part;t}}&amp;#8202;}} and {{mvar|{{sfrac|d&amp;rho;|dt}}&amp;#8202;}}, both of which we have seen before&amp;mdash;in equations ({{EquationNote|7d}}) and ({{EquationNote|7d'}}) above. Substituting from those equations then gives
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\rho\mathbf{v}
    \,=\, \rho\operatorname{div}\mathbf{v} \,+\, \mathbf{v}\cdot\nabla\rho \,,
  &lt;/math&gt;|{{EquationRef|17}}}}
where {{math|&amp;nabla;''&amp;rho;''}} can be taken as a gradient since {{mvar|&amp;rho;}} is scalar. This result is in fact an identity&amp;mdash;a ''product rule for the divergence''&amp;mdash;as we shall eventually confirm by another method.

{{cob}}

=== Generalized volume-integral theorem ===

{{cot}}

We can rewrite the fourth integral theorem ({{EquationNote|5L}}) in the &quot;dot del&quot; notation as
{{NumBlk|:|&lt;math&gt;\iint_S \mathbf{\hat{n}}\;\!{\cdot}\nabla\,\psi \;dS
    \,= \iiint_V \triangle\psi ~dV \,.
  &lt;/math&gt;|{{EquationRef|18L}}}}
Then, using notations ({{EquationNote|14}}), we can condense ''all four'' integral theorems ({{EquationNote|5g}}), ({{EquationNote|5c}}), ({{EquationNote|5d}}), and ({{EquationNote|18L}}) into the single equation
{{NumBlk|:|&lt;math&gt;\iint_S \mathbf{\hat{n}} * \psi \;dS
    \,= \iiint_V \nabla * \psi ~dV \,,
  &lt;/math&gt;|{{EquationRef|19}}}}
where the wildcard {{math|&amp;lowast;}} (conveniently pronounced &quot;star&quot;) is a generic binary operator which may be replaced by a null (direct juxtaposition of the operands) for theorem ({{EquationNote|5g}}), or a cross for ({{EquationNote|5c}}), or a dot for ({{EquationNote|5d}}), or&amp;#8202; {{math|'''&amp;sdot;'''&amp;nabla;}} for ({{EquationNote|18L}}); and the operand {{mvar|&amp;psi;}} is of a kind that makes the operator meaningful. This single equation is a ''generalized volume-integral theorem'', relating an integral over a volume to an integral over its enclosing surface.&lt;ref&gt;Kemmer ([[#kemmer-77|1977]], p.&amp;#8239;98, eq.&amp;#8239;4) gives an equivalent result for our first three integral theorems ({{EquationNote|5g}} to {{EquationNote|5d}}) only, and calls it the ''generalized divergence theorem'' because the divergence theorem is its most familiar special case.&lt;/ref&gt;

Theorem ({{EquationNote|19}}) is based on the following definitions, which have been found unambiguous:
* the ''gradient'' of a scalar field {{mvar|p}} is the closed-surface integral of&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}p\,&lt;/math&gt; per unit volume, where &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is the outward unit normal;
* the ''curl'' of a vector field is the skew surface integral per unit volume, also called the surface circulation per unit volume;
* the ''divergence'' of a vector field is the outward flux integral per unit volume; and
* the ''Laplacian'' is the closed-surface integral of the outward normal derivative, per unit volume.
The gradient maps a scalar field to a vector field; the curl maps a vector field to a vector field; the divergence maps a vector field to a scalar field; and the Laplacian maps a scalar field to a scalar field, or a vector field to a vector field, etc.

The ''gradient'' of {{mvar|p}}, as defined above, has been shown to be also
* the vector whose (scalar) component in any direction is the ''directional derivative'' of{{mvar| p}} in that direction (i.e. the derivative of{{mvar| p}} w.r.t. distance in that direction), and
* the vector whose direction is that in which the directional derivative of{{mvar| p}} is a maximum, and whose magnitude is that maximum.
Consistent with these alternative definitions of the gradient, we have defined the&amp;#8202;{{math| '''&amp;sdot;'''&amp;nabla;}} operator so that&amp;#8202; {{math|'''s&amp;#770;&amp;sdot;'''&amp;nabla;}} (for a ''unit'' vector {{math|'''s&amp;#770;'''}}) is the operator yielding the directional derivative in the direction of&amp;#8202; {{math|'''s&amp;#770;'''&amp;#8202;,}} and we have used that notation to bring theorem ({{EquationNote|5L}}) under theorem ({{EquationNote|19}}).

So far, we have said comparatively little about the curl. That imbalance will now be rectified.

{{cob}}

== Closed-circuit integrals per unit area ==

=== Instant integral theorems (on a condition) ===

{{cot}}

Theorems ({{EquationNote|5g}}) to ({{EquationNote|5L}}) are three-dimensional: each of them relates an integral over a volume {{mvar|V}}&amp;#8202; to an integral over its enclosing surface{{mvar| S}}. We now seek analogous ''two''-dimensional theorems, each of which relates an integral over a surface segment to an integral around its enclosing curve. For maximum generality, the surface segment should be allowed to be curved into a third dimension.{{efn|In mathematical jargon, it should be a two-dimensional ''manifold'' embedded in 3D Euclidean space.}} Theorems of the latter kind can be obtained as special cases of theorems ({{EquationNote|5g}}) to ({{EquationNote|5L}}) by suitably choosing {{mvar|V}} and {{mvar|S&amp;#8202;}}; this is another advantage of our &quot;volume first&quot; approach.

Let {{mvar|&amp;Sigma;}} be a surface segment enclosed by a curve {{mvar|C}} (a ''circuit'' or ''closed contour''), and let {{mvar|l}} be a parameter measuring arc length around {{mvar|C&amp;#8202;}}, so that a general element of{{mvar| C}}&amp;#8202; has length{{mvar| dl&amp;#8202;}}; and let a general element of the surface {{mvar|&amp;Sigma;}}&amp;#8202; have area {{mvar|d&amp;Sigma;}}. Let&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt; be the unit normal vector at a general point on {{mvar|&amp;Sigma;&amp;#8202;}}, and let &lt;math&gt;\mathbf{\hat{t}}&lt;/math&gt; be the unit ''tangent'' vector to{{mvar| C}} at a general point on {{mvar|C}}&amp;#8202; in the direction of increasing{{mvar| l}}. In the original case of a surface enclosing a volume, we had to decide whether the unit normal pointed into or out of the volume (we chose the latter). In the present case of a circuit enclosing a surface segment, we have to decide whether {{mvar|l}} is measured clockwise or counterclockwise as seen when looking in the direction of the unit normal, and we choose clockwise. So {{mvar|l }}''is measured clockwise about&lt;math&gt;~\boldsymbol{\hat{\nu}},&lt;/math&gt;'' and {{mvar|C }}is ''traversed'' clockwise about&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;.

From {{mvar|&amp;Sigma;}}&amp;#8202; we can construct obvious candidates for {{mvar|V}} and{{mvar| S}}. From every point on {{mvar|&amp;Sigma;&amp;#8202;}}, erect a perpendicular with a uniform ''small''&amp;#8202; height {{mvar|h}} in the direction of&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;. Then simply let {{mvar|V}} be the volume occupied by all the perpendiculars, and let {{mvar|S}} be its enclosing surface. Thus {{mvar|V}} is a (generally curved) thin slab of uniform thickness{{mvar| h}}, whose enclosing surface {{mvar|S}} consists of two close parallel (generally curved) broad faces connected by a perpendicular ''edge-face'' of uniform height{{mvar| h&amp;#8202;}}; and we can treat&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt; as a vector ''field''&amp;#8202; by extrapolating it perpendicularly from{{mvar| &amp;Sigma;}}. If we can arrange for {{mvar|h}} to cancel out, the volume{{mvar| V}}&amp;#8202; will serve as a 3D representation of the surface segment{{mvar| &amp;Sigma;}}&amp;#8202; while the ''edge-face'' will serve as a 2D representation of the curve{{mvar| C&amp;#8202;}}, so that our four theorems will relate an integral around {{mvar|C}}&amp;#8202; to an integral over {{mvar|&amp;Sigma;}}&amp;#8201; ''provided that there is no contribution from the broad faces to the integral over''{{mvar| S}}. For brevity, let us call this proviso the '''2D condition'''.

''If''&amp;#8202; the 2D condition is satisfied, an integral over the new {{mvar|S}}&amp;#8202; reduces to an integral over the edge-face, on which
:&lt;math&gt;dS = h\,dl \,,&lt;/math&gt;
so that the cancellation of{{mvar| h}} will leave an integral over {{mvar|C}}&amp;#8202; w.r.t. length. Meanwhile, in an integral over the new{{mvar| V}}, regardless of the 2D condition, we have
:&lt;math&gt;dV = h\,d\varSigma \,,&lt;/math&gt;
so that the cancellation of{{mvar| h}} will leave an integral over {{mvar|&amp;Sigma;}}&amp;#8202; w.r.t. area. So, substituting for {{mvar|dS}} and {{mvar|dV}}&amp;#8202; in ({{EquationNote|5g}}) to ({{EquationNote|5L}}), and canceling {{mvar|h}} as planned, we obtain respectively
{{NumBlk|:|&lt;math&gt;~~~~~\!\oint_C \mathbf{\hat{n}}~\!p \,dl
    \,= \iint_{\varSigma} \nabla p ~d\varSigma \qquad(?),
  &lt;/math&gt;|{{EquationRef|20g}}}}
{{NumBlk|:|&lt;math&gt;\oint_C \mathbf{\hat{n}}\times\mathbf{q} \,dl
    \,= \iint_{\varSigma} \operatorname{curl}\mathbf{q} ~d\varSigma \qquad(?),
  &lt;/math&gt;|{{EquationRef|20c}}}}
{{NumBlk|:|&lt;math&gt;~~\!\oint_C \mathbf{\hat{n}\cdot q} \,dl
    \,= \iint_{\varSigma} \operatorname{div}\mathbf{q} ~d\varSigma \qquad(?),
  &lt;/math&gt;|{{EquationRef|20d}}}}
{{NumBlk|:|&lt;math&gt;~~\oint_C \part_n \psi \;dl
    \,= \iint_{\varSigma} \triangle\psi ~d\varSigma \qquad(?),
  &lt;/math&gt;|{{EquationRef|20L}}}}
''all subject to the 2D condition'' (hence the question marks). In each equation, the circle on the left integral sign acknowledges that the integral is around a closed loop. The unit vector &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;, which ''was'' normal to the edge-face, is now normal to both &lt;math&gt;\mathbf{\hat{t}}&lt;/math&gt; and&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;; that is, &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is tangential to the surface segment {{mvar|&amp;Sigma;}}&amp;#8202; and projects perpendicularly outward from its bounding curve.

On the left side of ({{EquationNote|20g}}), the 2D condition is satisfied if (but not only if)&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}p&lt;/math&gt; takes equal-and-opposite values at any two opposing points on opposing broad faces of{{mvar| S&amp;#8202;,}} i.e. if {{mvar|p}} takes the ''same'' value at such points, i.e. if {{mvar|p}} has a zero directional derivative normal to{{mvar| &amp;Sigma;}}.

Skipping forward to ({{EquationNote|20L}}), we see that the 2D condition is satisfied if&lt;math&gt;~\part_n \psi&lt;/math&gt; takes equal-and-opposite values at any two opposing points on opposing broad faces of{{mvar| S&amp;#8202;,}} i.e. if&lt;math&gt;~\part_{\nu}\psi&lt;/math&gt; (where &lt;math&gt;\nu&lt;/math&gt; measures distance in the direction of&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;) takes the ''same'' value at such points, i.e. if&lt;math&gt;~\part^2_{\nu}\psi\!=\!0&lt;/math&gt;.

For ({{EquationNote|20c}}) and ({{EquationNote|20d}}), the 2D condition can be satisfied by construction, with more useful results&amp;mdash;as explained under the next two headings. To facilitate this process, we first make a minor adjustment to {{mvar|&amp;Sigma;}} and{{mvar| C}}. Noting that any curved surface segment can be approximated to any desired accuracy by a ''polyhedral'' surface enclosed by a ''polygon'', we shall indeed consider {{mvar|&amp;Sigma;}}&amp;#8202; to be a polyhedral surface made up of small planar elements, {{mvar|d&amp;Sigma;}}&amp;#8202; being the area of a general element, and we shall indeed consider {{mvar|C}} to be a polygon with short sides, {{mvar|dl}} being the length of a general side.{{efn|If any part of our argument requires {{mvar|&amp;Sigma;}} or {{mvar|C}} to be ''smooth'', this is not an impediment, because having approximated {{mvar|&amp;Sigma;}} or{{mvar| C}} to any desired accuracy by a polyhedron or polygon, we can then approximate the polyhedron or polygon to any desired ''higher'' accuracy by a smooth surface or curve!}} The benefit of this trick, as we shall see, is to make the unit normal &lt;math&gt;\boldsymbol{\hat{\nu}}&lt;/math&gt; uniform over each surface element, without forcing us to treat {{math|'''q'''}} (or any other field) as uniform over the same element. But, as the elements of{{mvar| C}}&amp;#8202; can ''independently'' be made as short as we like (dividing straight sides into shorter elements if necessary!), we can still consider &lt;math&gt;\boldsymbol{\hat{\nu}},&lt;/math&gt; {{math|'''q'''&amp;#8202;,}} and &lt;math&gt;\mathbf{\hat{t}}&lt;/math&gt; to be uniform over each element of{{mvar| C}}.

{{cob}}

=== Special case for the gradient ===

{{cot}}

In ({{EquationNote|20c}}), the 2D condition is satisfied by&lt;math&gt;~\mathbf{q}\!=\!p\boldsymbol{\hat{\nu}}&lt;/math&gt; (where {{mvar|p}} is a scalar field), because then the integrand on the left is zero on the broad faces of{{mvar| S&amp;#8202;}}, where {{math|'''n'''}} is parallel to&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;. Equation ({{EquationNote|20c}}) then becomes
{{NumBlk|:|&lt;math&gt;
    \oint_C \mathbf{\hat{n}}{\times}\boldsymbol{\hat{\nu}}~\!p \;dl \,=
    \iint_{\varSigma}\operatorname{curl}p\boldsymbol{\hat{\nu}}\;d\varSigma \,.
  &lt;/math&gt;|{{EquationRef|21n}}}}
Now on the left,&amp;#8201; &lt;math&gt;\mathbf{\hat{n}}\!\times\!\boldsymbol{\hat{\nu}}\!=\!-\mathbf{\hat{t}}~\!;\,&lt;/math&gt; and on the right, over each surface element, the unit normal &lt;math&gt;\boldsymbol{\hat{\nu}}&lt;/math&gt; is uniform so that, by ({{EquationNote|8p}}),&amp;#8201; &lt;math&gt;\operatorname{curl}p\boldsymbol{\hat{\nu}}=~\!\!\nabla p \!\times\!\boldsymbol{\hat{\nu}}=-\boldsymbol{\hat{\nu}}\!\times\!\nabla p&lt;/math&gt;.  With these substitutions, the minus signs cancel and we get
{{NumBlk|:|&lt;math&gt;
    \oint_C p\mathbf{\hat{t}} \,dl \,=
    \iint_{\varSigma} \boldsymbol{\hat{\nu}}\times\nabla p \;d\varSigma
  &lt;/math&gt;|{{EquationRef|21g}}}}
or, if we write&amp;#8201; &lt;math&gt;d\mathbf{r}\!=\!\mathbf{\hat{t}}~\!dl&lt;/math&gt;&amp;#8201; and&amp;#8201; &lt;math&gt;\boldsymbol{d\varSigma}\!=\!\boldsymbol{\hat{\nu}}\,d\varSigma~\!,&lt;/math&gt;
{{NumBlk|:|&lt;math&gt;
    \oint_C p \,d\mathbf{r} \,=
    \iint_{\varSigma} \big(\boldsymbol{d\varSigma}\times\!\nabla p\big) \,.
  &lt;/math&gt;|{{EquationRef|21r}}}}
This result, although well attested in the literature,&lt;ref&gt;E.g., [[#gibbs-1881-4|Gibbs, 1884]], &amp;sect;&amp;#8239;165, eq.&amp;#8239;(1); [[#wilson-1901|Wilson, 1901]], p.&amp;#8239;255, Ex.&amp;#8239;1; [[#kemmer-77|Kemmer, 1977]], p.&amp;#8239;99, eq.&amp;#8239;(6); [[#hsu-84|Hsu, 1984]], p.&amp;#8239;146, eq.&amp;#8239;(7.31).&lt;/ref&gt; does not seem to have a name&amp;mdash;unlike the next result.

{{cob}}

=== Special case for the curl ===

{{cot}}

In ({{EquationNote|20d}}), the 2D condition is satisfied if {{math|'''q'''}} is replaced by&lt;math&gt;~\boldsymbol{\hat{\nu}}{\times}\mathbf{q}~\!,\,&lt;/math&gt; because then (again) the integrand on the left is zero on the broad faces of{{mvar| S&amp;#8202;}}, where {{math|'''n'''}} is parallel to&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt;. Equation ({{EquationNote|20d}}) then becomes
{{NumBlk|:|&lt;math&gt;
    \oint_C \mathbf{\hat{n}}\cdot\boldsymbol{\hat{\nu}}{\times}\mathbf{q} \;dl
    \,=
    \iint_{\varSigma}
      \operatorname{div}(\boldsymbol{\hat{\nu}}\!\times\!\mathbf{q})
    \,d\varSigma \,.
  &lt;/math&gt;|{{EquationRef|22n}}}}
Now on the left, the integrand can be written&amp;#8201; &lt;math&gt;\mathbf{\hat{n}}{\times}\boldsymbol{\hat{\nu}}\!\cdot\!\mathbf{q}\!=\!-\mathbf{\hat{t}}\!\cdot\!\mathbf{q}~\!;\,&lt;/math&gt; and on the right,&amp;#8201; &lt;math&gt;\operatorname{div}(\boldsymbol{\hat{\nu}}\!\times\!\mathbf{q})\!=\!-\operatorname{div}(\mathbf{q}\!\times\!\boldsymbol{\hat{\nu}})\!=\!-\operatorname{curl}\mathbf{q}\cdot\boldsymbol{\hat{\nu}}\,&lt;/math&gt; by identity ({{EquationNote|8c}}), since &lt;math&gt;\boldsymbol{\hat{\nu}}&lt;/math&gt; is uniform over each surface element.  With these substitutions, the minus signs cancel and we get
{{NumBlk|:|&lt;math&gt;
    \oint_C \mathbf{q} \cdot \mathbf{\hat{t}} \,dl \,=
    \iint_{\varSigma}
      \operatorname{curl}\mathbf{q}\cdot\boldsymbol{\hat{\nu}}
    \,d\varSigma
  &lt;/math&gt;|{{EquationRef|22c}}}}
or, if we again write&amp;#8201; &lt;math&gt;d\mathbf{r}\!=\!\mathbf{\hat{t}}~\!dl&lt;/math&gt;&amp;#8201; and&amp;#8201; &lt;math&gt;\boldsymbol{d\varSigma}\!=\!\boldsymbol{\hat{\nu}}\,d\varSigma~\!,&lt;/math&gt;
{{NumBlk|:|&lt;math&gt;
    \oint_C \mathbf{q} \cdot d\mathbf{r} \,=
    \iint_{\varSigma}
      \operatorname{curl}\mathbf{q}\cdot\boldsymbol{d\varSigma} \,.
  &lt;/math&gt;|{{EquationRef|22r}}}}
This result&amp;mdash;the best-known theorem relating an integral over a surface segment to an integral around its enclosing curve, and the best-known theorem involving the curl&amp;mdash;is called ''[[w:Sir George Stokes, 1st Baronet|Stokes]]' theorem'' or, more properly, the '''[[w:Lord Kelvin|Kelvin]]&amp;ndash;Stokes theorem''',&lt;ref&gt;''Cf''. [[#katz-79|Katz, 1979]], pp.&amp;#8239;149–50.&lt;/ref&gt; or simply the ''curl theorem''.&lt;ref&gt;Although Hsu ([[#hsu-84|1984]], p.&amp;#8239;141) applies that name to our theorem ({{EquationNote|5c}}).&lt;/ref&gt;

The integral on the left of ({{EquationNote|22c}}) or ({{EquationNote|22r}}) is called the '''circulation''' of the vector field {{math|'''q'''}} around the closed curve{{mvar| C}}. So, &lt;span id=&quot;kelvin-stokes-verbal&quot;&gt;in words&lt;/span&gt;, the Kelvin&amp;ndash;Stokes theorem says that ''the circulation of a vector field around a closed curve is equal to the flux of the curl of that vector field through any surface spanning that closed curve''.

Now let a general element of {{mvar|&amp;Sigma;}} (with area {{mvar|d&amp;Sigma;&amp;#8202;}}) be enclosed by the curve {{mvar|&amp;delta;C}}, traversed in the same direction as the outer curve {{mvar|C}}. Then, applying ({{EquationNote|22c}}) to the single element, we have
:&lt;math&gt;
    \oint_{\delta C} \!\mathbf{q} \cdot \mathbf{\hat{t}} \,dl \,=\,
    \operatorname{curl}\mathbf{q}\cdot\boldsymbol{\hat{\nu}} \,d\varSigma \,;
  &lt;/math&gt;
that is,
{{NumBlk|:|&lt;math&gt;
    \operatorname{curl}\mathbf{q}\cdot\boldsymbol{\hat{\nu}} \,=\,
    \frac{1}{d\varSigma}\oint_{\delta C}\!\mathbf{q}\cdot\mathbf{\hat{t}}\,dl\,,
  &lt;/math&gt;|{{EquationRef|23c}}}}
where the right-hand side is simply the ''circulation per unit area''.

Equation ({{EquationNote|23c}}) is an alternative definition of the curl: it says that ''the curl of''{{math| '''q'''}} ''is the vector whose scalar component in any direction is the circulation of''{{math| '''q'''}} ''per unit area of a surface whose normal points in that direction''. For ''real''{{math| '''q''',}} this component has its maximum, namely {{math|{{abs|curl&amp;#8201;'''q'''}}&amp;#8202;,}} in the direction of{{math| curl&amp;#8239;'''q'''&amp;#8202;}}; thus ''the curl of''{{math| '''q'''}} ''is the vector whose direction is that which a surface must face if the circulation of''{{math| '''q'''}} ''per unit area of that surface is to be a maximum, and whose magnitude is that maximum''. This is the usual conceptual definition of the curl.&lt;ref&gt;E.g., [[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;#8239;61; [[#hsu-84|Hsu, 1984]], pp.&amp;#8239;117–18.&lt;/ref&gt;

[Notice, however, that our original volume-based definition ({{EquationNote|4c}}) is more succinct: the curl is the closed-surface circulation per unit volume, i.e. the skew surface integral per unit volume.]

It should now be clear where the curl gets its name (coined by [[w:James Clerk Maxwell|Maxwell]]), and why it is also called the ''rotation'' (indeed the {{math|curl}} operator is sometimes written &quot;{{math|rot}}&quot;, especially in Continental languages, in which &quot;rot&quot; does not have the same unfortunate everyday meaning as in English).

[[File:Vorticity_Figure_03_a-m.gif|thumb|Animation of a non-vortex-like velocity field whose curl (like its circulation around the red loop) is non-zero due to shear.]]
[[File:Vorticity_Figure_02_a-m.gif|thumb|Animation of a vortex-like velocity field whose curl is zero because the shear compensates for the rotation.]]

And it should now be unsurprising that ''a vector field with zero curl is described as '''irrotational''''' (which one must carefully pronounce differently from &quot;{{nowrap|irr''i&amp;#8202;''tational}}&quot;!), and that the curl of the velocity of a medium is called the '''vorticity'''.

However, a field does not need to be vortex-like in order to have a non-zero curl. For example, by identity ({{EquationNote|8p}}), in Cartesian coordinates, the velocity field {{math|''x'''''j'''}} has a curl equal to&amp;#8201; {{math|&amp;nabla;''x''&amp;#8201;&amp;times;&amp;#8239;'''j'''&amp;#8201;{{=}}&amp;#8201;'''i'''&amp;#8202;&amp;times;&amp;#8202;'''j'''&amp;#8201;{{=}}&amp;#8201;'''k'''&amp;#8202;,}}&amp;#8201; although it describes a ''shearing'' motion rather than a rotating motion. This is understandable because if you hold a pencil between the palms of your hands and slide one palm over the other (a shearing motion), the pencil rotates.

Conversely, we can have a vortex-like field whose curl is zero everywhere except on or near the axis of the vortex. For example, the '''Maxwell&amp;ndash;Amp&amp;egrave;re law''' in magnetostatics says that&amp;#8201; {{math|curl&amp;#8201;'''H'''&amp;#8201;{{=}}&amp;#8201;'''J'''&amp;#8202;,}} where {{math|'''H'''}} is the '''magnetizing field''' and {{math|'''J'''}} is the current density.{{efn|In the general case, there is an extra term {{math|{{sfrac|''&amp;part;''&amp;#8202;'''D'''|''&amp;part;t''}}}} on the right; but this term is zero in the magneto''static'' case.}} So if the current is confined to a wire, {{math|curl&amp;#8201;'''H'''&amp;#8202;}} is zero outside the wire&amp;mdash;although, as is well known, the field lines circle the wire. The resolution of the paradox is that {{math|'''H'''}} gets stronger as we approach the wire, making a shearing pattern, whose effect on the curl counteracts that of the rotation.

{{cob}}

=== The curl-grad and div-curl operators ===

{{cot}}

We have seen from ({{EquationNote|9L}}) that the Laplacian of a scalar field is the divergence of the gradient. Four more such second-order combinations make sense, namely the curl of the gradient (of a scalar field), and the divergence of the curl, the gradient of the divergence, and the curl of the curl (of a vector field). The first two&amp;#8239;&amp;mdash;&quot;curl grad&quot; and &quot;div curl&quot;&amp;mdash;&amp;#8239;can now be disposed of.

Let the surface segment {{mvar|&amp;Sigma;}} enclosed by the curve{{mvar| C}}&amp;#8202; be a segment of the closed surface {{mvar|S}} surrounding the volume{{mvar| V}}, and let {{mvar|&amp;Sigma;}} expand across {{mvar|S}} until it engulfs{{mvar| V}}, so that {{mvar|C}} shrinks to a point on the far side of{{mvar| S}}. Then, in the nameless theorem ({{EquationNote|21g}}) and the Kelvin&amp;ndash;Stokes theorem ({{EquationNote|22c}}), the integral on the left becomes zero while {{mvar|&amp;Sigma;}} and &lt;math&gt;\boldsymbol{\hat{\nu}}&lt;/math&gt; on the right become {{mvar|S}} and &lt;math&gt;\mathbf{\hat{n}},&lt;/math&gt; so that the theorems respectively reduce to
:&lt;math&gt;
    \iint_S \mathbf{\hat{n}}\times\nabla p \;dS
     \,=\, \mathbf{0}
  &lt;/math&gt;
and
:&lt;math&gt;
    \iint_S
      \mathbf{\hat{n}}\cdot\operatorname{curl}\mathbf{q}
    \;dS \,=\, 0 \,.
  &lt;/math&gt;
Applying theorem ({{EquationNote|5c}}) to the first of these two equations, and the divergence theorem ({{EquationNote|5d}}) to the second, we obtain respectively
:&lt;math&gt;\iiint_V \operatorname{curl}\nabla p \;dV ~\!=\, \mathbf{0} \,,&lt;/math&gt;
and
:&lt;math&gt;
    \iiint_{V}
      \operatorname{div}\operatorname{curl}\mathbf{q}
    \;dV ~\!=\, 0 \,.
  &lt;/math&gt;
As the integrals vanish for ''any'' volume {{mvar|V}}&amp;#8202; in which the integrands are defined, the integrands must be zero wherever they are defined; that is,
{{NumBlk|:|&lt;math&gt;
    \operatorname{curl}\nabla p \equiv \mathbf{0}
  &lt;/math&gt;|{{EquationRef|24c}}}}
and
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\operatorname{curl}\mathbf{q} \equiv 0 \,.
  &lt;/math&gt;|{{EquationRef|24d}}}}
In words, ''the curl of the gradient is zero'', and ''the divergence of the curl is zero''; or, more concisely, ''any gradient is irrotational'', and ''any curl is solenoidal''.

We might well ask whether the converses are true. Is every irrotational vector field the gradient of something? And is every solenoidal vector field the curl of something? The answers are affirmative, but the proofs require more preparation.

Meanwhile we may note, as a mnemonic aid, that when the left-hand sides of the last two equations are rewritten in the del-cross and del-dot notations, they become&amp;#8201; {{math|&amp;nabla;&amp;#8201;&amp;times;&amp;#8201;&amp;nabla;''p''}}&amp;#8201; and&amp;#8201; {{math|&amp;nabla;&amp;#8201;'''&amp;sdot;'''&amp;#8201;&amp;nabla;&amp;#8202;&amp;times;&amp;#8202;'''q'''&amp;#8202;,}} respectively. The former ''looks like'' (but isn't) a cross-product of two parallel vectors, and the latter ''looks like'' (but isn't) a scalar triple product with a repeated factor, so that each expression ''looks like'' it ought to be zero (and it is). But such appearances can lead one astray, because {{math|&amp;nabla;}} is an operator, not a self-contained vector quantity; for example,&amp;#8201; {{math|&amp;nabla;''p''&amp;#8201;&amp;times;&amp;#8201;&amp;nabla;''&amp;phi;''}}&amp;#8201; is ''not'' identically zero, because two gradients are not necessarily parallel.&lt;ref&gt;''Cf''. [[#feynman-63|Feynman, 1963]], vol.&amp;#8239;2, &amp;sect;2-8.&lt;/ref&gt;

We should also note, to tie a loose end, that identity ({{EquationNote|24d}}) was to be expected from our [[#kelvin-stokes-verbal|verbal statement]] of the Kelvin&amp;ndash;Stokes theorem ({{EquationNote|22c}}). That statement implies that the flux of the curl through any two surfaces spanning the same closed curve is the same. So if we make a ''closed'' surface from two spanning surfaces, the flux into one spanning surface is equal to the flux out of the other, i.e. the net flux out of the closed surface is zero, i.e. the integral of the divergence over the enclosed volume is zero; and since ''any'' simple volume in which the divergence is defined can be enclosed this way, the divergence itself (of the curl) must be zero wherever it is defined.

{{cob}}

== Change per unit length ==

{{cot}}

Continuing (and concluding) the trend of reducing the number of dimensions, we now seek ''one''-dimensional theorems, each of which relates an integral over a ''path'' to values at the endpoints of the path. For maximum generality, the path should be allowed to be curved into a second and a third dimension.

We ''could'' do this by further specializing theorems ({{EquationNote|5g}}) to ({{EquationNote|5L}}). We could take a curve {{math|&amp;Gamma;}} with a unit tangent vector{{math| '''s&amp;#770;'''}}. At every point on{{math| &amp;Gamma;}} we could mount a circular disk with a uniform ''small'' area{{mvar| &amp;alpha;&amp;#8202;,}} centered on{{math| &amp;Gamma;}} and orthogonal to it. We could let {{mvar|V}} be the volume occupied by all the disks and let {{mvar|S}} be its enclosing surface; thus {{mvar|V}} would be a thin right circular cylinder, except that its axis could be curved. If we could arrange for {{mvar|&amp;alpha;}} to cancel out, our four theorems would indeed be reduced to the desired form, ''provided'' that there were no contribution from the curved face of the &quot;cylinder&quot; to the integral over{{mvar| S}} (the &quot;1D proviso&quot;). But, as it turns out, this exercise yields only one case in which the &quot;1D proviso&quot; can be satisfied by a construction involving {{math|'''s&amp;#770;'''}} and a general field, and we have already ''almost'' discovered that case by a simpler and more conventional argument&amp;mdash;which we shall now continue.

{{cob}}

=== Fundamental theorem ===

{{cot}}

Equation ({{EquationNote|9g}}) is applicable where {{math|''p''('''r''')}} is a scalar field,&amp;#8201; {{mvar|s}} is a parameter measuring arc length along a curve{{math| &amp;Gamma;,}} and {{math|'''s&amp;#770;'''}} is the unit tangent vector to{{math| &amp;Gamma;}} in the direction of increasing{{mvar| s}}. Let {{mvar|s}} take the values {{math|''s''&lt;sub&gt;1&lt;/sub&gt;}} and {{math|''s''&lt;sub&gt;2&lt;/sub&gt;}} at the endpoints of{{math| &amp;Gamma;,}} where the position vector {{math|'''r'''}} takes the values {{math|'''r'''&lt;sub&gt;1&lt;/sub&gt;}} and {{math|'''r'''&lt;sub&gt;2&lt;/sub&gt;}} respectively. Then, integrating ({{EquationNote|9g}}) w.r.t.{{mvar| s}} from {{math|''s''&lt;sub&gt;1&lt;/sub&gt;}} to {{math|''s''&lt;sub&gt;2&lt;/sub&gt;}} and applying the fundamental theorem of calculus, we get
{{NumBlk|:|&lt;math&gt;
    \int_{s_1}^{s_2} \nabla p \cdot \mathbf{\hat{s}} \,ds
      \,=\, p(\mathbf{r}_2) - p(\mathbf{r}_1) \,.
  &lt;/math&gt;|{{EquationRef|25g}}}}
This is our third integral theorem involving the gradient, and the best-known of the three: it is commonly called simply the '''[[w:gradient theorem|gradient theorem]]''',&lt;ref&gt;Although Hsu ([[#hsu-84|1984]], p.&amp;#8239;141) applies that name to our theorem ({{EquationNote|5g}}).&lt;/ref&gt; or the ''fundamental theorem of the gradient'', or the ''fundamental theorem of line integrals''; it generalizes the fundamental theorem of calculus to a curved path.&lt;ref&gt;''Cf''. [[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;sect;&amp;#8239;50,&amp;#8239;59; presumably this is one reason why Gibbs called the gradient simply the ''derivative''.&lt;/ref&gt; If we write {{math|''d'''''r'''}}&amp;#8202; for&amp;#8239; {{math|'''s&amp;#770;'''&amp;#8239;''ds''}} (the change in the position vector), we get the theorem in the alternative form
{{NumBlk|:|&lt;math&gt;
    \int_{\mathbf{r}_1}^{\mathbf{r}_2} \nabla p \cdot d\mathbf{r}
      \,=\, p(\mathbf{r}_2) - p(\mathbf{r}_1) \,.
  &lt;/math&gt;|{{EquationRef|25r}}}}
As the right-hand side of ({{EquationNote|25g}}) or ({{EquationNote|25r}}) obviously depends on the endpoints but ''not on the path in between'', so does the integral on the left. This integral is commonly called the '''work integral''' of{{math| &amp;nabla;''p''}} over the path&amp;mdash;because if {{math|&amp;nabla;''p''}} is a force, the integral is the work done by the force over the path. So, in words, the gradient theorem says that ''the change in value of a scalar field from one point to another is the work integral of the gradient of that field field over any path from the one to the other''.

Applying ({{EquationNote|25r}}) to a single element of the curve, we get
{{NumBlk|:|&lt;math&gt;\nabla p \cdot d\mathbf{r} = dp \,,
  &lt;/math&gt;|{{EquationRef|26g}}}}
which is reminiscent of&amp;#8201; &lt;math&gt;y'(x)~\!dx\,{=}\,dy\,&lt;/math&gt; in elementary calculus.&lt;ref&gt;''Cf''. [[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;sect;&amp;#8239;50,&amp;#8239;51; presumably this is another reason why Gibbs called the gradient the ''derivative''.&lt;/ref&gt; Alternatively, we could have obtained ({{EquationNote|26g}}) by multiplying both sides of ({{EquationNote|9g}}) by{{mvar| ds}}, and then obtained ({{EquationNote|25r}}) by adding ({{EquationNote|26g}}) over all the elemental displacements{{math| ''d'''''r'''}} on any path from {{math|'''r'''&lt;sub&gt;1&lt;/sub&gt;}} to{{math| '''r'''&lt;sub&gt;2&lt;/sub&gt;}}.

If we ''close'' the path by setting&amp;#8201; {{math|'''r'''&lt;sub&gt;2 &lt;/sub&gt;{{=}} '''r'''&lt;sub&gt;1&lt;/sub&gt;&amp;#8202;,}} the gradient theorem reduces to
{{NumBlk|:|&lt;math&gt;\oint \nabla p \cdot d\mathbf{r} \,=\, 0 \,,
  &lt;/math&gt;|{{EquationRef|27g}}}}
where the integral is around ''any'' closed loop. Applying the Kelvin&amp;ndash;Stokes theorem then gives
{{NumBlk|:|&lt;math&gt;
    \iint_{\varSigma}
      \operatorname{curl}\nabla p \cdot \boldsymbol{\hat{\nu}}
    \,d\varSigma \,=\, 0 \,,
  &lt;/math&gt;|{{EquationRef|28g}}}}
where {{mvar|&amp;Sigma;}}&amp;#8202; is any surface spanning the loop and&lt;math&gt;~\boldsymbol{\hat{\nu}}&lt;/math&gt; is the unit normal to{{mvar| &amp;Sigma;}}.&amp;#8201; As this applies to any loop spanned by any surface on which the integrand is defined,&amp;#8201; {{math|curl&amp;#8201;&amp;nabla;''p''}}&amp;#8202; must be zero wherever it is defined. This is a second proof (more conventional than the first) of theorem ({{EquationNote|24c}}).

{{cob}}

=== Scalar potential: field with given gradient ===

{{cot}}

'''Lemma''':  If&amp;#8201; {{math|curl&amp;#8239;'''q'''&amp;#8201;{{=}}&amp;#8201;'''0'''}}&amp;#8201; in a simply connected region{{mvar| V}},&amp;#8201; then&amp;#8239; &lt;math&gt;\textstyle\int\!\mathbf{q}\!\cdot\!d\mathbf{r}\,&lt;/math&gt; over any path in{{mvar| V}}&amp;#8239; depends only on the endpoints of the path.

''Proof:''  Suppose, on the contrary, that there are two paths {{math|&amp;Gamma;}} and {{math|&amp;Lambda;}} in{{mvar| V}},&amp;#8202; with a common starting point and a common finishing point, such that
:&lt;math&gt;\textstyle\int_{\Gamma}\mathbf{q}\cdot d\mathbf{r}
       \,\neq \textstyle\int_{\Lambda}\mathbf{q}\cdot d\mathbf{r} \,.&lt;/math&gt;
Let&amp;#8201; {{math|&amp;minus;&amp;Lambda;}} denote {{math|&amp;Lambda;}} traversed backwards. Then for every {{math|''d'''''r'''}} on {{math|&amp;Lambda;}}&amp;#8201; there is an equal and opposite{{math| ''d'''''r'''}} on&amp;#8202; {{math|&amp;minus;&amp;Lambda;&amp;#8202;,}} and vice versa, so that we have
:&lt;math&gt;\textstyle\int_{\Gamma}\mathbf{q}\cdot d\mathbf{r} \,\neq\,
       \textstyle-\!\int_{-\Lambda}\mathbf{q}\cdot d\mathbf{r} \,,&lt;/math&gt;
i.e.
:&lt;math&gt;\textstyle\int_{\Gamma}\mathbf{q}\cdot d\mathbf{r} \,+
  \textstyle\int_{-\Lambda}\mathbf{q}\cdot d\mathbf{r} \,\neq\, 0 \,,&lt;/math&gt;
where the left-hand side is now a work integral of{{math| '''q'''}} around a closed loop in{{mvar| V}}.&amp;#8201; By the simple connectedness of{{mvar| V}},&amp;#8202; this loop is spanned by some surface{{mvar| &amp;Sigma;}} in{{mvar| V}}.&amp;#8201; So we can apply the Kelvin&amp;ndash;Stokes theorem and conclude that the flux integral of&amp;#8202; {{math|curl&amp;#8239;'''q'''}}&amp;#8202; through{{mvar| &amp;Sigma;}}&amp;#8202; is non-zero, in which case&amp;#8202; {{math|curl&amp;#8239;'''q'''}}&amp;#8202; must be non-zero somewhere on{{mvar| &amp;Sigma;&amp;#8202;,}} hence somewhere in{{mvar| V}}&amp;#8201;&amp;mdash;&amp;#8239;contradicting the hypothesis of the lemma. &amp;#9724;

'''Corollary''':  If&amp;#8201; {{math|curl&amp;#8239;'''q'''&amp;#8201;{{=}}&amp;#8201;'''0'''}}&amp;#8201; in a simply connected region{{mvar| V}},&amp;#8202; there exists a scalar field {{mvar|p}} such that&amp;#8239; {{math|'''q'''&amp;#8201;{{=}}&amp;#8201;&amp;nabla;''p''}}&amp;#8201; in{{mvar| V}}.

''Proof:''  We shall show that a suitable candidate is
:&lt;math&gt;p(\mathbf{r}) \,=
    \int_{\mathbf{r}_0}^{\mathbf{r}} \!\mathbf{q}\cdot d\boldsymbol{\rho} \,,
  &lt;/math&gt;
where {{math|'''r'''&lt;sub&gt;0&lt;/sub&gt;}} is the position vector of any fixed point in{{mvar| V}},&amp;#8202; and {{mvar|'''&amp;rho;'''}} is the position vector of a general point on the path of integration, which may be any path in{{mvar| V}}. First note that {{math|''p''('''r''')}} is unambiguous because, by the preceding lemma, it is independent of the path for given {{math|'''r'''&lt;sub&gt;0&lt;/sub&gt;}} and{{math| '''r''',}} provided that the path is in{{mvar| V}}.&amp;#8201; Now to find&amp;#8202; {{math|&amp;nabla;''p''('''r'''),}}&amp;#8202; let {{mvar|&amp;sigma;}} be the arc length along the path from {{math|'''r'''&lt;sub&gt;0&lt;/sub&gt;}} to{{mvar| '''&amp;rho;'''&amp;#8202;}}, so that {{mvar|&amp;sigma;}} ranges from 0 to (say){{mvar| s}}&amp;#8201; as {{mvar|'''&amp;rho;'''}} ranges from {{math|'''r'''&lt;sub&gt;0&lt;/sub&gt;}} to{{math| '''r'''&amp;#8202;}}; and let {{math|'''s&amp;#770;'''}} be the unit vector tangential to the path at{{mvar| '''&amp;rho;'''&amp;#8202;}}, in the direction of increasing{{mvar| &amp;sigma;}}.&amp;#8201; Then&amp;#8239; {{math|''d'''&amp;rho;'''''&amp;#8201;{{=}}&amp;#8201;'''s&amp;#770;'''&amp;#8239;''d&amp;sigma;''&amp;#8202;,}} so that the above equation becomes
:&lt;math&gt;p\big(\mathbf{r}(s)\big) \,=
    \int_0^s \!\mathbf{q}\cdot\mathbf{\hat{s}} \,d\sigma \,.
  &lt;/math&gt;
Differentiating w.r.t.{{mvar| s}} gives
:&lt;math&gt;\part_s p = \mathbf{q}\cdot\mathbf{\hat{s}} \,,&lt;/math&gt;
where {{math|'''s&amp;#770;'''}} is evaluated at&amp;#8201; {{mvar|&amp;sigma;&amp;#8201;{{=}}&amp;#8201;s}}&amp;#8201; and is therefore in the direction in which the path reaches{{math| '''r'''}}.&amp;#8201; By the generality of the path, this can be ''any'' direction. So the last equation says that {{math|'''q'''}} is the vector whose (scalar) component in any direction is the derivative of{{mvar| p}} w.r.t. arc length in that direction; that is, {{math|'''q'''&amp;#8201;{{=}}&amp;#8201;&amp;nabla;''p''&amp;#8202;,}} as required. &amp;#9724;

This is the promised converse of theorem ({{EquationNote|24c}}). ''But'', given an irrotational vector field {{math|'''q'''&amp;#8202;,}} we usually prefer to find a scalar field whose ''negative'' gradient is{{math| '''q'''&amp;#8202;}};&amp;#8201; that is, we usually prefer a scalar field &lt;math&gt;\varphi&lt;/math&gt; such that&amp;#8201;
&lt;math&gt;\mathbf{q}~\!\!=\!-\nabla\varphi&lt;/math&gt;.&amp;#8201; Such a field &lt;math&gt;\varphi&lt;/math&gt; is called a '''scalar potential''' for{{math| '''q'''}}.&amp;#8201; From the above expression for {{math|''p''('''r'''),}} a suitable candidate is
{{NumBlk|:|&lt;math&gt;\varphi(\mathbf{r}) \,=\,
    -\!\int_{\mathbf{r}_0}^{\mathbf{r}} \!\mathbf{q}\cdot d\boldsymbol{\rho} \,.
  &lt;/math&gt;|{{EquationRef|29}}}}

A scalar field has zero gradient if and only if it is uniform, so that adding a uniform field, but ''only'' a uniform field, to a given scalar field leaves its gradient unchanged. Thus ''the scalar potential is determined up to an arbitrary additive uniform field''. This would be the case with or without the minus sign in front of the gradient. The reason for preferring the minus sign appears next.

{{cob}}

=== Conservative fields ===

{{cot}}

An irrotational vector field&amp;mdash;or, equivalently, a field that is (plus or minus) the gradient of something&amp;mdash;is described as '''conservative''', because if the field is a force, it does zero work around a closed loop, and consequently ''conserves energy'' around the loop (at least if the field does not change during traversal of the loop).

If the only force acting on a particle is&amp;#8201; {{math|'''F'''&amp;#8201;{{=}}&amp;#8201;&amp;minus;&amp;nabla;''U'',}}&amp;#8201; then, by the gradient theorem, the work done on the particle over a path is the increase in {{mvar|&amp;minus;U}},&amp;#8201; i.e. the ''decrease'' in{{mvar| U&amp;#8202;}}; and this work is the increase in the particle's kinetic energy{{mvar| T}}.&amp;#8201; Hence, if we identify {{mvar|U}} with the ''potential'' energy, the total energy&amp;#8202; {{mvar|U&amp;#8202;+&amp;#8202;T}}&amp;#8201; is conserved. This interpretation of the scalar potential is possible only if the force is ''minus'' the gradient of the potential.

The minus sign is also used if the conservative vector field is an '''electric field''' (force per unit charge) or a gravitational acceleration (force per unit mass); the scalar potential is potential energy per unit charge, or potential energy per unit mass, respectively.

{{cob}}

== Some special fields ==

=== The 1/''r'' scalar potential ===

{{cot}}

For the potential energy field
{{NumBlk|:|&lt;math&gt;U = \frac{1}{\,r\,} \,&lt;/math&gt;|{{EquationRef|30}}}}
where {{mvar|r}} is the distance from the origin (and {{math|''r''&amp;#8201;&amp;ne;&amp;#8201;0}}), let us find the corresponding force&amp;#8201; {{math|'''F'''&amp;#8201;{{=}}&amp;#8201;&amp;minus;&amp;nabla;''U''}}.&amp;#8201; The direction of&amp;#8202; {{math|&amp;nabla;''U''}}&amp;#8202; is that of the steepest increase of{{mvar| U}}, which, by the spherical symmetry, can only be parallel or antiparallel to &lt;math&gt;\mathbf{\hat{r}}&lt;/math&gt; (the unit vector pointing away from the origin). So
:&lt;math&gt;\nabla U = \big(\nabla U \cdot \mathbf{\hat{r}}\big)~\!\mathbf{\hat{r}}
                = \part_r U \,\mathbf{\hat{r}}
                = \frac{d}{dr}\Big(\!\frac{1}{\,r\,}\!\Big)~\!\mathbf{\hat{r}}
                = -\frac{1}{\,r^2}~\!\mathbf{\hat{r}} \,,&lt;/math&gt;
whence
{{NumBlk|:|&lt;math&gt;\mathbf{F} = \frac{\mathbf{\hat{r}}}{\,r^2} \,.
  &lt;/math&gt;|{{EquationRef|31}}}}
So the negative gradient of the {{math|1/''r''}}&amp;#8202; scalar potential ({{EquationNote|30}}) is the unit '''inverse-square radial vector field'''. Multiplying the numerator and denominator by {{mvar|r}} gives the alternative form
:&lt;math&gt;\mathbf{F} = \frac{\mathbf{r}}{\,r^3} \,,&lt;/math&gt;
which is convenient if the center of the force is shifted from the origin to position{{math| '''r&amp;prime;'''}}: in that case we simply replace {{math|'''r'''}} by {{math|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;''',}} and {{mvar|r}} by {{math|{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}},}} so that the force becomes
:&lt;math&gt;\mathbf{F} = \frac{\mathbf{r}\!-\!\mathbf{r}'}
                         {|\mathbf{r}\!-\!\mathbf{r}'|^3}&lt;/math&gt;
and the corresponding scalar potential becomes
:&lt;math&gt;U = \frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|} \,.&lt;/math&gt;

{{cob}}

=== Inverse-square radial vector field ===

{{cot}}

We derived the vector field ({{EquationNote|31}}) as the negative gradient of the scalar potential ({{EquationNote|30}}). Conversely, given the inverse-square radial vector field ({{EquationNote|31}}), we could derive its scalar potential from ({{EquationNote|29}}). At a general point on the path, let the position vector be&amp;#8201; &lt;math&gt;\boldsymbol{\rho}~\!\!=\!\rho\boldsymbol{\hat{\rho}}\,&lt;/math&gt; so that, by ({{EquationNote|31}}),&amp;#8201; &lt;math&gt;\mathbf{F}\!=\!\boldsymbol{\hat{\rho}}/\rho^2&lt;/math&gt;.&amp;#8201; Then ({{EquationNote|29}}) becomes
:&lt;math&gt;\begin{align}U(\mathbf{r})
    \,&amp;=\, -\!\int_{\mathbf{r}_0}^{\mathbf{r}} \!
               \mathbf{F} \cdot d\boldsymbol{\rho} \\[1ex]
      &amp;=\, -\!\int_{\mathbf{r}_0}^{\mathbf{r}} \!
               \tfrac{1}{\,\rho^2}~\!
              \boldsymbol{\hat{\rho}} \!\cdot\! d\boldsymbol{\rho} \\[1ex]
      &amp;=\, -\!\int_{r_0}^r \! \tfrac{1}{\,\rho^2} \,d\rho
     \ =\, \tfrac{1}{\,\rho\,}\bigg|^r_{r_0}
     \,=\, \frac{1}{\,r\,}-\frac{1}{\,r_0} \,,
  \end{align}&lt;/math&gt;
so that, if we choose&amp;#8201; {{math|''r''&lt;sub&gt;0&lt;/sub&gt;&amp;#8202;&amp;rarr;&amp;#8239;&amp;infin;&amp;#8202;,}} we recover ({{EquationNote|30}}).

Because {{math|'''F''',}} given by ({{EquationNote|31}}), has a scalar potential,&amp;#8201; {{math|curl&amp;#8201;'''F'''}}&amp;#8202; must be zero. This is independently obvious in that the spherical symmetry of{{math|&amp;#8202; '''F'''}} seems to rule out any resemblance of rotation or shear&amp;mdash;even at the origin, where {{math|'''F'''}} becomes infinite. On the last point, let us check whether&amp;#8201; {{math|curl&amp;#8201;'''F'''}}&amp;#8202; has a meaningful integral over a volume containing the origin. If the volume {{mvar|V}}&amp;#8202; is enclosed by the surface {{mvar|S}}&amp;#8202; whose outward unit normal is &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;, then, by theorem ({{EquationNote|5c}}),
:&lt;math&gt;\iiint_V \operatorname{curl}\mathbf{F} ~dV
    \,= \iint_S \mathbf{\hat{n}}\times\mathbf{F} \,dS
    \,= \iint_S \mathbf{\hat{n}}\times\frac{\mathbf{\hat{r}}\,}{r^2} \,dS \,.
  &lt;/math&gt;
If {{mvar|V}} contains the origin, then, because&amp;#8201; {{math|curl&amp;#8201;'''F'''}}&amp;#8239; is zero everywhere ''except'' at the origin, the volume {{mvar|V}}&amp;#8202; can be replaced by any ''element'' of{{mvar| V}}&amp;#8202; containing the origin, whatever the shape of that element may be. If we choose that element to be a spherical ball centered on the origin, then &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is parallel to &lt;math&gt;\mathbf{\hat{r}}&lt;/math&gt;, so that the cross-product in the integrand on the right is zero. Thus the volume integral on the left is not only meaningful, but is ''zero'', even if the volume contains the point where the integrand is infinite. In this sense, the field {{math|'''F'''}} is ''so'' irrotational that its curl may be taken as zero even where the field itself is undefined!

The situation concerning the ''divergence'' of{{math|&amp;#8202; '''F'''}} is more complicated. Again, let the volume {{mvar|V}}&amp;#8202; be enclosed by the surface {{mvar|S}} whose outward unit normal is &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;.&amp;#8201; By the divergence theorem ({{EquationNote|5d}}),
:&lt;math&gt;\begin{align}\iiint_V \operatorname{div}\mathbf{F} ~dV
    \,= \iint_S \mathbf{\hat{n}}\cdot\mathbf{F} \,dS
   \,&amp;= \iint_S \mathbf{\hat{n}}\cdot\frac{\mathbf{\hat{r}}\,}{r^2} \,dS\\[1ex]
     &amp;= \iint_S \frac{\mathbf{\hat{r}}\cdot\mathbf{\hat{n}}\,dS}{\,r^2} \\[1ex]
     &amp;= \iint_S d\Omega \,,
  \end{align}&lt;/math&gt;
where {{math|''d''&amp;Omega;}} is the ''solid angle'' subtended at the origin by the surface element of area{{mvar| dS&amp;#8202;}}, and is taken as positive if the outward unit normal &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; has a positive component ''away from'' the origin &lt;math&gt;(\mathbf{\hat{r}}\!\cdot\!\mathbf{\hat{n}}&gt;0)&lt;/math&gt;, and negative if&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; has a positive component ''toward'' the origin &lt;math&gt;(\mathbf{\hat{r}}\!\cdot\!\mathbf{\hat{n}}&lt;0)&lt;/math&gt;. If the volume enclosed by {{mvar|S}} does ''not'' include the origin, then for every positive contribution {{math|''d''&amp;Omega;}}&amp;#8201; there is a compensating negative contribution, so that the integral of&amp;#8201; {{math|div&amp;#8201;'''F'''}}&amp;#8202; over the volume is zero. As this applies to every such volume,&amp;#8201; {{math|div&amp;#8201;'''F'''}}&amp;#8202; must be zero everywhere except at the origin. If, on the contrary, the volume ''does'' include the origin, then the contributions {{math|''d''&amp;Omega;}} add up to the total solid angle subtended by the enclosing surface, which is{{math| 4''&amp;pi;''}}. In summary,
{{NumBlk|:|&lt;math&gt;\mathrm{div}\Big(\frac{\mathbf{\hat{r}}}{\,r^2}\Big)
  =~\! 4\pi~\!\delta(\mathbf{r}) \,,&lt;/math&gt;|{{EquationRef|32d}}}}
where {{math|''&amp;delta;''('''r'''),}} the 3D '''unit delta function''', is zero everywhere except at the origin, but has an integral of&amp;#8202; {{math|1}} over any volume that includes the origin. For example, a unit point-mass at the origin has the density {{math|''&amp;delta;''('''r''')}}, and a point-mass {{mvar|m}} at position {{math|'''r&amp;prime;'''}} has the density&amp;#8202; {{math|''m&amp;delta;''('''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;''')}}. As the argument of&amp;#8202; {{math|div}}&amp;#8202; in ({{EquationNote|32d}}) is&amp;#8202; {{math|&amp;minus;&amp;nabla;(1/''r''),}} we also have
{{NumBlk|:|&lt;math&gt;\triangle\Big(\frac{1}{\,r\,}\Big)
  = -4\pi~\!\delta(\mathbf{r}) \,.&lt;/math&gt;|{{EquationRef|32L}}}}
If we shift the centers from the origin to {{math|'''r&amp;prime;''',}} the last two results become
{{NumBlk|:|&lt;math&gt;
    \mathrm{div}\bigg(\frac{ \mathbf{r}\!-\!\mathbf{r}'}
                           {|\mathbf{r}\!-\!\mathbf{r}'|^3} \!\bigg)
    =~\! 4\pi~\!\delta(\mathbf{r}\!-\!\mathbf{r}')
  &lt;/math&gt;|{{EquationRef|33d}}}}
and
{{NumBlk|:|&lt;math&gt;
    \triangle\bigg(\frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|}\bigg)
    = -4\pi~\!\delta(\mathbf{r}\!-\!\mathbf{r}') \,.
  &lt;/math&gt;|{{EquationRef|33L}}}}

{{cob}}

=== Field with given divergence (and zero curl) ===

{{cot}}

It follows from '''Coulomb's law''' that the electric field due to a point-charge {{mvar|Q}} at the origin, in a vacuum, is
:&lt;math&gt;\mathbf{E}
  = \frac{Q}{4\pi\epsilon_0 r^2} ~\!\mathbf{\hat{r}} \,,&lt;/math&gt;
where {{math|''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;}} is a physical constant (called the '''vacuum permittivity''' or simply the '''electric constant'''). In a ''vacuum'', the '''electric displacement field''', denoted by{{math| '''D'''&amp;#8202;,}} is {{math|''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;'''E'''}}.&amp;#8201; So it is convenient to multiply the above equation by {{math|''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8202;,}} obtaining
:&lt;math&gt;\mathbf{D}
  = \frac{Q}{4\pi} ~\!\frac{\mathbf{\hat{r}}\,}{r^2} \,.&lt;/math&gt;
This is a inverse-square radial vector field and therefore has zero curl.

Now suppose that, instead of a charge {{mvar|Q}} at the origin, we have a static ''charge density'' {{math|''&amp;rho;''('''r&amp;prime;''')}} in a general elemental volume {{mvar|dV&amp;prime;}}&amp;#8202; at position{{math| '''r&amp;prime;'''}} (the standard symbol for ''charge'' density being unfortunately the same as for ''mass'' density). Then the contribution from that element to the field{{math| '''D'''}} at position{{math| '''r'''}}&amp;#8202; is
:&lt;math&gt;d\mathbf{D}(\mathbf{r}) ~\!=~\!
    \frac{\,\rho(\mathbf{r}')\,dV'}{4\pi}\,
    \frac{\mathbf{r}\!-\!\mathbf{r}'}{|\mathbf{r}\!-\!\mathbf{r}'|^3}
  &lt;/math&gt;
provided that, for each {{math|'''r''',}} the dimensions of each volume element are small compared with {{math|{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}}}}. This contribution likewise has zero curl. The total field due to static charges is then the sum of the contributions:
{{NumBlk|:|&lt;math&gt;\mathbf{D}(\mathbf{r}) \,= \iiint
    \frac{\,\rho(\mathbf{r}')}{4\pi}\,
    \frac{\mathbf{r}\!-\!\mathbf{r}'}{|\mathbf{r}\!-\!\mathbf{r}'|^3} \,dV'
  &lt;/math&gt;|{{EquationRef|34}}}}
where the integral is over all space. And {{math|'''D'''('''r''')}} has zero curl because all the contributions have zero curl.

Independently of the physical significance of&amp;#8202; {{math|'''D'''('''r'''),}} we can take its divergence &quot;term by term&quot; (or &quot;under the integral sign&quot;), obtaining
:&lt;math&gt;\begin{align}\operatorname{div}\mathbf{D}(\mathbf{r})
  \,&amp;= \iiint \frac{\,\rho(\mathbf{r}')}{4\pi}~\!\mathrm{div}\bigg(
         \frac{\mathbf{r}\!-\!\mathbf{r}'}{|\mathbf{r}\!-\!\mathbf{r}'|^3}
       \!\bigg) ~\!dV' \\[3pt]
    &amp;= \iiint \frac{\,\rho(\mathbf{r}')}{4\pi}\,
         4\pi~\!\delta(\mathbf{r}\!-\!\mathbf{r}') \,dV'
         \quad \big[\mathsf{by~eq.(33d)}\big] \\[3pt]
    &amp;= \iiint \rho(\mathbf{r}')\,\delta(\mathbf{r}\!-\!\mathbf{r}')\,dV'\\[3pt]
    &amp;= \iiint \rho(\mathbf{r})\,\delta(\mathbf{r}\!-\!\mathbf{r}') \,dV'
           ~~~ \begin{bmatrix}~\!\!
                 \mathsf{since}~\delta(\mathbf{r}\!-\!\mathbf{r}')\!=\!0\\
                 \mathsf{unless}~\,\mathbf{r}'{=}~\!\mathbf{r}
               ~\!\!\end{bmatrix} \\
    &amp;= \,\rho(\mathbf{r})\!\iiint\delta(\mathbf{r}\!-\!\mathbf{r}')\,dV'\\[3pt]
    &amp;= \,\rho(\mathbf{r})\!\iiint\delta(\mathbf{r}'{-}~\!\mathbf{r})\,dV',
  \end{align}&lt;/math&gt;
where the last step is permitted because the volume integral of the delta function of{{math| '''r&amp;prime;'''}} is not changed by a &quot;[[w:point reflection|point reflection]]&quot; (inversion) across {{math|'''r'''}}.&amp;#8201; As the volume of integration (all space) includes the shifted origin of the delta function, the integral is simply{{math| 1&amp;#8202;,}} so that
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\mathbf{D}=\rho \,,
  &lt;/math&gt;|{{EquationRef|35}}}}
where both sides are evaluated at{{math| '''r'''}}.

Mathematically, this result is an identity which applies if&amp;#8202; {{math|'''D'''}} is given by ({{EquationNote|34}}); substituting for{{math| '''D'''&amp;#8202;,}} we can write the identity in full as
{{NumBlk|:|&lt;math&gt;\rho(\mathbf{r}) \,\equiv\, \mathrm{div}\iiint
    \frac{\,\rho(\mathbf{r}')}{4\pi}\,
    \frac{\mathbf{r}\!-\!\mathbf{r}'}{|\mathbf{r}\!-\!\mathbf{r}'|^3}
  \,dV',&lt;/math&gt;|{{EquationRef|36}}}}
where the integral is over all space, or at least all of the space in which {{mvar|&amp;rho;}} may be non-zero. Subject to the convergence of the integral, this shows that ''we can construct an irrotational vector field whose divergence is a given scalar field''{{math| ''&amp;rho;''('''r''')}}. And of course, by theorem ({{EquationNote|24d}}), ''any curl'' can be added to that vector field without changing its divergence.

In ''electrostatics'', ({{EquationNote|34}}) is a generalization of Coulomb's law; and ({{EquationNote|35}}), which follows from ({{EquationNote|34}}), is '''Gauss's law''' expressed in ''differential form''. If we integrate ({{EquationNote|35}}) over a volume enclosed by a surface{{mvar| S}} (with outward unit normal &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt;) and apply the divergence theorem on the left, we get the ''integral form'' of Gauss's law:
{{NumBlk|:|&lt;math&gt;
    \iint_S \mathbf{D}\cdot\mathbf{\hat{n}}\,dS \,=\, Q_{\mathrm{e}} \,,
  &lt;/math&gt;|{{EquationRef|37}}}}
where {{math|''Q''&lt;sub&gt;e&lt;/sub&gt;}} is the total charge ''enclosed''&amp;#8202; by{{mvar| S}}.

{{cob}}

=== Field with given Laplacian ===

{{cot}}

In ({{EquationNote|36}}), we can recognize the {{math|'''r'''}}-dependent factor&amp;#8202; {{math|{{sfrac|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''|{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}}&lt;sup&gt;3&lt;/sup&gt;}}}}&amp;#8201; as&amp;#8202; {{math|&amp;minus;&amp;nabla;{{sfrac|1|&amp;#8202;{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}}&amp;#8202;}}}}&amp;#8201; and take the gradient operator outside the integral, obtaining
&lt;div style=&quot;margin-top: 1em&quot;&gt;
:&lt;math&gt;\rho(\mathbf{r}) \,\equiv\, \mathrm{div}\bigg(\!{-}\nabla\!\iiint
    \frac{\,\rho(\mathbf{r}')}{4\pi}\,
    \frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|}
  \,dV' \!\bigg) \,,&lt;/math&gt;
&lt;/div&gt;
i.e.
{{NumBlk|:|&lt;math&gt;\rho(\mathbf{r}) \,\equiv\, \triangle\bigg(\!{-}\!\iiint
    \frac{\,\rho(\mathbf{r}')}{4\pi}\,
    \frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|}
  \,dV' \!\bigg) \,,&lt;/math&gt;|{{EquationRef|38}}}}
where again the integral is over all space, or at least all of the space in which {{mvar|&amp;rho;}} may be non-zero. Subject to the convergence of the integral, this shows that ''we can construct a field whose Laplacian is a given field''. More precisely, it shows that we can construct a ''scalar'' field whose Laplacian is a given ''scalar''&amp;#8202; field{{math| ''&amp;rho;''('''r''')}}. But, due to the linearity of the Laplacian, the same applies to any given linear combination of scalar fields, including any combination whose coefficients are uniform vectors, uniform matrices, or uniform tensors of any order; that is, the same applies to any field that we can express with a uniform basis.

Mathematically, ({{EquationNote|38}}) is simply an identity. To find its significance in electrostatics, we can multiply it by&amp;#8202; {{math|&amp;minus;1&amp;#10744;''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8239;,}} obtaining
{{NumBlk|:|&lt;math&gt;-\frac{\rho(\mathbf{r})}{\epsilon_0} \,\equiv\,
  \triangle\iiint
    \frac{\,\rho(\mathbf{r}')}{4\pi\epsilon_0}\,
    \frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|}
  \,dV',&lt;/math&gt;|{{EquationRef|39}}}}
which is also an identity. But the negative gradient of the expression after the integral sign is
:&lt;math&gt;\frac{\,\rho(\mathbf{r}')\,dV'}{4\pi\epsilon_0}\,
  \frac{\mathbf{r}\!-\!\mathbf{r}'}{|\mathbf{r}\!-\!\mathbf{r}'|^3}\,,&lt;/math&gt;
which is the contribution to the electric field at position{{math| '''r'''}} due to a charge&amp;#8202; {{math|''&amp;rho;''('''r&amp;prime;''')&amp;#8239;''dV&amp;prime;''}}&amp;#8239; at position{{math| '''r&amp;prime;'''}} in a vacuum. So the expression after the integral sign is the corresponding contribution to the electrostatic potential, and the whole integral is the whole electrostatic potential. Denoting this by &lt;math&gt;\varphi~\!,\,&lt;/math&gt; we can rewrite ({{EquationNote|39}}) as
{{NumBlk|:|&lt;math&gt;
    \triangle\varphi = -\frac{\rho}{\,\epsilon_0} \,.
  &lt;/math&gt;|{{EquationRef|40}}}}
This is '''Poisson's equation''' in electrostatics, treating the medium as a vacuum (so that {{mvar|&amp;rho; }}must be taken as the ''total'' charge density, including any contributions caused by the effect of the field on the medium). In a region in which&amp;#8201; {{math|''&amp;rho;''&amp;#8201;{{=}}&amp;#8201;0&amp;#8202;,}}&amp;#8201; Poisson's equation ({{EquationNote|40}}) reduces to
{{NumBlk|:|&lt;math&gt;
    \triangle\varphi = 0 \,,
  &lt;/math&gt;|{{EquationRef|41}}}}
which is '''Laplace's equation''' in electrostatics.

{{cob}}

=== The wave equation ===

{{cot}}

It is an empirical fact that a compressible fluid, such as air, carries waves of a mechanical nature: sound waves. In establishing the unambiguity of the gradient and the divergence, we have already derived equations dealing with the inertia and continuity (mass-conservation) of non-viscous fluids. So, by introducing a relation describing the compressibility, and eliminating variables, we should be able to get ''one'' equation (the &quot;wave equation&quot;) in ''one'' scalar or vector field (the &quot;wave function&quot;), with recognizably &quot;wavelike&quot; solutions. And we should expect this equation to be analogous to equations describing other kinds of waves.

If we suppose, for simplicity, that the only force acting on an element of fluid is the pressure force, then the applicable equation of motion is ({{EquationNote|6g}}). But, for reasons which will soon be apparent, let us call the pressure{{mvar| P}}, so that ({{EquationNote|6g}}) becomes
:&lt;math&gt;
    \rho\,\frac{d\mathbf{v}}{dt} = -\nabla P \,.
  &lt;/math&gt;
Then at ''equilibrium'' we have
:&lt;math&gt;
    0 ~\!= -\nabla P_0 \,,
  &lt;/math&gt;
where {{math|''P''&lt;sub&gt;0&lt;/sub&gt;}} is the equilibrium pressure. Subtracting this equation from the previous one and defining
:&lt;math&gt;p = P - P_0 \,,&lt;/math&gt;
we get
:&lt;math&gt;
    \rho\,\frac{d\mathbf{v}}{dt} = -\nabla p \,,
  &lt;/math&gt;
which looks like ({{EquationNote|6g}}), except that {{mvar|p}} is now the '''sound pressure''' (also called &quot;acoustic pressure&quot;, or sometimes &quot;excess pressure&quot;), i.e. the pressure rise above equilibrium.

For the equation of continuity we can use ({{EquationNote|7d'}}), which we repeat for convenience:
:&lt;math&gt;
    \rho\operatorname{div}\mathbf{v} = -\frac{d\rho}{dt} \,.
  &lt;/math&gt;

Eliminating {{math|'''v'''}} between the last two equations is fraught because {{math|'''v''' }}is evaluated at a moving point in the former and at a fixed point in the latter; and introducing any relation between {{mvar|p}} and {{mvar|&amp;rho;}} is similarly fraught because {{mvar|p}} is evaluated at a fixed point and {{mvar|&amp;rho;}} at a moving point. The obvious remedy is to apply the advection rule ({{EquationNote|16}}) to the last two equations, obtaining respectively
:&lt;math&gt;\begin{align}
    \rho\Big(\tfrac{\part\mathbf{v}}{\part t}
             + \mathbf{v}\cdot\nabla\mathbf{v}\Big) \,&amp;=\, -\nabla p ~; \\
    \rho\operatorname{div}\mathbf{v} \,&amp;=\,
      -\tfrac{\part\rho}{\part t} - \mathbf{v}\cdot\nabla\rho \,.
  \end{align}&lt;/math&gt;
That gets all the variables evaluated at fixed points, at the cost of making the equations more complicated and more obviously non-linear. But the equations and be simplified and linearized by '''small-amplitude approximations'''. In the parentheses in the first equation, the first term is proportional to the amplitude of the vibrations while the second term is a product of ''two'' factors proportional to the amplitude, so that, for sufficiently small amplitudes, the second term is negligible. Similarly, in the second equation, for sufficiently small amplitudes and a ''homogeneous medium'', we can neglect the second term on the right. Then, on the left side of each equation, we are left with a factor proportional to the amplitude, multiplied by{{mvar| &amp;rho;}}. But {{mvar|&amp;rho;}} is not proportional to the amplitude; only its deviation from the equilibrium density is so proportional. Hence, for small amplitudes,&amp;#8201; {{mvar|&amp;rho; }}can be replaced by the equilibrium density, which we shall call{{math| ''&amp;rho;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8202;,}} which is independent of time and (in a homogeneous medium) independent of position. With these approximations, our equations of motion and continuity become
:&lt;math&gt;\begin{align}
    \rho_0 \mathbf{\dot{v}} &amp;= -\nabla p \,, \\[.5ex]
    \rho_0 \operatorname{div}\mathbf{v} &amp;= -\dot{\rho} \,,
  \end{align}&lt;/math&gt;
where, for brevity, we use an overdot to denote ''partial'' differentiation w.r.t. time (i.e., at a ''fixed'' point, not a point moving with the fluid).

Now we can eliminate {{math|'''v'''}}. Taking divergences in the first equation, and differentiating the second ''partially'' w.r.t. time (which can be done inside the {{math|div}} operator, which represents a linear combination), we get
:&lt;math&gt;\begin{align}
    \rho_0 \operatorname{div}\mathbf{\dot{v}} &amp;= -\triangle p \,, \\[.5ex]
    \rho_0 \operatorname{div}\mathbf{\dot{v}} &amp;= -\ddot{\rho} \,,
  \end{align}&lt;/math&gt;
so that we can equate the right-hand sides, obtaining
{{NumBlk|:|&lt;math&gt;\ddot{\rho} = \triangle p \,.&lt;/math&gt;|{{EquationRef|42}}}}

Maintaining the small-amplitude assumption, we can now consider compressibility. For ''small'' compressions in a ''homogeneous'' medium, we may suppose that the pressure change {{mvar|dp}} is some constant times the density change{{mvar| d&amp;rho;}}. It is readily verified that such a constant must have the dimension of velocity squared. So we can say&amp;#8201; {{math|''dp''&amp;#8201;{{=}}&amp;#8201;''c''&amp;sup2;&amp;#8202;''d&amp;rho;''&amp;#8202;,}} where {{mvar|c}} is a constant with the units of velocity.{{efn|When a gas is compressed, work is done on it, causing its temperature to rise, so that the ratio of {{mvar|dp}} to{{mvar| d&amp;rho;}} is higher than if the compression were isothermal. In sound waves, there is typically not enough time for a significant part of the heat of compression to be conducted away; that is, the compression is near enough to '''adiabatic'''. The words &quot;not enough time&quot; may suggest that the adiabatic approximation is a high-frequency approximation. But in fact, in free air, it is a ''low''-frequency approximation, because as the frequency is reduced, the equalization of temperature is hindered more by the longer wavelength than it is helped by the longer period. Only in a confined space, which limits the required distance of conduction, does the adiabatic assumption require the frequency to be ''above'' some lower limit. In a musical wind instrument, that lower limit tends to be far below the audible range. Meanwhile the upper limit, due to easier heat conduction within a shorter wavelength, tends to be very far above the audible range. Thus, under typical conditions, for the purpose of calculating{{mvar| c&amp;#8202;}}, the adiabatic assumption is reasonable. (See [[#fletcher-74|Fletcher, 1974]].)}} Dividing by {{mvar|dt}} gives&amp;#8201; &lt;math&gt;\dot{p}\!=\!c^2\dot{\rho}~\!,\,&lt;/math&gt; whence
{{NumBlk|:|&lt;math&gt;\ddot{p} = c^2~\!\ddot{\rho} \,.&lt;/math&gt;|{{EquationRef|43}}}}
Substituting from  ({{EquationNote|42}}) then gives the desired '''wave equation''':
{{NumBlk|:|&lt;math&gt;\ddot{p} = c^2 \triangle p \,.&lt;/math&gt;|{{EquationRef|44}}}}

This is the 3D classical wave equation with the sound pressure {{mvar|p}} as the wave function. For a generic wave function {{mvar|&amp;psi;&amp;#8202;,}} in a homogeneous isotropic medium, we would expect the equation to be
{{NumBlk|:|&lt;math&gt;\ddot{\psi} = c^2 \triangle\psi \,,&lt;/math&gt;|{{EquationRef|45}}}}
which may be written more compactly as
{{NumBlk|:|&lt;math&gt;\Box\psi =~\! 0 \,,&lt;/math&gt;|{{EquationRef|46}}}}
where {{math|&amp;#9744;,}} pronounced &quot;wave&quot; or &quot;box&quot;,{{efn|Or sometimes &quot;quabla&quot;, by analogy with &quot;nabla&quot;.}} is called the '''D'Alembertian''' operator and is defined by
{{NumBlk|:|&lt;math&gt;
    \Box\psi := \triangle\psi - \frac{1}{\,c^2}\frac{\part^2 \psi}{\part t^2}
  &lt;/math&gt;|{{EquationRef|47}}}}
in this paper, although other conventions exist.{{efn|In particular, some authorities change the sign, defining {{math|&amp;#9744;}} as&amp;#8201; &lt;math&gt;\tfrac{1}{\,c^2}\tfrac{\part^2}{\part t^2}\!-\!\triangle&lt;/math&gt;&amp;#8239;,&amp;#8201; and some write the operator (however defined) as{{math| &amp;#9744;&lt;sup&gt;2&lt;/sup&gt;}}.}}

In a ''static'' situation, the second term on the right of ({{EquationNote|47}}) is zero. So one advantage of definition ({{EquationNote|47}}), over any alternative definition that changes the sign or the scale factor, is that ''in the static case, the D'Alembertian is reduced to the Laplacian'', making it especially obvious that ''in the static case, the wave equation is reduced to Laplace's equation'' [compare ({{EquationNote|46}}) and ({{EquationNote|41}})]. Also notice that the D'Alembertian, being a linear combination of two linear operators, is itself ''linear''.

{{cob}}

=== Spherical waves ===

{{cot}}

Having established that there are wavelike time-dependent fields described by equation ({{EquationNote|45}}), in which the constant {{mvar|c}} has the units of velocity, we can now make an informed guess at an elementary solution of the equation. Consider the candidate
{{NumBlk|:|&lt;math&gt;
    \psi(\mathbf{r},t) = \tfrac{1}{\,r\,}~\!f\big(t-r/c\big) \,,
  &lt;/math&gt;|{{EquationRef|48}}}}
where&amp;#8201; &lt;math&gt;\mathbf{r}=r\mathbf{\hat{r}}&lt;/math&gt;&amp;#8201; is the position vector (so that {{mvar|r}} is distance from the origin),&amp;#8201; {{mvar|f}}&amp;#8239; is an arbitrary function (arbitrary except that it will need to be twice differentiable),&amp;#8201; {{mvar|t }}is time, and {{mvar|c }}is a constant (and obviously {{mvar|&amp;psi; }}is not defined at the origin even if {{mvar|f&amp;#8202; }}is.)

If, at the origin, the function {{mvar|f}}&amp;#8239; has a certain argument at time&amp;#8201; {{math|''t&amp;#8201;{{=}}&amp;#8201;&amp;tau;''&amp;#8202;,}}&amp;#8201; then at any distance{{mvar| r}}&amp;#8202; from the origin, it has the same argument at time&amp;#8201; {{math|''t&amp;#8201;{{=}}&amp;#8201;&amp;tau;&amp;#8239;+&amp;#8202;r''&amp;#10744;''c''&amp;#8202;,}}&amp;#8201; which is&amp;#8201; {{math|''r''&amp;#10744;''c'' }}&amp;#8239;''later''&amp;#8202; than at the origin. Hence, if {{mvar|f}}&amp;#8239; has a certain feature (e.g., a zero-crossing) at the origin, the time taken for that feature to reach any distance{{mvar| r}}&amp;#8239; is{{math| ''r''&amp;#10744;''c''&amp;#8202;,}}&amp;#8239; implying that the feature travels outward from the origin at speed{{mvar| c}}.&amp;#8201; Another way to perceive this is to set the argument of{{mvar| f}}&amp;#8201; equal to a constant (corresponding to some feature of the function) and differentiate w.r.t.{{mvar| t&amp;#8202;,}} obtaining&amp;#8201; &lt;math&gt;\dot{r}\!=\!c&lt;/math&gt;&amp;#8202; (the speed at which the feature recedes from the origin). Thus equation ({{EquationNote|48}}) describes ''waves''&amp;#8202; radiating outward from the origin with speed{{mvar| c}}. (The symbol {{mvar|c}} comes from a general-purpose Latin word for speed, but has become the usual symbol for ''wave'' speed.)

Equation ({{EquationNote|48}}) further implies that there are surfaces over which the wave function {{mvar|&amp;psi;}}&amp;#8239; is uniform&amp;mdash;namely surfaces of constant{{mvar| r}},&amp;#8201; i.e. spheres centered on the origin. These are the '''wavefronts'''. So ({{EquationNote|48}}) describes '''spherical waves'''.

Because the surface area of a sphere is proportional to the square of its radius, we should expect the radiated '''intensity''' (power per unit area) to satisfy an ''inverse-square law'' (if the medium is ''lossless''&amp;mdash;neither absorbing nor scattering the radiated power). That does ''not'' mean that the wave function itself should satisfy an inverse-square law. In a traveling wave in 3D space, there will be an &quot;effort&quot; variable (e.g., sound pressure) and a &quot;flow&quot; variable (e.g., fluid velocity), and the instantaneous intensity will be proportional to the product of the two. If the two are proportional to each other, the instantaneous intensity will be proportional to the square of one or the other. Hence if the instantaneous intensity falls off like{{math| 1/''r''&amp;#8202;&amp;sup2;,}} the effort and flow variables&amp;mdash;and the wave function, if it is proportional to one or the other&amp;mdash;will fall off like{{math| 1/''r''}}. That suggests the attenuation factor {{math|1/''r''}}&amp;#8202; in ({{EquationNote|48}}).

But there are big ''if''&amp;#8202;s in that argument. For all we know so far, the relation between effort and flow could involve a lag, so that the ''instantaneous'' product of the two could swing negative although it averages to something positive. And for all we know so far, the lag could vary with{{mvar| r}}, allowing at least one of the two (effort or flow) to depart from the {{math|1/''r''}}&amp;#8202; law, even if their average product still falls off like{{math| 1/''r''&amp;#8202;&amp;sup2;}}. The {{math|1/''r''}}&amp;#8202; factor in ({{EquationNote|48}}) is therefore only an &quot;informed guess&quot;. Notwithstanding these complications, we have also guessed that the form of the function {{mvar|f}}&amp;#8202; (the '''waveform''') does not change as {{mvar|r}} increases; we have not considered whether this behavior might depend on the medium, or the waveform, or the geometry of the wavefronts.

So let us carefully check whether ({{EquationNote|48}}) satisfies ({{EquationNote|45}}) or, equivalently, ({{EquationNote|46}}).

As a first step, and as a useful inquiry in its own right, we find {{math|&amp;#9651;''&amp;psi;''}} from definition ({{EquationNote|4L}}), given that {{mvar|&amp;psi;}} is a function of {{math|(''r'',&amp;#8202;''t'') }}only. For the surface {{mvar|&amp;delta;S}}&amp;#8202; let us start with
* a cone (''not'' a double cone) with its apex at the origin, subtending a ''small'' solid angle {{mvar|&amp;omega;}} at the origin,
* a sphere centered on the origin, with radius {{math|''r''}}, and
* a sphere centered on the origin, with radius {{mvar|r&amp;#8202;+&amp;#8202;dr&amp;#8202;}};
and let the volume element be the region inside the cone and between the spheres, so that its enclosing surface {{mvar|&amp;delta;S}}&amp;#8202; has three faces: a segment of the cone, a segment of the inner sphere with area{{math| ''r''&amp;#8202;&amp;sup2;''&amp;#8202;&amp;omega;''&amp;#8202;,}} and a segment of the outer sphere with area{{math| (''r&amp;#8202;+&amp;#8202;dr'')&lt;sup&gt;2&lt;/sup&gt;''&amp;omega;''&amp;#8202;}}. By the symmetry of{{mvar| &amp;psi;&amp;#8202;}}, the outward normal derivative {{mvar|&amp;part;&lt;sub&gt;n&lt;/sub&gt;&amp;#8202;&amp;psi;}}&amp;#8202; is equal to zero on the conical face,&amp;#8201; {{math|+''&amp;part;&lt;sub&gt;r&lt;/sub&gt;&amp;#8202;&amp;psi;''(''r&amp;#8202;+&amp;#8202;dr'',&amp;#8202;''t'')}} on the outer spherical face, and&amp;#8202; {{math|&amp;minus;''&amp;part;&lt;sub&gt;r&lt;/sub&gt;&amp;#8202;&amp;psi;''(''r'',&amp;#8202;''t'')}} on the inner spherical face. The volume of the element is&amp;#8201; {{math|''dV''&amp;#8201;{{=}}&amp;#8201;''r''&amp;#8202;&amp;sup2;''&amp;#8202;&amp;omega;&amp;#8239;dr''}}. So, assembling the pieces of definition ({{EquationNote|4L}}), we get
:&lt;!-- SUBSCRIPTS ENLARGED FOR LEGIBILITY: --&gt;&lt;math&gt;\begin{align}\triangle\psi
    &amp;= \frac{1}{r^2 \omega \,dr}\Big(\!
         (r\!+\!dr)^2 \omega ~\!\part_{\textstyle r} \psi(r\!+\!dr,t)
         - r^2 \omega ~\!\part_{\textstyle r} \psi(r,t)
       \!\Big) \\[1ex]
    &amp;= \frac{1}{\,r^2}~\!
       \frac{(r\!+\!dr)^2 \part_{\textstyle r}\psi(r\!+\!dr,t)
             - r^2 \part_{\textstyle r}\psi(r,t)}{dr} \\[.5ex]
    &amp;= \frac{1}{\,r^2}~\!
       \frac{\part}{\part r}\Big(r^2 \part_{\textstyle r}\psi(r,t)\Big) \,,
  \end{align}&lt;/math&gt;
i.e.
{{NumBlk|:|&lt;math&gt;
    \triangle\psi(r,t)
    \equiv \frac{1}{\,r^2}~\!\frac{\part}{\part r}
             \Big(r^2 \frac{\part\psi}{\part r}\Big)
             \qquad \big[\mathsf{if}\,\,r\!\neq~\!\!0\big]\,.
  &lt;/math&gt;|{{EquationRef|49}}}}

Now we can verify our &quot;informed guess&quot;. Differentiating ({{EquationNote|48}}) twice w.r.t.{{mvar| t}}&amp;#8202; by the chain rule gives
{{NumBlk|:|&lt;math&gt;
    \frac{\part^2\psi}{\part t^2} = \frac{1}{\,r\,}~\!f''\!\big(t-r/c\big) \,,
  &lt;/math&gt;|{{EquationRef|50}}}}
where each prime {{math|(&amp;prime;)}} denotes differentiation of the function w.r.t. its own argument. Differentiating ({{EquationNote|48}}) once w.r.t.{{mvar| r}}&amp;#8202; by the product rule and chain rule, we get
{{NumBlk|:|&lt;math&gt;
    \frac{\part\psi}{\part r}
    \,=\, -\frac{1}{cr}~\!f'\!\big(t-r/c\big)
          -\frac{1}{\,r^2}~\!f\big(t-r/c\big) \,.
  &lt;/math&gt;|{{EquationRef|51}}}}
Proceeding as specified in ({{EquationNote|49}}), we multiply this by {{math|''r''&amp;#8202;&amp;sup2;}}, differentiate again w.r.t.{{mvar| r}} (giving three terms, of which two cancel), and divide by {{math|''r''&amp;#8202;&amp;sup2;}}, obtaining
{{NumBlk|:|&lt;math&gt;
    \triangle\psi = \frac{1}{c^2 r}~\!f''\!\big(t-r/c\big) \,.
  &lt;/math&gt;|{{EquationRef|52}}}}
Then if we substitute ({{EquationNote|52}}) and ({{EquationNote|50}}) into ({{EquationNote|47}}), we obviously get&amp;#8201; {{math|&amp;#9744;''&amp;psi;''&amp;#8201;{{=}}&amp;#8201;0&amp;#8202;,}} satisfying ({{EquationNote|46}}). So we have guessed correctly.

Having shown that the D'Alembertian of{{mvar| &amp;psi;&amp;#8202;}}, as given by ({{EquationNote|48}}), is zero everywhere except at the origin (where it is not defined), let us now find its integral over a volume{{mvar| V}} (enclosed by a surface{{mvar| S}}) that includes the origin. From ({{EquationNote|47}}),
:&lt;math&gt;\begin{align}
    \iiint_V \Box\psi \,dV
      \,&amp;= \iiint_V \triangle\psi \,dV
           - \frac{1}{\,c^2}\iiint_V \frac{\part^2 \psi}{\part t^2}\,dV\\[.5em]
        &amp;= \iint_S \part_n \psi \,dS
           - \frac{1}{\,c^2}\iiint_V \frac{\part^2 \psi}{\part t^2}\,dV\,,
  \end{align}&lt;/math&gt;
where the second equality follows from theorem ({{EquationNote|5L}}). Now because the integrand on the left is zero except at the origin, ''any''{{mvar| V}} containing the origin will give the same integral. So for convenience, let {{mvar|V}} be a spherical ball of radius{{mvar| R}} centered on the origin. Then, by the spherical symmetry of{{mvar| &amp;psi;&amp;#8202;,}} integration over{{mvar| S}} reduces to multiplication by{{math| 4''&amp;pi;R''&amp;#8202;&lt;sup&gt;2&lt;/sup&gt;,}} and {{mvar|&amp;part;&lt;sub&gt;n&lt;/sub&gt;}} is equivalent to{{mvar| &amp;part;&lt;sub&gt;r&lt;/sub&gt;&amp;#8202;,}} and {{mvar|dV}} can be taken as{{math| 4''&amp;pi;r''&lt;sup&gt;&amp;#8202;2&lt;/sup&gt;''dr''}}. With these substitutions we have
:&lt;math&gt;
    \iiint_V \Box\psi \,dV \,=\,
      4\pi R^2 \frac{\part\psi}{\part r}\bigg|_{r=R} \!
      - \frac{1}{\,c^2}\!\int_0^R \!\frac{\part^2 \psi}{\part t^2}\,4\pi r^2\,dr
  &lt;/math&gt;
or, substituting from ({{EquationNote|51}}) and ({{EquationNote|50}}),
:&lt;math&gt;\begin{align}
    \iiint_V \!\Box\psi \,dV ~\!\!
     =\,&amp; 4\pi R^2 \!\Big(\!{-}\tfrac{1}{cR} f'\!\big(t\!-\!R/c\big)
                   - \tfrac{1\,}{R^2} f\big(t\!-\!R/c\big)\!\Big) \\
        &amp;- \tfrac{1}{\,c^2}\!\int_0^R \!\tfrac{1}{\,r\,}~\!
                                      f''\!\big(t-r/c\big)\,4\pi r^2\,dr \\[1ex]
     =\,&amp;-\tfrac{4\pi R}{\,c\,}~\!f'\!\big(t\!-\!R/c\big)
           - 4\pi f\big(t\!-\!R/c\big) \\
        &amp;- \tfrac{4\pi}{\,c^2}\!\int_0^R \!rf''\!\big(t-r/c\big) \,dr \,.
  \end{align}&lt;/math&gt;
Again noting that any {{mvar|V}} containing the origin will give the same volume integral, we can let {{mvar|R}} approach zero, with the result that the right-hand side approaches&amp;#8239;{{math| &amp;minus;4''&amp;pi;f''&amp;#8202;(''t'')}}. This is the integral of{{math| &amp;#9744;''&amp;psi;''}} over any volume containing the origin, for {{mvar|&amp;psi;}} given by ({{EquationNote|48}}). Meanwhile {{math|&amp;#9744;''&amp;psi;''}} is zero everywhere except that the origin. In summary,
{{NumBlk|:|&lt;math&gt;
    \Box~\!\Big\{\!\tfrac{1}{\,r\,}~\!f\big(t-r/c\big)\!\Big\}
    \equiv -4\pi f(t)\,\delta(\mathbf{r}) \,.
  &lt;/math&gt;|{{EquationRef|53}}}}
Shifting the center of the spherical waves from the origin to position{{math| '''r&amp;prime;''',}} we get
{{NumBlk|:|&lt;math&gt;
    \Box~\!\Big\{\tfrac{1}{|\mathbf{r}-\mathbf{r}'|}
                 ~\!f\big(t-\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big)\Big\}
    \equiv -4\pi f(t)\,\delta(\mathbf{r}\!-\!\mathbf{r}') \,.
  &lt;/math&gt;|{{EquationRef|54}}}}

We shall refer to the field given by ({{EquationNote|48}}) as the wave function due to a '''monopole''' source with '''strength''' {{math|''f''&amp;#8202;(''t'')}} at the origin. The D'Alembertian of this wave function is given by ({{EquationNote|53}}).&lt;ref&gt;Our definition of ''strength'' follows the old convention used by Baker &amp;amp; Copson ([[#baker-copson-39|1939, p.&amp;#8239;42]]), Born &amp;amp; Wolf ([[#born-wolf-02|2002, p.&amp;#8239;421]]), and Larmor ([[#larmor-1904|1904, p.&amp;#8239;5]]). The newer convention followed by Miller ([[#miller-91|1991, p.&amp;#8239;1371]]) would use the denominator {{math|4''&amp;pi;r''}} instead of our {{mvar|r}} in ({{EquationNote|48}}); this would have the advantage of eliminating the factor{{math| 4''&amp;pi;''}} from the D'Alembertian of the wave function, and the disadvantage of introducing that factor into the (denominator of the) wave function itself.&lt;/ref&gt; Hence the field whose D'Alembertian is given by ({{EquationNote|54}}) is the wave function due to a monopole source with strength {{math|''f''&amp;#8202;(''t'')}} at position{{math| '''r&amp;prime;'''}}. In each case, the D'Alembertian is zero everywhere except at the source; that is, the field satisfies the wave equation except at the source.

''A note in passing:&amp;#8202;'' The above verification that the field ({{EquationNote|48}}) satisfies the wave equation (except at the origin) did not depend on whether {{mvar|c}} was positive or negative. Nor did the demonstration that its D'Alembertian is given by ({{EquationNote|53}}). Hence we may replace{{mvar| c}} by{{mvar| &amp;minus;c}} in ({{EquationNote|54}}) and conclude that, even for positive{{mvar| c&amp;#8202;}}, the field
{{NumBlk|:|&lt;math&gt;
    \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}
    ~\!f\big(t+\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big)
  &lt;/math&gt;|{{EquationRef|54a}}}}
also satisfies the wave equation (except at{{math| '''r&amp;prime;'''}}), and has the same D'Alembertian as in ({{EquationNote|54}}) and the same limiting behavior as{{math| '''r'''&amp;rightarrow;'''r&amp;prime;'''}} and the argument of{{mvar| f}}&amp;#8201; approaches{{mvar| t}}. In this case, however, for positive{{mvar| c&amp;#8202;}}, we can hardly speak of a &quot;source&quot; at{{math| '''r&amp;prime;'''}}, because expression ({{EquationNote|54a}}) describes ''inward''-bound spherical waves converging on{{math| '''r&amp;prime;'''}}, which would violate causality if {{math| '''r&amp;prime;'''}} were the location of the source. But even if ({{EquationNote|54a}}) is dismissed as an &quot;acausal&quot; or &quot;unphysical&quot; solution of the wave equation, it nevertheless ''is'' a solution, and we are free to exploit this fact (such as it is) in derivations and proofs.

{{cob}}

=== Field with given D'Alembertian ===

{{cot}}

Now suppose that, instead of a monopole wave source with strength {{math|''f''&amp;#8202;(''t'')}} at the general position{{math| '''r&amp;prime;''',}} we have at that position a source strength ''density&lt;math&gt;~w(\mathbf{r}'\!,t)&lt;/math&gt;'' in an elemental volume {{mvar|dV&amp;prime;}}, whose (causal!) contribution to the wave function {{mvar|&amp;psi;}} at position{{math| '''r'''}}&amp;#8202; is therefore
:&lt;math&gt;d\psi(\mathbf{r},t)
    = \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}~\!
      w\big(\mathbf{r}',\,t-\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big)\,dV' ,
  &lt;/math&gt;
where for each {{math|'''r''',}} the dimensions of each volume element are small compared with {{math|{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}}}}. Then the total wave function is the sum of the contributions:
{{NumBlk|:|&lt;math&gt;\psi(\mathbf{r},t)
    = \iiint \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}~\!
      w\big(\mathbf{r}',\,t-\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big)\,dV' ,
  &lt;/math&gt;|{{EquationRef|55}}}}
where the integral is over all space.

Independently of the physical significance of{{math| ''&amp;psi;''('''r''',&amp;#8202;''t''),}} we can take its D'Alembertian &quot;under the integral sign&quot; by rule ({{EquationNote|54}}), obtaining
:&lt;math&gt;\begin{align}\Box\psi(\mathbf{r},t)
   &amp;= \iiint \Big({-}4\pi~\!w(\mathbf{r}'\!,t)\,
                  \delta(\mathbf{r}\!-\!\mathbf{r}')\Big)\,dV' \\[.5ex]
   &amp;= \iiint \Big({-}4\pi~\!w(\mathbf{r},t)\,
                  \delta(\mathbf{r}\!-\!\mathbf{r}')\Big)\,dV' \\[.5ex]
   &amp;= -4\pi~\!w(\mathbf{r},t)\!
       \iiint\delta(\mathbf{r}\!-\!\mathbf{r}')\,dV' \\[.5ex]
   &amp;= -4\pi~\!w(\mathbf{r},t)\!
       \iiint\delta(\mathbf{r}'{-}~\!\mathbf{r})\,dV' ;
  \end{align}&lt;/math&gt;
that is,
{{NumBlk|:|&lt;math&gt;
    \Box\psi(\mathbf{r},t) = -4\pi~\!w(\mathbf{r},t) \,.
  &lt;/math&gt;|{{EquationRef|56}}}}

Mathematically, equation ({{EquationNote|56}}) is an identity which applies if {{math|''&amp;psi;''('''r''',&amp;#8202;''t'')}} is given by ({{EquationNote|55}}). Substituting from ({{EquationNote|55}}) and solving for&lt;math&gt;~w,&lt;/math&gt; we can write the identity in full as
{{NumBlk|:|&lt;math&gt;w(\mathbf{r},t) \equiv \Box\bigg({-}\tfrac{1}{4\pi}\!\iiint
      \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}~\!
      w\big(\mathbf{r}',\,t-\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big)
    \,dV'\bigg) \,,
  &lt;/math&gt;|{{EquationRef|57}}}}
where the integral is over all space, or at least all of the space in which&lt;math&gt;~w&lt;/math&gt; may be non-zero. Subject to the convergence of the integral, this shows that ''we can construct a wave function with a given D'Alembertian''.

Physically, equation ({{EquationNote|56}}) gives the D'Alembertian of the wave function for a source density&amp;nbsp;&lt;math&gt;w&lt;/math&gt;. It is the ''inhomogeneous wave equation'', which applies in the presence of an arbitrary source density&amp;mdash;in contrast to the ''homogeneous wave equation'' ({{EquationNote|46}}), which applies in a region where the source density is zero. In this context the word ''homogeneous'' or ''inhomogeneous'' describes the equation, not the medium (which has been assumed homogeneous and isotropic).

In a ''static'' situation, in which the D'Alembertian is reduced to the Laplacian, the inhomogeneous wave equation ({{EquationNote|56}}) is reduced to the form of Poisson's equation ({{EquationNote|40}}). As written, equation ({{EquationNote|40}}) is Poisson's equation in electro''statics''; it applies to the charge density{{math| ''&amp;rho;''('''r''')}}, for which the scalar potential [in ({{EquationNote|39}})] is
:&lt;math&gt;\varphi(\mathbf{r}) = \tfrac{1}{4\pi\epsilon_0}
    \iiint \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}
    \rho(\mathbf{r}') \,dV'.
  &lt;/math&gt;
In electro''dynamics'', which takes time-dependence into account, the scalar potential due to the charge density{{math| ''&amp;rho;''('''r''',&amp;#8202;''t'')}} is
:&lt;math&gt;\varphi(\mathbf{r},t) = \tfrac{1}{4\pi\epsilon_0}
    \iiint \tfrac{1}{|\mathbf{r}-\mathbf{r}'|}
    \rho\big(\mathbf{r}',\,t-\tfrac{|\mathbf{r}-\mathbf{r}'|\,}{c}\big) \,dV',
  &lt;/math&gt;
where the wave speed {{mvar|c}} is the speed of light; this is the same as in the static case except for the delay {{math|{{sfrac|&amp;#8202;{{abs|'''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;'''}}&amp;#8202;|''c''}}&amp;#8202;,}} indicating that the influence of the change density at{{math| '''r&amp;prime;'''}} travels outward from that point at the speed of light. In the dynamic case, by rule ({{EquationNote|57}}), the D'Alembertian of the scalar potential is
:&lt;math&gt;
    \Box\varphi = -\frac{\rho(\mathbf{r},t)}{\,\epsilon_0} \,.
  &lt;/math&gt;
This result is the inhomogeneous wave equation in the scalar potential&amp;mdash;the equation which, in the electro''static'' case, reduces to Poisson's equation ({{EquationNote|40}}).

In electro''dynamics'', however, the electric field&amp;#8202; {{math|'''E'''}} is ''not'' simply&lt;math&gt;\,\,{-}\nabla\varphi~\!,\,&lt;/math&gt; but&lt;math&gt;~\,{-}\nabla\varphi~\!\!-\!\tfrac{\part\mathbf{A}}{\part t}~\!,\,&lt;/math&gt; where {{math|'''A'''}} is the '''magnetic vector potential''', whose defining property is that its curl is the '''magnetic flux density''':
:&lt;math&gt;\mathbf{B} = \operatorname{curl}\mathbf{A} \,.&lt;/math&gt;
By identity ({{EquationNote|24d}}), this property implies
:&lt;math&gt;\operatorname{div}\mathbf{B} = 0 \,,&lt;/math&gt;
which is '''Gauss's law for magnetism'''. We have noted in passing&amp;mdash;but not yet proven&amp;mdash;that ({{EquationNote|24d}}) has a converse, whereby the solenoidality of{{math|&amp;#8202; '''B'''}} implies the ''existence'' of the vector potential{{math| '''A'''}}.  Precedents suggest we might be able to prove this by finding a vector field whose curl is a delta function&amp;mdash;perhaps through new identities relating it to a field whose divergence is a delta function&amp;mdash;and using it to construct a vector field with a given curl. In fact we shall prove our &quot;converse&quot; differently, but we shall still need some new identities for the purpose. And to obtain those identities (among others), we must take the detour that we have made a virtue of ''not'' taking until now&amp;hellip;

{{cob}}

== Cartesian coordinates ==

=== Indicial notation; implicit summation ===

{{cot}}

Considering that a scalar field is a function of three coordinates, while a vector field has three components each of which is a function of three coordinates, we can readily imagine that coordinate-based derivations of vector-analytic identities are likely to be excruciatingly repetitive&amp;mdash;unless perhaps we choose a notation that concisely specifies the repetition. So, instead of writing the Cartesian coordinates as {{math|''x'',&amp;#8239;''y'',&amp;#8239;''z''&amp;#8202;,}}&amp;#8239; we shall usually write them as {{mvar|x&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; where&amp;#8202; {{math|''i''&amp;#8201;{{=}}&amp;#8239;1,&amp;#8202;2,&amp;#8202;3&amp;#8202;,}}&amp;#8201; respectively;&amp;#8202; and instead of writing the unit vectors in the directions of the respective axes as {{math| '''i''',&amp;#8202;'''j''','''k'''&amp;#8202;,}}&amp;#8239; we shall usually write them as {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; And for partial differentiation w.r.t.{{math| ''x&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;,}} instead of writing {{mvar|{{sfrac|&amp;part;|&amp;part;x&lt;sub&gt;i&lt;/sub&gt;}}}} or even {{math|''&amp;part;&lt;sub&gt;x&lt;sub&gt;i&lt;/sub&gt;&lt;/sub&gt;''&amp;#8202;,}} we shall write {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;}}.

Now comes a stroke of genius for which we are indebted to Einstein (although he used it in a more sophisticated context!). Instead of writing the position vector as
:&lt;math&gt;\mathbf{r} = x_1\mathbf{e}_1 + x_2\mathbf{e}_2 + x_3\mathbf{e}_3&lt;/math&gt;
or even as
:{{big|&lt;math&gt;\mathbf{r} = \textstyle\sum_i x_i \mathbf{e}_i \,,&lt;/math&gt;}}
we shall write it simply as
:{{big|&lt;math&gt;\mathbf{r} = x_i \mathbf{e}_i \,,&lt;/math&gt;}}
where it is ''understood''&amp;#8202; that we ''sum over the repeated index''. More generally, we shall write the vector field {{math|'''q'''}} as
:{{big|&lt;math&gt;\mathbf{q} = q_i \mathbf{e}_i&lt;/math&gt;}}
with implicit summation, and the vector field {{math|'''v'''}} as
:{{big|&lt;math&gt;\mathbf{v} = v_i \mathbf{e}_i&lt;/math&gt;}}
with implicit summation, and so on. (By that nomenclature, the position vector in Cartesian coordinates should be, and often is, called {{math|'''x'''&amp;#8202;}}; but we called it {{math|'''r'''}} because we wanted to call its magnitude {{mvar|r}}, for ''radius''.)

Implicit summation not only avoids writing the {{big|{{math|&amp;Sigma;}}}} symbol and specifying the index of summation, but also allows a summation over ''two'' repeated indices, say {{mvar|i}} and {{mvar|j&amp;#8202;}}, to be considered as summed first over {{mvar|i}} and then over {{mvar|j}} or vice versa, removing the need for an explicit regrouping of terms. Of course, if we hide messy details behind a notation, we need to make sure that it handles those details correctly. In particular, when we perform an operation on an implicit sum, we implicitly perform it ''term-by-term'', and must therefore make sure that the operation is valid when interpreted that way.

{{cob}}

=== Formulation of operators ===

{{cot}}

'''Gradient''':  Putting&amp;#8201; {{mvar|s&amp;#8201;{{=}}&amp;#8201;x&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8201; in ({{EquationNote|9g}}), we find that the scalar component of{{math|&amp;#8202; &amp;nabla;''p''}} in the direction of each {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;}}&amp;#8202; is{{mvar|&amp;#8202; &amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8201;p}}.&amp;#8201; To obtain the vector component in that direction, we multiply by {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; Assembling the components, we have (with implicit summation)
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla p = \mathbf{e}_i ~\!\part_i p
  &lt;/math&gt;}}|{{EquationRef|58g}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla =~\! \mathbf{e}_i \part_i
  &lt;/math&gt;}}|{{EquationRef|58o}}}}
or, in traditional longhand notation,
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla =~\! \mathbf{i}~\!\tfrac{\part}{\part x}
           +~\! \mathbf{j}~\!\tfrac{\part}{\part y}
           +~\! \mathbf{k}~\!\tfrac{\part}{\part z} \,.
  &lt;/math&gt;}}|{{EquationRef|58t}}}}
It is also worth noting, from ({{EquationNote|58g}}), that the squared magnitude of{{math|&amp;#8202; &amp;nabla;''p'' &amp;#8239;}}is
{{NumBlk|:|{{big|&lt;math&gt;
    |\nabla p|^2 =~\! \part_i p \;\part_i p \,,
  &lt;/math&gt;}}|{{EquationRef|58s}}}}
where we write&amp;#8239; {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p&amp;#8201;&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p}}&amp;#8201; rather than {{math|(''&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p'')&lt;sup&gt;2&lt;/sup&gt;}}&amp;#8239; to ensure that implicit summation applies.

As reported by Tai ([[#tai-94|1994]]), there are unfortunately some textbooks in which the del operator is defined as
:{{big|&lt;math&gt;\nabla =~\!
    \tfrac{\part}{\part x}~\!\mathbf{i} +
    \tfrac{\part}{\part y}~\!\mathbf{j} +
    \tfrac{\part}{\part z}~\!\mathbf{k} \quad\qquad
  &lt;/math&gt;}}{{big|1=[''sic!''&amp;#8239;]}}
&amp;mdash;which, on its face, is not an operator at all, but a self-contained expression whose value is the zero vector (because it is a sum of derivatives of constant vectors). Among the offenders is Erwin Kreyszig, who, in the 6th edition of his bestselling ''Advanced Engineering Mathematics'' ([[#kreyszig-62-|1988]], p.&amp;#8239;486), misdefines the del operator thus and then rewrites the gradient of{{mvar|&amp;#8202; f}}&amp;#8239; as {{math|&amp;nabla;&amp;#8202;''f'',}} apparently imagining that the differentiation operators look ''through'' the constant vectors rather than ''at''&amp;#8202; them. Six pages later, he defines the divergence in Cartesian coordinates (which we shall do shortly) and then immediately informs us that &quot;Another common notation for the divergence of{{math| '''v'''}} is {{math|&amp;nabla;'''&amp;sdot;&amp;#8202;v'''}},&quot; where {{math|&amp;nabla;}} is defined as before, but the resulting {{math|&amp;nabla;'''&amp;sdot;&amp;#8202;v'''}} is apparently not identically zero!&lt;ref&gt;The latter passage, as it appears in the 5th edition (p.&amp;#8239;397), is the one cited by Tai ([[#tai-94|1994]], p.&amp;#8239;6).&lt;/ref&gt; These errors persist in the 10th edition ([[#kreyszig-62-|2011]], pp.&amp;#8239;396,&amp;#8239;402–3). Tai finds similar howlers in mathematics texts by Wilfred Kaplan, Ladis D. Kovach, and Merle C. Potter, and in electromagnetics texts by William H. Hayt and Martin A. Plonus.&lt;ref&gt;Quoted by Tai ([[#tai-94|1994]]), in alphabetical order within each category. For Kovach he could have added p.&amp;#8239;308.&amp;#8201; Potter he misnames as Porter.&lt;/ref&gt;&amp;#8201; Knudsen &amp;amp; Katz, in ''Fluid Dynamics and Heat Transfer'' (1958), avoid the misdefinition of{{math| &amp;nabla;,}} but implicitly define the divergence of{{math| '''V'''}} as {{math|1='''V&amp;sdot;'''&amp;nabla;}}&amp;#8202; (which, as we have seen, is actually an operator), and then somehow reduce it to the correct expression for{{math|&amp;#8202; div&amp;#8239;'''V'''}}.&amp;#8239;&lt;ref&gt;Quoted by Tai ([[#tai-94|1994]], p.&amp;#8239;23).&lt;/ref&gt;  But I digress.

'''Curl and divergence''':  Expressing the operand of the curl in components, and noting that the unit vectors are ''uniform'', we can apply ({{EquationNote|8p}}):
:{{big|&lt;math&gt;\begin{align}
    \operatorname{curl}\mathbf{q}
    &amp;= ~\!\mathrm{curl}(q_j ~\!\mathbf{e}_j) &amp;&amp; \\
    &amp;= \nabla q_j \times \mathbf{e}_j &amp;&amp; [\mathsf{\scriptstyle by~eq.(8p)}] \\
    &amp;= ~\!\mathbf{e}_i \part_i q_j \times \mathbf{e}_j
       &amp;&amp; [\mathsf{\scriptstyle by~eq.(58g)}] \\
    &amp;= ~\!\mathbf{e}_i ~\!\!\times \part_i ~\!q_j \mathbf{e}_j \,. &amp;&amp;
  \end{align}&lt;/math&gt;}}
If we sum over {{mvar|j}} first, this is
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}\mathbf{q} =~\! \mathbf{e}_i ~\!\!\times\part_i\mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|59c}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl} =~\! \mathbf{e}_i ~\!\!\times\part_i
  &lt;/math&gt;}}|{{EquationRef|59o}}}}
or, in traditional longhand,
:{{big|&lt;math&gt;
    \operatorname{curl}
    \,=\, \mathbf{i} \times ~\!\!\tfrac{\part}{\part x}
     +~\! \mathbf{j} \times ~\!\!\tfrac{\part}{\part y}
     +~\! \mathbf{k} \times ~\!\!\tfrac{\part}{\part z} \,.
  &lt;/math&gt;}}
For the ''divergence'' we proceed as for the curl except that, instead of ({{EquationNote|8p}}), we use ({{EquationNote|8g}}):
:{{big|&lt;math&gt;\begin{align}
    \operatorname{div}\mathbf{q}
    &amp;= ~\!\mathrm{div}(q_j ~\!\mathbf{e}_j) &amp;&amp; \\
    &amp;= \nabla q_j \cdot \mathbf{e}_j &amp;&amp; [\mathsf{\scriptstyle by~eq.(8g)}] \\
    &amp;= ~\!\mathbf{e}_i \part_i q_j \cdot \mathbf{e}_j
       &amp;&amp; [\mathsf{\scriptstyle by~eq.(58g)}] \\
    &amp;= ~\!\mathbf{e}_i ~\!\!\cdot \part_i ~\!q_j \mathbf{e}_j \,; &amp;&amp;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div}\mathbf{q} =~\! \mathbf{e}_i ~\!\!\cdot \part_i\mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|60d}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div} =~\! \mathbf{e}_i ~\!\!\cdot \part_i
  &lt;/math&gt;}}|{{EquationRef|60o}}}}
or, in traditional longhand,
:{{big|&lt;math&gt;
    \operatorname{div}
    \,=\, \mathbf{i} \cdot \tfrac{\part}{\part x}
     +~\! \mathbf{j} \cdot \tfrac{\part}{\part y}
     +~\! \mathbf{k} \cdot \tfrac{\part}{\part z} \,.
  &lt;/math&gt;}}
It follows from ({{EquationNote|59c}}) and ({{EquationNote|60d}}), if it was not already obvious, that ''a uniform vector field has zero curl and zero divergence''.

Although the above expressions for the divergence and curl will surprise many modern readers, they match the ''initial definitions'' of the divergence and curl given by the founder of vector analysis as we know it, [[w:Josiah Willard Gibbs|J.&amp;#8239;Willard Gibbs]] ([[#gibbs-1881-4|1881]], &amp;sect;&amp;#8239;54). Gibbs even uses the {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; notations on the left sides of the defining equations, and only ''after''&amp;#8202; the equations (albeit immediately after) does he announce that&amp;#8202; &quot;{{math|&amp;#8202;&amp;nabla;'''&amp;sdot;'''&amp;#8202;''&amp;omega;''}} is called the ''divergence'' of{{mvar| &amp;omega;}}&amp;#8239; and {{math|&amp;nabla;&amp;#8202;&amp;times;''&amp;omega;''}}&amp;#8239; its ''curl''.&quot; (He uses Greek letters for vectors.) Our notation and Cartesian expression for the gradient ({{EquationNote|58g}}) also match Gibbs ([[#gibbs-1881-4|1881]], &amp;sect;&amp;#8239;52). Hence, using the Gibbs notations, we can merge definitions ({{EquationNote|58g}}), ({{EquationNote|59c}}), and ({{EquationNote|60d}}) into the general Cartesian formula
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla~\!\! * \psi =~\! \mathbf{e}_i ~\!\! * \part_i \psi
  &lt;/math&gt;}}|{{EquationRef|60s}}}}
(with implicit summation), where the {{math|&amp;lowast;}} operator may be a null (for the gradient), a cross (for the curl), or a dot (for the divergence).

Gibbs does not offer any justification for the {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; notations, but nor is it difficult to find such a justification based on his definitions. As{{math| '''e'''&lt;sub&gt;''i''&lt;/sub&gt;}} is a ''uniform'' vector, we can rewrite ({{EquationNote|59c}}) ''rigorously'' as
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}\mathbf{q} = \part_i(\mathbf{e}_i ~\!\!\times\mathbf{q})
  &lt;/math&gt;}}|{{EquationRef|61c}}}}
and thence ''operationally'' as
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}\mathbf{q} =~\! \mathbf{e}_i\part_i \times \mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|61o}}}}
or, recalling ({{EquationNote|58o}}),
:&lt;math&gt;
    \operatorname{curl}\mathbf{q} ~\!= \nabla \times \mathbf{q} \,,
  &lt;/math&gt;
which can be evaluated in the usual manner as
:&lt;math&gt;
    \operatorname{curl}\mathbf{q} \,=\,
      \begin{vmatrix}
        \mathbf{i} &amp; \part_x &amp; q_x \\
        \mathbf{j} &amp; \part_y &amp; q_y \\
        \mathbf{k} &amp; \part_z &amp; q_z
      \end{vmatrix} \,,
  &lt;/math&gt;
where {{mvar|q&lt;sub&gt;x&lt;/sub&gt;}} is the {{mvar|x}} component of{{math| '''q'''&amp;#8202;}}, etc. This indeed is how one evaluates the curl of a given field in Cartesian coordinates, although we shall find ({{EquationNote|59c}}) more convenient for deriving identities. Similarly, we can rewrite ({{EquationNote|60d}}) ''rigorously'' as
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div}\mathbf{q} = \part_i(\mathbf{e}_i ~\!\!\cdot \mathbf{q})
  &lt;/math&gt;}}|{{EquationRef|62d}}}}
and thence ''operationally'' as
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div}\mathbf{q} =~\! \mathbf{e}_i\part_i \cdot \mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|62o}}}}
or, recalling ({{EquationNote|58o}}),
:&lt;math&gt;
    \operatorname{div}\mathbf{q} ~\!= \nabla \!\cdot \mathbf{q} ~.
  &lt;/math&gt;
For evaluating the divergence of a given field, however, we simplify ({{EquationNote|62d}}) to
:{{big|&lt;math&gt;
    \operatorname{div}\mathbf{q} = \part_i q_i
  &lt;/math&gt;}}
or, in traditional longhand,
:&lt;math&gt;
    \operatorname{div}\mathbf{q}
    ~\!= \frac{\part q_x}{\part x}
       + \frac{\part q_y}{\part y}
       + \frac{\part q_z}{\part z} \,,
  &lt;/math&gt;
although we shall find ({{EquationNote|60d}}) more convenient for deriving identities. But the longhand form makes it especially obvious that if{{math|&amp;#8202; '''r'''}} is the position vector,
{{NumBlk|:|&lt;math&gt;
    \operatorname{div}\mathbf{r} = 3 \,.
  &lt;/math&gt;|{{EquationRef|62r}}}}

Notice that we can get from ({{EquationNote|62o}}) back to ({{EquationNote|60d}}) by permuting the {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}} with the dot, and from ({{EquationNote|61o}}) back to ({{EquationNote|59c}}) by permuting the {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}} with the cross, as if the differentiation operator could, as it were, look through the dot or the cross&amp;mdash;or, as Gibbs's student [[w:Edwin Bidwell Wilson|Edwin B.&amp;#8201;Wilson]] puts it, &quot;pass by&quot; the dot and the cross, yielding Gibbs's original definitions.&lt;ref&gt;[[#wilson-1901|Wilson, 1901]], p.&amp;#8239;150.&lt;/ref&gt; Hence Wilson considers it helpful to regard Gibbs's {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; and {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; notations as &quot;the (formal) scalar product and the (formal) vector product of{{math|&amp;#8202; &amp;nabla;}} into&quot; the operand, or &quot;the symbolic scalar and vector products of{{math|&amp;#8202; &amp;nabla;}} into&quot; the operand, and to regard {{math|&amp;nabla;}} as a &quot;symbolic vector&quot;&lt;ref&gt;[[#wilson-1901|Wilson, 1901]], pp.&amp;#8239;150,&amp;#8239;152. Wilson does not announce this idea in his preface (p.&amp;#8239;xii), although Tai ([[#tai-95|1995, p.&amp;#8239;26]]) gets the contrary impression by omitting a comma from the relevant quote.&lt;/ref&gt; (not to be confused with Tai's symbolic vector&lt;math&gt;~\nabla\!\!\!\!^{\textstyle_-}&lt;/math&gt;).

Tai ([[#tai-94|1994]], [[#tai-95|1995]]) rejects Wilson's argument together with the entire tradition of treating {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; as compound operators. Of formal products, Tai says that the concept &quot;has had a tremendously detrimental effect upon the learning of vector analysis&quot;; he calls such a product a &quot;meaningless assembly&quot;.&lt;ref&gt;[[#tai-95|Tai, 1995]], pp.&amp;#8239;26,&amp;#8239;38.&lt;/ref&gt; Of the &quot;pass by&quot; step, he complains that &quot;standard books on mathematical analysis do not have such a theorem.&quot;&lt;ref&gt;[[#tai-95|Tai, 1995]], p.&amp;#8239;28.&lt;/ref&gt;

I submit, however, that the intermediate steps ({{EquationNote|61c}}) and ({{EquationNote|62d}}), after which we take the constant multiplier outside the operator (eqs. {{EquationNote|61o}} &amp;amp; {{EquationNote|62o}}), support Wilson's &quot;pass by&quot; argument. In any event the reader may write out the sums on the right-hand sides of ({{EquationNote|59c}}) and ({{EquationNote|60d}}) and verify that they agree with the formal products {{math|&amp;nabla;&amp;#8202;&amp;times;&amp;#8202;'''q'''}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;&amp;#8239;q'''}}&amp;#8202; respectively&amp;mdash;and may notice that in the evaluation of each formal product, the cross or dot is eventually eliminated, leaving nothing to &quot;pass by&quot;.&lt;ref&gt;The latter observation is made, or at least suggested, by Kemin et al. ([[#kemin-et-al-00|2000]], p.&amp;#8239;605).&lt;/ref&gt; I further submit that the great generality of our derivation of equations ({{EquationNote|14}}), above, compels us to treat the {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; notations as more than mere notations. But the kicker is that Tai himself, having found the form of the del operator in ''general'' coordinates ([[#tai-95|1995]], p.&amp;#8239;64, eq.&amp;#8239;9.33), derives original corresponding forms of the {{math|div}} and {{math|curl}} operators (his eqs.&amp;#8239;9.35 &amp;amp; 9.40) which, upon reversal of the forbidden &quot;pass by&quot;, become del-dot and del-cross! Indeed his three equations, just cited, are reminiscent of our ({{EquationNote|58o}}), ({{EquationNote|60o}}), and ({{EquationNote|59o}}) respectively. That being said, I shall find some points of agreement with Tai, and some reasons to criticize Wilson.

'''Laplacian''':  If {{mvar|&amp;psi;}} is a ''scalar'' field, then
:{{big|&lt;math&gt;\begin{align}\triangle\psi
    &amp;= \operatorname{div}\nabla\psi
       &amp;&amp; [\mathsf{\scriptstyle by~eq.(9L')}] \\
    &amp;= \part_i(\mathbf{e}_i \cdot \nabla\psi)
       &amp;&amp; [\mathsf{\scriptstyle by~eq.(62d)}] \\
    &amp;= \part_i(\part_i \psi) \,; &amp;&amp;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \triangle\psi = \part_i \part_i \psi \,,
  &lt;/math&gt;}}|{{EquationRef|63L}}}}
where we write&amp;#8202; {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;&amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; rather than {{math|''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&lt;sup&gt;2&lt;/sup&gt;}}&amp;#8202; in order to maintain implicit summation. In traditional longhand, ({{EquationNote|63L}}) becomes
:&lt;math&gt;\triangle\psi ~\!= \frac{\part^2 \psi}{\part x ^2}
                        + \frac{\part^2 \psi}{\part y ^2}
                        + \frac{\part^2 \psi}{\part z ^2}
  &lt;/math&gt;
or, in operational terms,
:&lt;math&gt;\triangle ~\!= \frac{\part^2}{\part x ^2}
                    + \frac{\part^2}{\part y ^2}
                    + \frac{\part^2}{\part z ^2}
  &lt;/math&gt;
or, by comparison with ({{EquationNote|58t}}),
:&lt;math&gt;\triangle = \nabla{\cdot}\nabla &lt;/math&gt;
&amp;mdash;as expected.

By the linearity of the Laplacian, the same applies if {{mvar|&amp;psi;}} is any field expressible in terms of a uniform basis. For example, if {{mvar|&amp;psi;}} is a ''vector'' field given by&amp;#8239; {{math|''&amp;psi;&lt;sub&gt;j&lt;/sub&gt;''&amp;#8239;'''e'''&lt;sub&gt;''j''&lt;/sub&gt;}}&amp;#8239; (with implicit summation), then
:{{big|&lt;math&gt;\begin{align}
    \triangle\psi
    &amp;= \triangle(\psi_j ~\!\mathbf{e}_j) \\
    &amp;= \mathbf{e}_j ~\!\triangle\psi_j \\
    &amp;= \mathbf{e}_j \part_i \part_i \psi_j \\
    &amp;= \part_i \part_i (\psi_j ~\!\mathbf{e}_j)
     = \part_i \part_i \psi \,,
  \end{align}&lt;/math&gt;}}
where the third line follows from ({{EquationNote|63L}}) as applied to a scalar field. Thus ({{EquationNote|63L}}) is quite general.

After listing theorems ({{EquationNote|5g}}) to ({{EquationNote|5L}}) above, we gave reasons for describing {{math|&amp;nabla;,}} {{math|curl,}} and {{math|div}} as ''differential operators'', and {{math|&amp;#9651;}} as a ''2nd-order'' differential operator&amp;mdash;the implication being that the others are only 1st-order. We now have the promised &quot;additional reason&quot; for these descriptions: when expressed in Cartesian coordinates, the {{math|&amp;#9651;}} operator involves second derivatives, while the others involve (only) first derivatives. In the meantime we have acquired the {{math|'''q&amp;sdot;'''&amp;nabla;}} operator, which is also 1st-order, as we shall now confirm.

'''Advection, directional derivative, etc.''':   If {{mvar|&amp;psi;}} is a ''scalar'' field, then
:{{big|&lt;math&gt;\begin{align}
    \mathbf{q}\;\!{\cdot}\nabla\,\psi
    &amp;= (\mathbf{q}) \cdot (\nabla\psi) \\
    &amp;= (q_i~\!\mathbf{e}_i) \cdot (\mathbf{e}_j ~\!\part_j \psi) \\
    &amp;= \;\!\mathbf{e}_i{\cdot}\;\!\mathbf{e}_j \;q_i \part_j \psi \,.
  \end{align}&lt;/math&gt;}}
In this double summation, the only non-zero terms are those for which&amp;#8202; {{mvar|j&amp;#8239;{{=}}&amp;#8201;i&amp;#8202;}},&amp;#8239; in which case&amp;#8239; {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;'''&amp;sdot;&amp;#8239;e'''&lt;sub&gt;''j''&lt;/sub&gt;&amp;#8201;{{=}}&amp;#8239;1}}.&amp;#8201; So we have
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\psi = q_i ~\!\part_i \psi
  &lt;/math&gt;}}|{{EquationRef|64}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla = q_i ~\!\part_i
  &lt;/math&gt;}}|{{EquationRef|64o}}}}
or, in traditional longhand,
:{{big|&lt;math&gt;\mathbf{q}\;\!{\cdot}\nabla
    =~\! q_x\tfrac{\part}{\part x}
    +~\! q_y\tfrac{\part}{\part y}
    +~\! q_z\tfrac{\part}{\part z} \,,
  &lt;/math&gt;}}
which indeed is the &quot;formal&quot; or &quot;symbolic&quot; dot-product of&amp;#8202; {{math|'''q'''}} and{{math| &amp;nabla;}}.&amp;#8202; By the linearity of the directional derivative in ({{EquationNote|11}}), the same result applies if {{mvar|&amp;psi;}} is a vector field or any field expressible in terms of a uniform basis. In particular, if{{math| '''r'''}} is the position vector, we have
:{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\mathbf{r}
    = q_i ~\!\part_i \mathbf{r}
    = q_i ~\!\mathbf{e}_i \,,
  &lt;/math&gt;}}
i.e.,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\mathbf{r} = \mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|64r}}}}
&amp;mdash;which is also deducible from ({{EquationNote|11}}).

For convenience in the following discussion, we shall refer to the scaled-directional-derivative operator {{math|'''q&amp;sdot;'''&amp;nabla;}} as an &quot;advection&quot; operator although, physically, it represents advection only if {{math|'''q'''}} is the material velocity.

{{cob}}

=== Identities without pain ===

{{cot}}

In deriving the Cartesian expressions for the gradient, curl, divergence, Laplacian, and advection operators, we used the preceding identities ({{EquationNote|9g}}), ({{EquationNote|8p}}), ({{EquationNote|8g}}), ({{EquationNote|9L'}}), and ({{EquationNote|11}}) respectively, the last being a definition generalizing ({{EquationNote|9g}}). Thus we could have derived the Cartesian expressions quite early in the exposition, although we did not find that option convenient. The other vector-analytic identities that we have previously mentioned are:
* ({{EquationNote|8c}}), which showed the unambiguity of the curl;
* ({{EquationNote|8q}}), which has a question mark after it;
* ({{EquationNote|17}}), a product rule for the divergence, which is yet to be proven as a general identity;
* ({{EquationNote|24c}}) and ({{EquationNote|24c}}), concerning &quot;curl grad&quot; and &quot;div curl&quot;; and
* the identities showing that we can construct a field with a given divergence ({{EquationNote|36}}), Laplacian ({{EquationNote|38}}), or D'Alembertian ({{EquationNote|57}}).

The above list exposes the following shortcomings:
* we have not yet investigated &quot;grad div&quot; and &quot;curl curl&quot;;
* we have only one ''product rule''&amp;#8202;&amp;mdash;the unverified identity ({{EquationNote|17}})&amp;mdash;in which ''both'' factors are spatially variable fields; this needs to be verified and identities ({{EquationNote|8c}}) and ({{EquationNote|8p}}) need to be generalized;
* our collection of product rules does not yet include the curl of a cross-product, or the gradient of a dot-product or of a product of scalars, or the advection of a product; and
* we do not yet have any ''chain rules'' involving {{math|&amp;nabla;,}} {{math|curl,}} or {{math|div}}.
With the aid of the Cartesian forms of the various operators, we may now fill these gaps.

&lt;br /&gt;

The &quot;'''grad div'''&quot; and &quot;'''curl curl'''&quot; operators turn out to be related:
:{{big|&lt;math&gt;\begin{align}
    \operatorname{curl}\operatorname{curl}\mathbf{q} \;\!
    &amp;= \mathbf{e}_i \times\part_i(\operatorname{curl}\mathbf{q}) \\
    &amp;= \mathbf{e}_i \times\part_i(\mathbf{e}_j \times\part_j\mathbf{q}) \\
    &amp;= \mathbf{e}_i \times(\mathbf{e}_j \times\part_i\part_j\mathbf{q}) \,,
  \end{align}&lt;/math&gt;}}
whence expanding the vector triple product gives
:{{big|&lt;math&gt;\operatorname{curl}\operatorname{curl}\mathbf{q} \;\!
    = \mathbf{e}_i \!\cdot~\!\!\part_i\part_j\mathbf{q} ~\mathbf{e}_j
      - \mathbf{e}_i {\cdot}~\!\mathbf{e}_j \,\part_i\part_j\mathbf{q} \,.
  &lt;/math&gt;}}
In the first term on the right, we can switch the order of partial differentiation; and in the second term&amp;mdash;which, like the first, is a double summation&amp;mdash;the only non-zero contributions are those for which&amp;#8202; {{mvar|j&amp;#8239;{{=}}&amp;#8201;i}}&amp;#8239; and&amp;#8239; {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;'''&amp;sdot;&amp;#8239;e'''&lt;sub&gt;''j''&lt;/sub&gt;&amp;#8201;{{=}}&amp;#8239;1}}.&amp;#8201; So we have
:{{big|&lt;math&gt;\begin{align}
    \operatorname{curl}\operatorname{curl}\mathbf{q} \;\!
    &amp;= \mathbf{e}_i \!\cdot~\!\!\part_j\part_i\mathbf{q} ~\mathbf{e}_j
       - \part_i\part_i\mathbf{q} \\
    &amp;= \mathbf{e}_j \,\part_j(\mathbf{e}_i \!\cdot~\!\!\part_i\mathbf{q})
      - \part_i\part_i\mathbf{q} \,;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}\operatorname{curl}\mathbf{q}
    ~\!\equiv \nabla\operatorname{div}\mathbf{q} - \triangle\mathbf{q} \,.
  &lt;/math&gt;}}|{{EquationRef|65}}}}
This result may be memorized as &quot;''curl curl is grad div minus del squared''&amp;#8239;&quot; and written as
{{NumBlk|:|{{big|{{math|&amp;nabla;&amp;#8239;&amp;times;&amp;#8201;(&amp;nabla;&amp;#8239;&amp;times;&amp;#8201;'''q''') &amp;equiv; &amp;nabla; &amp;nabla;'''&amp;sdot;&amp;#8239;q''' &amp;minus; &amp;nabla;&lt;sup&gt;2&lt;/sup&gt;&amp;#8202;'''q'''}}&amp;#8201;,}}|{{EquationRef|66}}}}
which ''looks like'' the expansion of a vector triple product; and the key step in the above derivation, based on the Gibbs definitions of the operators,  ''really is''&amp;#8202; the expansion of a vector triple product.

&lt;br /&gt;

We now turn to ''product rules'' in which neither factor is assumed uniform.

The '''curl of a cross-product''' is
:{{big|&lt;math&gt;\begin{align}
    &amp;\operatorname{curl}(\mathbf{a}\!\times\!\mathbf{b}) \\
    &amp;~= \mathbf{e}_i ~\!\!\times\part_i(\mathbf{a}\times\mathbf{b}) \\
    &amp;~= \mathbf{e}_i ~\!\!\times(\part_i\mathbf{a}\times\mathbf{b} +
                                 \mathbf{a}\times\part_i\mathbf{b}) \\
    &amp;~= \mathbf{e}_i ~\!\!\times(\part_i\mathbf{a}\times\mathbf{b}) +
        \mathbf{e}_i ~\!\!\times(\mathbf{a}\times\part_i\mathbf{b}) \\
    &amp;~= \mathbf{e}_i{\cdot}~\!\mathbf{b} \,\part_i\mathbf{a}
        - \mathbf{e}_i{\cdot}~\!\part_i\mathbf{a} \;\mathbf{b}
        + \mathbf{e}_i{\cdot}~\!\part_i\mathbf{b} \;\mathbf{a}
        - \mathbf{e}_i{\cdot}~\!\mathbf{a} \,\part_i\mathbf{b} \\
    &amp;~= b_i\part_i\mathbf{a} - (\operatorname{div}\mathbf{a})~\!\mathbf{b}
        + (\operatorname{div}\mathbf{b})~\!\mathbf{a} - a_i\part_i\mathbf{b} \\
    &amp;~= \mathbf{b}~\!{\cdot}\nabla\,\mathbf{a}
        - \mathbf{b}\operatorname{div}\mathbf{a}
        + \mathbf{a}\operatorname{div}\mathbf{b}
        - \mathbf{a}~\!{\cdot}\nabla\,\mathbf{b} \,,
  \end{align}&lt;/math&gt;}}
i.e.,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}(\mathbf{a}\!\times\!\mathbf{b})
    \equiv \mathbf{a}\operatorname{div}\mathbf{b}
         - \mathbf{b}\operatorname{div}\mathbf{a}
         + \mathbf{b}~\!{\cdot}\nabla\,\mathbf{a}
         - \mathbf{a}~\!{\cdot}\nabla\,\mathbf{b} \,.
  &lt;/math&gt;}}|{{EquationRef|67c}}}}

The '''divergence of a cross-product''', as we might expect, is simpler:
:{{big|&lt;math&gt;\begin{align}\operatorname{div}(\mathbf{a}\!\times\!\mathbf{b})
    &amp;= \mathbf{e}_i ~\!\!\cdot\part_i(\mathbf{a}\times\mathbf{b}) \\
    &amp;= \mathbf{e}_i ~\!\!\cdot(\part_i\mathbf{a}\times\mathbf{b} +
                               \mathbf{a}\times\part_i\mathbf{b}) \\
    &amp;= \mathbf{e}_i ~\!\!\cdot\part_i\mathbf{a}\times\mathbf{b} +
       \mathbf{e}_i ~\!\!\cdot\mathbf{a}\times\part_i\mathbf{b} \\
    &amp;= \mathbf{e}_i ~\!\!\cdot\part_i\mathbf{a}\times\mathbf{b} -
       \mathbf{e}_i ~\!\!\cdot\part_i\mathbf{b}\times\mathbf{a} \\
    &amp;= \mathbf{b}\cdot\mathbf{e}_i ~\!\!\times\part_i\mathbf{a} -
       \mathbf{a}\cdot\mathbf{e}_i ~\!\!\times\part_i\mathbf{b} \,;
  \end{align}&lt;/math&gt;}}
i.e.,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div}(\mathbf{a}\!\times\!\mathbf{b})
    \equiv \mathbf{b}\cdot\operatorname{curl}\mathbf{a}
         - \mathbf{a}\cdot\operatorname{curl}\mathbf{b} \,.
  &lt;/math&gt;}}|{{EquationRef|67d}}}}
In particular, in electromagnetics,&amp;#8201; {{math|div('''E'''&amp;#8239;&amp;times;&amp;#8239;'''H''') &amp;equiv; '''H&amp;#8239;&amp;sdot;'''&amp;#8201;curl&amp;#8201;'''E''' &amp;minus; '''E&amp;#8239;&amp;sdot;'''&amp;#8201;curl&amp;#8201;'''H'''}}&amp;#8239;;&amp;#8239; this is the identity on which [[w:Poynting's theorem|Poynting's theorem]] is based. But if&amp;#8202; {{math|'''b'''}} in ({{EquationNote|67d}}) is uniform, then ({{EquationNote|67d}}) reduces to ({{EquationNote|8c}}).

The '''gradient of a dot-product''', by comparison, is surprisingly messy:
:{{big|&lt;math&gt;\begin{align}\nabla\,\mathbf{a}{\cdot}\mathbf{b}
    &amp;= \mathbf{e}_i
       \part_i(\mathbf{a}\!\cdot\!\mathbf{b}) \\
    &amp;= \mathbf{e}_i
       (\mathbf{a}\!\cdot\!\part_i\mathbf{b} +
        \mathbf{b}\!\cdot\!\part_i\mathbf{a}) \\
    &amp;= \mathbf{a}\!\cdot\!\part_i\mathbf{b}
       \;\mathbf{e}_i +
       \mathbf{b}\!\cdot\!\part_i\mathbf{a}
       \;\mathbf{e}_i \,.
  \end{align}&lt;/math&gt;}}
Now the first term on the right can be recognized as&amp;#8201; {{math|'''a'''&amp;#8202;&amp;times;&amp;#8239;('''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8202;&amp;times;&amp;#8239;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''b''')&amp;#8201;+&amp;#8201;'''a&amp;sdot;&amp;#8202;e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8201;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''b'''&amp;#8202;}};&amp;#8201; that is,&amp;#8201; {{math|'''a'''&amp;#8202;&amp;times;&amp;#8239;('''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8202;&amp;times;&amp;#8239;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''b''')&amp;#8201;+&amp;#8201;''a&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''b'''&amp;#8202;}};&amp;#8201; that is,&amp;#8201; &lt;math&gt;\mathbf{a}\!\times\!\operatorname{curl}\mathbf{b}+\mathbf{a}~\!{\cdot}\nabla\,\mathbf{b}&lt;/math&gt;.&amp;#8201; Similarly, the second term is&amp;#8201; &lt;math&gt;\mathbf{b}\!\times\!\operatorname{curl}\mathbf{a}+\mathbf{b}~\!{\cdot}\nabla\,\mathbf{a}&lt;/math&gt;.&amp;#8201; Thus we have
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla\,\mathbf{a}{\cdot}\mathbf{b}
    \equiv \mathbf{a}\!\times\!\operatorname{curl}\mathbf{b}
         + \mathbf{b}\!\times\!\operatorname{curl}\mathbf{a}
	 + \mathbf{a}~\!{\cdot}\nabla\,\mathbf{b}
	 + \mathbf{b}~\!{\cdot}\nabla\,\mathbf{a} \,.
  &lt;/math&gt;}}|{{EquationRef|68}}}}
For ''uniform''&amp;#8202; {{math|'''b'''&amp;#8202;,}} the first and third terms on the right vanish, and we can solve for the first term on the right, obtaining
:&lt;math&gt;\mathbf{b}\times\operatorname{curl}\mathbf{a} ~\!=
    \nabla\,\mathbf{b{\cdot}a} - \mathbf{b}~\!{\cdot}\nabla\,\mathbf{a}
  \qquad&lt;/math&gt;[&amp;#8202;for uniform {{math|'''b'''}}] ,
so that we can now drop the question mark after ({{EquationNote|8q}}). If we write the curl operator as&amp;#8202; {{math|&amp;nabla;&amp;#8202;&amp;times;&amp;#8239;,}}&amp;#8239; the last equation [or ({{EquationNote|8q}})]&amp;#8202; ''looks like'' the expansion of a vector triple product; but the identity is valid only for uniform{{math| '''b'''}}.

The '''gradient of a product of scalars''', unlike that of a dot-product, is as simple as the product rule for ordinary differentiation:
:{{big|&lt;math&gt;\begin{align}\nabla(p\varphi)
    &amp;= \mathbf{e}_i \part_i(p\varphi) \\
    &amp;= \mathbf{e}_i(p~\!\part_i\varphi + \varphi~\!\part_i p) \\
    &amp;= p~\!\mathbf{e}_i\part_i\varphi + \varphi~\!\mathbf{e}_i\part_i p \,;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla(p\varphi) \equiv p\;\!\nabla\varphi + \varphi\;\!\nabla p \,.
  &lt;/math&gt;}}|{{EquationRef|69}}}}

The '''advection of a product''' is equally simple, ''regardless of the type of product'', except that the order of a cross-product matters. Let {{mvar|&amp;psi;}} and {{mvar|&amp;chi;}} be scalar or vector fields, and let {{math|''&amp;psi;''&amp;#8202;&amp;lowast;''&amp;chi;''}} denote any meaningful product of the two. Then, by ({{EquationNote|64}}),
:{{big|&lt;math&gt;\begin{align}
    \mathbf{q}\;\!{\cdot}\nabla\,(\psi~\!\! * \!\chi)
    &amp;= q_i \part_i (\psi~\!\! * \!\chi) \\
    &amp;= q_i (\psi * \part_i \chi + \part_i \psi * \chi) \\
    &amp;= \psi * q_i \part_i \chi + q_i \part_i \psi * \chi \,;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,(\psi~\!\! * \!\chi)
    \equiv \psi * (\mathbf{q}\;\!{\cdot}\nabla\chi) +
           (\mathbf{q}\;\!{\cdot}\nabla\psi) * \chi \,.
  &lt;/math&gt;}}|{{EquationRef|70}}}}
The {{math|'''q&amp;sdot;'''&amp;nabla;}} operator is a ''scalar'' operator in the sense that it maps the operand field to a field of the same order&amp;mdash;a scalar field to a scalar field, a vector field to a vector field, a matrix field to a matrix field, etc.&amp;mdash;&amp;#8202;''as if''&amp;#8202; it were multiplication by a scalar or differentiation w.r.t. a scalar; and indeed a differentiation w.r.t. path length appears in the coordinate-free definition ({{EquationNote|11}}) of the operator. Moreover, we did not need coordinates to obtain rule ({{EquationNote|70}}); as the reader may verify, the same rule can be obtained directly from the definition ({{EquationNote|11}}) in a similar manner. From these points of view, the simplicity of the rule is unsurprising.

The '''curl of the product of a scalar and a vector''' is
:{{big|&lt;math&gt;\begin{align}\operatorname{curl}p\mathbf{b}
    &amp;= \mathbf{e}_i ~\!\!\times \part_i(p\mathbf{b}) \\
    &amp;= \mathbf{e}_i ~\!\!\times(p~\!\part_i\mathbf{b}+\part_i p\;\mathbf{b}) \\
    &amp;= p~\!\mathbf{e}_i{\times}~\!\part_i\mathbf{b} +
       \mathbf{e}_i\part_i p \times\mathbf{b} \\
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}p\mathbf{b}
    \,\equiv\, p\operatorname{curl}\mathbf{b} ~\!+ \nabla p \times\mathbf{b} \,.
  &lt;/math&gt;}}|{{EquationRef|71c}}}}
For uniform {{math|'''b'''&amp;#8202;,}} this reduces to ({{EquationNote|8p}}), which was used to derive the Cartesian form of the curl ({{EquationNote|59c}}).

For the '''divergence of the product of a scalar and a vector''', we proceed likewise except that we use a dot instead of a cross. The result is
{{NumBlk|:|{{big|&lt;math&gt;
     \operatorname{div}p\mathbf{b}
     \,\equiv\, p\operatorname{div}\mathbf{b} ~\!+ \nabla p \cdot \mathbf{b} \,,
  &lt;/math&gt;}}|{{EquationRef|71d}}}}
which has the same form as ({{EquationNote|17}}), delivering the promised confirmation that ({{EquationNote|17}}) is an identity. For uniform {{math|'''b'''&amp;#8202;,}}&amp;#8201; ({{EquationNote|71d}}) reduces to ({{EquationNote|8g}}), which was used to derive the Cartesian form of the divergence ({{EquationNote|60d}}).

That exhausts the first-order product rules. For curiosity's sake, we shall also derive one second-order rule.

The '''Laplacian of the product of a scalar field and a generic field''', by ({{EquationNote|63L}}), is
:{{big|&lt;math&gt;\begin{align}\triangle(p\psi)
    &amp;= \part_i \part_i (p\psi) \\
    &amp;= \part_i (p~\!\part_i \psi + \psi~\!\part_i p) \\
    &amp;= p\,\part_i \part_i \psi + \part_i \psi\,\part_i p +
       \psi~\!\part_i \part_i p + \part_i p\,\part_i \psi \\
    &amp;= p\,\part_i \part_i \psi + 2\part_i p\,\part_i \psi +
       \psi~\!\part_i \part_i p \\
    &amp;= p~\!\triangle\psi + 2\part_i p\,\part_i \psi +
       \psi~\!\triangle p \,.
  \end{align}&lt;/math&gt;}}
In the middle term, by ({{EquationNote|58g}}), {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p}}&amp;#8202; is the {{mvar|i&amp;#8202;}}th component of{{math|&amp;#8202; &amp;nabla;''p''}}&amp;#8239; so that, by ({{EquationNote|64o}}),&amp;#8201; {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p&amp;#8239;&amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8239; is the {{math|'''q&amp;sdot;'''&amp;nabla;}} operator for&amp;#8239; {{math|'''q'''&amp;#8239;{{=}}&amp;#8239;&amp;nabla;''p''}}.&amp;#8201; So we have
{{NumBlk|:|{{big|&lt;math&gt;
     \triangle(p\psi)
     \equiv p~\!\triangle\psi + 2(\nabla p \cdot~\!\! \nabla)\psi +
              \psi~\!\triangle p \,.
  &lt;/math&gt;}}|{{EquationRef|72}}}}
The argument assumes a scalar{{mvar| p}} but is indifferent to whether {{mvar|&amp;psi;}} is a scalar or a vector or a higher-order tensor.

&lt;br /&gt;

Finally we turn to ''chain rules''&amp;#8202;&amp;mdash;&amp;#8202;especially the simple cases of the gradient, curl, divergence, advection, and Laplacian of a function of a scalar field{{mvar| u}}. As usual, let {{mvar|p}} denote a scalar field, {{math|'''q'''}} a vector field, and {{mvar|&amp;psi;}} a generic field.

'''Gradient&amp;#8202;&amp;#10744;&amp;#8202;curl&amp;#8202;&amp;#10744;&amp;#8202;divergence of a function of a scalar''':  By the general Cartesian formula ({{EquationNote|60s}}) and the chain rule for{{math| ''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;,}}
:{{big|&lt;math&gt;\begin{align}\nabla~\!\! * \big(\psi(u)\big)
    &amp;= \mathbf{e}_i ~\!\! * \part_i \big(\psi(u)\big) \\
    &amp;= \mathbf{e}_i ~\!\! * \psi'~\!\!(u) ~\!\part_i u \\
    &amp;= \mathbf{e}_i \part_i u * \psi'~\!\!(u) \,;
  \end{align}&lt;/math&gt;}}
i.e., by ({{EquationNote|58g}}),
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla~\!\! * \big(\psi(u)\big)
    \equiv \nabla u * \psi'~\!\!(u) \,.
  &lt;/math&gt;}}|{{EquationRef|73}}}}
In particular, if&amp;#8202; {{math|&amp;lowast;}} is a null,
{{NumBlk|:|&lt;math&gt;
    \nabla\big(p(u)\big) \equiv \nabla u \;p'~\!\!(u) \,;
  &lt;/math&gt;|{{EquationRef|73g}}}}
and if&amp;#8202; {{math|&amp;lowast;}} is a cross,
{{NumBlk|:|&lt;math&gt;
    \mathrm{curl}\big(\mathbf{q}(u)\big)
    \equiv \nabla u \times \mathbf{q}'~\!\!(u) \,;
  &lt;/math&gt;|{{EquationRef|73c}}}}
and if&amp;#8202; {{math|&amp;lowast;}} is a dot,
{{NumBlk|:|&lt;math&gt;
    \mathrm{div}\big(\mathbf{q}(u)\big)
    \equiv \nabla u \cdot \mathbf{q}'~\!\!(u) \,.
  &lt;/math&gt;|{{EquationRef|73d}}}}

'''Advection of a function of a scalar''':
:&lt;!-- SUBSCRIPTS ENLARGED FOR LEGIBILITY: --&gt;&lt;math&gt;\begin{align}
    \mathbf{q}\;\!{\cdot}\nabla\,\big(\psi(u)\big)
    &amp;= q_{\textstyle i} \part_{\textstyle i} \big(\psi(u)\big) \\
    &amp;= q_{\textstyle i} ~\!\psi'~\!\!(u) ~\!\part_{\textstyle i} u \\
    &amp;= q_{\textstyle i} \part_{\textstyle i} u \;\psi'~\!\!(u) \,;
  \end{align}&lt;/math&gt;
i.e.,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\big(\psi(u)\big)
    \equiv~\! \mathbf{q}\;\!{\cdot}\nabla u ~\psi'~\!\!(u) \,.
  &lt;/math&gt;}}|{{EquationRef|73q}}}}
This fits into the pattern set by ({{EquationNote|73}}) in that the gradient operator in ({{EquationNote|73g}}) is replaced by an advection operator.

Of the last four results, only ({{EquationNote|73c}}) is dependent on the order of the {{math|&amp;lowast;}} product; the others could equally well be written
{{NumBlk|:|&lt;math&gt;\begin{align}
    \nabla\big(p(u)\big) &amp;\equiv~\! p'~\!\!(u) ~\!\nabla u \\
    \mathrm{div}\big(\mathbf{q}(u)\big)
    &amp;\equiv~\! \mathbf{q}'~\!\!(u) \cdot~\!\! \nabla u \\
    \mathbf{q}\;\!{\cdot}\nabla\,\big(\psi(u)\big)
    &amp;\equiv~\! \psi'~\!\!(u)\;\mathbf{q}\;\!{\cdot}\nabla u ~.
  \end{align}&lt;/math&gt;|{{EquationRef|73z}}}}

The '''Laplacian of a function of a scalar''' departs from the above pattern.
:{{big|&lt;math&gt;\begin{align}\triangle\big(\psi(u)\big)
    &amp;= \part_i \part_i \big(\psi(u)\big) \\
    &amp;= \part_i\big(\psi'~\!\!(u) ~\!\part_i u\big) \\
    &amp;= \psi'~\!\!(u) ~\!\part_i\part_i u
       + \psi''~\!\!(u) ~\!\part_i u \,\part_i u \,,
  \end{align}&lt;/math&gt;}}
where the last line follows from the product rule for {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; and, in the second term, the chain rule for{{mvar| &amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; In that second term, the implicit sum&amp;#8202; {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;u&amp;#8239;&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;u}}&amp;#8239; can be recognized as&amp;#8202; {{math|{{abs|&amp;nabla;''u''}}&lt;sup&gt;2&lt;/sup&gt;}}&amp;#8202; by ({{EquationNote|58s}}). So we have
{{NumBlk|:|{{big|&lt;math&gt;
    \triangle\big(\psi(u)\big)
    \equiv \psi'~\!\!(u)~\!\triangle u + \psi''~\!\!(u)~\!\big|\nabla u\big|^2.
  &lt;/math&gt;}}|{{EquationRef|74}}}}

'''Multivariate chain rule''':  The foregoing chain rules involve ''one'' intermediate function of ''one'' scalar variable. It will be useful to have an elementary chain rule that can handle more than one of each. Let {{math|''p''('''r''')}} be a smooth scalar field, and let {{math|'''r'''}} in turn be a smooth function of several variables, one of which, say{{mvar| t&amp;#8202;}}, is allowed to vary while the others are held constant, so that {{math|'''r'''}} changes by {{math|''d'''''r'''}} when {{mvar|t}} changes by {{mvar|dt}}. Then dividing ({{EquationNote|26g}}) by {{mvar|dt}}&amp;#8202; gives
:&lt;math&gt;\part_t p = \nabla p \cdot \part_t \mathbf{r}&lt;/math&gt;
or, in indicial Cartesian coordinates with implicit summation,
:{{big|&lt;math&gt;\part_t p = \part_i p \,\part_t x_i&lt;/math&gt;}}
or, in traditional longhand,
:{{big|&lt;math&gt;\tfrac{\part}{\part t}~\!p(x,y,z)
    = \tfrac{\part p}{\part x}~\!\tfrac{\part x}{\part t}
    + \tfrac{\part p}{\part y}~\!\tfrac{\part y}{\part t}
    + \tfrac{\part p}{\part z}~\!\tfrac{\part z}{\part t} \,.
  &lt;/math&gt;}}
This is the desired multivariate chain rule for a scalar function of three intermediate real variables. The assumption that these variables are Cartesian coordinates is not a loss of generality, because any three real quantities can be suitably scaled and represented by perpendicular axes, so that any scalar function of them becomes a function of position, to which ({{EquationNote|26g}}) applies; and then the scaling can be reversed without changing the products in the last equation. Moreover, by the linearity of{{mvar| &amp;part;&lt;sub&gt;t&lt;/sub&gt;&amp;#8239;}}, the scalar field {{mvar|p}} may be replaced by any field expressible in terms of a uniform basis. For example, for a vector field{{math| '''q'''&amp;#8202;}},
:{{big|&lt;math&gt;\begin{align}\part_t \mathbf{q}
    &amp;= \part_t (q_j \mathbf{e}_j) \\
    &amp;= \mathbf{e}_j \part_t q_j \\
    &amp;= \mathbf{e}_j \part_i q_j \,\part_t x_i \\
    &amp;= \part_i (q_j \mathbf{e}_j) ~\!\part_t x_i
     = \part_i \mathbf{q} \,\part_t x_i \,,
  \end{align}&lt;/math&gt;}}
where the third line is obtained by applying the multivariate chain rule for a scalar field. Thus, for a generic field {{mvar|&amp;psi;&amp;#8202;}},
{{NumBlk|:|{{big|&lt;math&gt;
    \part_t \psi = \part_i \psi \,\part_t x_i
  \qquad&lt;/math&gt;}}[&amp;#8202;for generic {{mvar|&amp;psi;}} and {{mvar|x&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;}}].|{{EquationRef|75}}}}

'''Gradient&amp;#8202;&amp;#10744;&amp;#8202;curl&amp;#8202;&amp;#10744;&amp;#8202;divergence of a function of a scaled position vector''':  We end this subsection by deriving a lemma for use in the next subsection. If{{mvar| k}} is a uniform scalar multiplier and {{math|'''r'''}} is the position vector,
:{{big|&lt;math&gt;
    \nabla * \psi(k\mathbf{r})
    = \mathbf{e}_i * \part_i \psi(k\mathbf{r})
    = k\mathbf{e}_i * \part_{(kx_{\scriptstyle i})} \psi(k\mathbf{r}) \,,
  &lt;/math&gt;}}
where the third expression is obtained by from the second by multiplying each denominator (change in{{mvar| x&lt;sub&gt;i&lt;/sub&gt;}}) by{{mvar| k}}&amp;#8239; and compensating. But now we have
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla * \psi(k\mathbf{r}) = k\,(\nabla {*}~\! \psi)\Big|_{k\mathbf{r}} \,,
  &lt;/math&gt;}}|{{EquationRef|76}}}}
where the vertical bar and subscript indicate that the gradient, curl, or divergence is evaluated at{{math| ''k''&amp;#8202;'''r'''}}. We shall be interested in the curl (for which {{math|&amp;lowast;}} is a cross).

{{cob}}

=== Field with given curl ===

{{cot}}

Consider the vector field
{{NumBlk|:|&lt;math&gt;
    \mathbf{v}(\mathbf{r}) = \mathbf{q}(\mathbf{r})\times\mathbf{r} \,,
  &lt;/math&gt;|{{EquationRef|77}}}}
where {{math|'''q'''}} is a ''solenoidal''&amp;#8202; vector field and {{math|'''r''' }}is the position vector. By identity ({{EquationNote|67c}}),
:&lt;math&gt;\operatorname{curl}\mathbf{v}
  =   \mathbf{q}\operatorname{div}\mathbf{r}
    - \mathbf{r}\operatorname{div}\mathbf{q}
    + \mathbf{r}~\!{\cdot}\nabla\,\mathbf{q}
    - \mathbf{q}~\!{\cdot}\nabla\,\mathbf{r}
  &lt;/math&gt;
where, by hypothesis, {{math|div&amp;#8239;'''q'''}}&amp;#8202; is zero. Applying identities ({{EquationNote|62r}}) and ({{EquationNote|64r}}) then yields
:&lt;math&gt;\begin{align}
  \operatorname{curl}\mathbf{v}
  &amp;= 3\mathbf{q} + \mathbf{r}~\!{\cdot}\nabla\,\mathbf{q} - \mathbf{q} \\
  &amp;= 2\mathbf{q} + \mathbf{r}~\!{\cdot}\nabla\,\mathbf{q} \,.
  \end{align}&lt;/math&gt;

[[File:Vorticity_Figure_01_a-m.gif|thumb|Animation of a rigid-body-like velocity field, whose curl is twice the angular velocity.]]

In the special case in which {{math|'''q'''}} is the ''angular velocity'' {{math|'''&amp;omega;'''}} of a '''rigid body''' about an axis through the origin,&amp;#8201; {{math|'''v''' }}is the velocity field ({{math|'''&amp;omega;'''&amp;#8239;&amp;times;&amp;#8239;'''r'''}}) and {{math|'''&amp;omega;'''}} is uniform, so that the last result reduces to&amp;#8201; {{math|curl&amp;#8239;'''v'''&amp;#8201;{{=}}&amp;#8201;2'''&amp;omega;'''&amp;#8202;}}; that is, ''the vorticity is twice the angular velocity''. As the vorticity in this case is uniform and therefore independent of position relative to the axis, it does not change if the axis is shifted, provided that the angular velocity has the same magnitude and direction. And because a uniform velocity field has zero curl, the vorticity is also unchanged if a translational motion is superposed on the rotation. This is the most direct connection that we have seen between curl and rotation. But again I digress.

Returning to the more general case in which {{math|'''q''' }}is not necessarily uniform, but merely solenoidal,&lt;ref&gt;The following explanation takes some hints from Christopher Ford's note on &quot;Vector Potentials&quot; at [https://www.maths.tcd.ie/~houghton/231/Notes/ChrisFord/vp.pdf maths.tcd.ie/~houghton/231/Notes/ChrisFord/vp.pdf] (2006).&lt;/ref&gt; we have
:&lt;math&gt;\operatorname{curl}\mathbf{v}(\mathbf{r})
    = 2\mathbf{q}(\mathbf{r})
      + \mathbf{r}~\!{\cdot}\nabla\,\mathbf{q}(\mathbf{r}) \,,
  &lt;/math&gt;
to which we can apply our lemma ({{EquationNote|76}}) with a uniform real factor {{mvar|t&amp;#8202;}}, obtaining
:&lt;math&gt;
    \operatorname{curl}\mathbf{v}(t\mathbf{r})
    = 2t\mathbf{q}(t\mathbf{r})
       + t\mathbf{r}~\!{\cdot}\nabla\,\mathbf{q}(t\mathbf{r}) \,.
  &lt;/math&gt;
On the left we can recall ({{EquationNote|77}}); and on the right we can apply ({{EquationNote|11}}), noting that the magnitude of{{math| {{abs|'''r'''}}}} is{{mvar| r&amp;#8202;}}, which measures distance in the direction of{{math| '''r'''}}. Thus we obtain
:&lt;math&gt;
  \mathrm{curl}\big(\mathbf{q}(t\mathbf{r})\times t\mathbf{r}\big)
    = 2t\mathbf{q}(t\mathbf{r})
      + tr~\! \part_{\textstyle r} \mathbf{q}(t\mathbf{r}) \,.
  &lt;/math&gt;
Now if the direction of{{math| '''r'''}} is held constant,&amp;#8201; {{math|'''q'''(''t''&amp;#8202;'''r''')}} is a function of{{mvar| tr&amp;#8239;}}; and in general&amp;#8201; {{math|''r&amp;#8202;&amp;part;&lt;sub&gt;r&lt;/sub&gt;&amp;#8201;f''&amp;#8202;(''tr'')&amp;#8201;{{=}}&amp;#8201;''t&amp;#8202;&amp;part;&lt;sub&gt;t&lt;/sub&gt;&amp;#8201;f''&amp;#8202;(''tr'')}}.&amp;#8201; So we have
:&lt;math&gt;\begin{align}
  \mathrm{curl}\big(\mathbf{q}(t\mathbf{r})\times t\mathbf{r}\big)
    &amp;= 2t\mathbf{q}(t\mathbf{r}) + t^2 \part_t \mathbf{q}(t\mathbf{r}) \\
    &amp;= \part_t \big(t^2 \mathbf{q}(t\mathbf{r})\big) \,.
  \end{align}&lt;/math&gt;
Integrating w.r.t. {{mvar|t}}&amp;#8239; from 0 to 1 gives
:&lt;math&gt;
  \int_0^1\!\mathrm{curl}\big(\mathbf{q}(t\mathbf{r})\times t\mathbf{r}\big)\,dt
    = \big(t^2 \mathbf{q}(t\mathbf{r})\big)\Big|_0^1
    =~\! \mathbf{q}(\mathbf{r}) \,;
  &lt;/math&gt;
that is,
{{NumBlk|:|&lt;math&gt;\mathbf{q}(\mathbf{r}) \equiv
  \mathrm{curl}\int_0^1\!\mathbf{q}(t\mathbf{r})\!\times\!t\mathbf{r}\;dt\qquad
  &lt;/math&gt;[&amp;#8202;for solenoidal {{math|'''q'''&amp;#8202;}}].|{{EquationRef|78}}}}
Thus for any solenoidal vector field{{math| '''q'''}}&amp;#8239; we can construct a '''vector potential'''&amp;mdash;that is, a field whose curl is{{math| '''q'''&amp;#8202;}}; such a field is given by the integral on the right. This is the long-promised proof of the &quot;converse&quot; of identity ({{EquationNote|24d}}). Of course the vector potential is not unique, because any conservative field&amp;mdash;but ''only'' a conservative field&amp;mdash;can be added to it without changing its curl. Hence the existence of ''one'' vector potential implies the existence of infinitely many. The above integral gives us ''one''.

The proof of ({{EquationNote|78}}) assumes that {{math|'''q''' }}is solenoidal not only at position{{math| '''r'''&amp;#8202;,}} but also at{{math| ''t''&amp;#8202;'''r'''}}&amp;#8202; where&amp;#8239; {{math|0&amp;#8239;&amp;leq;&amp;#8239;''t''&amp;#8239;&amp;leq;&amp;#8202;1}}, i.e. at every point on the line-segment from the origin to{{math| '''r'''}}.&amp;#8201; A '''star-shaped''' region is one that contains a point{{mvar| O}}&amp;#8202; such that for every point{{mvar| P}} in the region, the line-segment {{mvar|OP}} is entirely contained in the region. We may choose any such {{mvar|O}}&amp;#8202; as the origin in the proof of ({{EquationNote|78}}). So the proof tells us that if a vector field is solenoidal within a star-shaped region, it has a vector potential in that region. As a special case, a vector field that is solenoidal everywhere has a vector potential everywhere.

{{cob}}

=== Notes on the curl of the curl ===

{{cot}}

Identity ({{EquationNote|65}}), namely
:&lt;math&gt;
    \operatorname{curl}\operatorname{curl}\mathbf{q}
    ~\!\equiv \nabla\operatorname{div}\mathbf{q} - \triangle\mathbf{q}
  &lt;/math&gt;
(&quot;curl curl is grad div minus del squared&quot;), has at least three implications worth noting here.

First, it can be rearranged as
{{NumBlk|:|&lt;math&gt;\triangle\mathbf{q}
    ~\!\equiv \nabla\operatorname{div}\mathbf{q}
              - \operatorname{curl}\operatorname{curl}\mathbf{q}
  &lt;/math&gt;|{{EquationRef|79}}}}
(&quot;del squared is grad div minus curl curl&quot;). This would serve as a coordinate-free definition of the Laplacian of a vector, if we did not already have one.&lt;ref&gt;''Cf''. [[#gibbs-1881-4|Gibbs, 1881]], &amp;sect;&amp;#8202;71, and [[#moon-spencer-65|Moon &amp;amp; Spencer, 1965]], p.&amp;#8239;235; quoted in [[#tai-95|Tai, 1995]], pp.&amp;#8239;18,&amp;#8239;43.&lt;/ref&gt; But we do: we started with a coordinate-free definition ({{EquationNote|4L}}) for a generic field, established its unambiguity via ({{EquationNote|9L}}), and found its Cartesian form ({{EquationNote|63L}}), which we used in the derivation of ({{EquationNote|79}}). Wherever we start, we may properly assert by way of contrast that the Laplacian of a ''vector''&amp;#8202; is given by ({{EquationNote|79}}), whereas the Laplacian of a ''scalar''&amp;#8202; is given by the divergence of the gradient. But we should ''not'' conclude, as Moon &amp;amp; Spencer do, that representing the scalar and vector Laplacians by the same symbol is &quot;poor practice&amp;hellip; since the two are basically quite different&quot;,&lt;ref&gt;[[#moon-spencer-65|Moon &amp;amp; Spencer, 1965]], p.&amp;#8239;236.&lt;/ref&gt; because in fact the two have a common definition which is succinct, unambiguous, and coordinate-free: the Laplacian (of anything) is the closed-surface integral of the outward normal derivative, per unit volume.{{efn|Tai ([[#tai-95|1995]], pp.&amp;#8239;43–4) also disagrees with Moon &amp;amp; Spencer, but for a different reason: he regards the Laplacian as the divergence of the gradient even if the operand is a ''vector'' field. For better or worse, we do not consider the gradient of a vector in the present paper&amp;mdash;although the reader can probably work out how to modify ({{EquationNote|26g}}) if&amp;#8202; {{math|''d'''''r'''}} is written as a column vector and&amp;#8202; {{mvar|dp}}&amp;#8202; is ''replaced''&amp;#8202; by a column vector (compare the later footnote on ''dyadics'').}}

Second, by reason of identity ({{EquationNote|38}}) and the remarks thereunder, a given vector field{{math| '''v'''}} can be written
:&lt;math&gt;\mathbf{v}(\mathbf{r}) \,\equiv\, \triangle\bigg(\!{-}\!\iiint
    \frac{\,\mathbf{v}(\mathbf{r}')}{4\pi}\,
    \frac{1}{|\mathbf{r}\!-\!\mathbf{r}'|}
  \,dV' \!\bigg) \,,&lt;/math&gt;
where the integral is over all space, or at least all of the space in which {{math|'''v'''}} may be non-zero. So, subject to the convergence of the integral, there exists a vector field{{math| '''q'''}} such that
:&lt;math&gt;\mathbf{v} = \triangle\mathbf{q} \,;&lt;/math&gt;
that is, by ({{EquationNote|79}}), there exists{{math| '''q'''}} such that
:&lt;math&gt;\mathbf{v}
    = \nabla\operatorname{div}\mathbf{q}
      - \operatorname{curl}\operatorname{curl}\mathbf{q} \,,
  &lt;/math&gt;
which implies the existence of a scalar field, say&lt;math&gt;\,\varphi~\!,\,&lt;/math&gt; and a vector field, say{{math| '''&amp;Psi;'''}}, such that
:&lt;math&gt;\mathbf{v}
    = -\nabla\varphi+\operatorname{curl}\boldsymbol{\Psi}
  &lt;/math&gt;
(namely&amp;#8201; &lt;math&gt;\varphi\!=\!-\!\operatorname{div}\mathbf{q}\,&lt;/math&gt; and&amp;#8201; {{math|'''&amp;Psi;'''&amp;#8201;{{=}}&amp;#8201;&amp;minus;&amp;#8202;curl&amp;#8239;'''q'''}}). In short, subject to the convergence of the said integral,
* ''a given vector field can be resolved into [minus] a gradient plus a curl''.
Such a resolution is called a '''Helmholtz decomposition''', and the proposition that it exists is the ''Helmholtz decomposition theorem''. Of course the gradient is irrotational and the curl is solenoidal so that, subject to the same convergence,
* ''a given vector field can be resolved into an irrotational field plus a solenoidal field''.
This is a second statement of the theorem, and follows from the first. And the first follows from the second because an irrotational field has a scalar potential by ({{EquationNote|29}}) and a solenoidal field has a vector potential by ({{EquationNote|78}}).

Third, if {{math|'''q'''}} is ''solenoidal'', the term&amp;#8201; {{math|&amp;nabla;&amp;#8201;div&amp;#8239;'''q'''}}&amp;#8239; in ({{EquationNote|65}}) or ({{EquationNote|79}}) vanishes. Hence ''for a solenoidal field, the curl of the curl is minus the Laplacian''. For example, in the ''dynamic'' case, in a ''vacuum'', the Maxwell&amp;ndash;Amp&amp;egrave;re law says that&amp;#8201; {{math|curl&amp;#8201;'''H''' {{=}} ''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8202;'''E&amp;#775;'''}}.&amp;#8201; Multiplying this by the physical constant {{math|''&amp;mu;''&lt;sub&gt;0&lt;/sub&gt;}} (called the '''vacuum permeability''' or simply the '''magnetic constant''') gives&amp;#8201; {{math|curl&amp;#8201;'''B''' {{=}} ''&amp;mu;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8202;''&amp;epsiv;''&lt;sub&gt;0&lt;/sub&gt;&amp;#8239;'''E&amp;#775;'''&amp;#8202;,}}&amp;#8239; whence
:&lt;math&gt;\operatorname{curl}\operatorname{curl}\mathbf{B}
    = \mu_0\epsilon_0\operatorname{curl}\mathbf{\dot{E}} \,.
  &lt;/math&gt;
But, by Gauss's law for magnetism, {{math|'''B'''}} is solenoidal so that, by ({{EquationNote|65}}), the left-hand side of the above is&amp;#8201; {{math|&amp;minus;&amp;#9651;'''B'''}}.&amp;#8201; And by '''Faraday's law''',&amp;#8201; &lt;math&gt;\operatorname{curl}\mathbf{E}=-\mathbf{\dot{B}}&lt;/math&gt;,&amp;#8201; so that&amp;#8201; &lt;math&gt;\operatorname{curl}\mathbf{\dot{E}}=-\mathbf{\ddot{B}}&lt;/math&gt;.&amp;#8201; Making these substitutions, we get&amp;#8201; &lt;math&gt;-\triangle\mathbf{B}=-\mu_0\epsilon_0\mathbf{\ddot{B}}~\!,\,&lt;/math&gt; i.e.
:&lt;math&gt;\mathbf{\ddot{B}}=\frac{1}{\mu_0\epsilon_0}~\!\triangle\mathbf{B} \,.&lt;/math&gt;
By comparison with ({{EquationNote|45}}), this is the wave equation with
:&lt;math&gt;c=\frac{1}{\sqrt{\mu_0\epsilon_0}} \,.&lt;/math&gt;
Thus the Maxwell&amp;ndash;Amp&amp;egrave;re law, Gauss's law for magnetism, and Faraday's law, with the aid of ({{EquationNote|65}}), predict the existence of '''electromagnetic waves''' together with their speed.

For these reasons, especially the last, one could hardly overstate the importance of identity ({{EquationNote|65}}).

{{cob}}

=== Digression: Proofs from formal products ===

{{cot}}

We have seen that Wilson ([[#wilson-1901|1901]], pp.&amp;#8239;150,&amp;#8239;152) interprets the divergence and curl as &quot;formal&quot; or &quot;symbolic&quot; scalar and vector products with the {{math|&amp;nabla; }}operator.&amp;#8201; {{nowrap|C.-T. Tai}}, in his [[#tai-95|1995 report]] (pp.&amp;#8239;26–9), alleges that this interpretation began with Wilson and not with Gibbs. Here I shall submit, on the contrary, that while the terminology may not be attributable to Gibbs, the concept certainly is.

Later in the same report, Tai confuses the picture by citing the first volume of [[w:Oliver Heaviside|Heaviside]]'s ''Electromagnetic Theory'' (1893), where Heaviside, although his notations for the scalar and vector products differ from those of Gibbs, nevertheless considers the {{math|&amp;nabla;}} operator as a factor in such products. Tai continues:
&lt;blockquote&gt;At the time of his writing he [Heaviside] was already aware of Gibbs' pamphlets on vector analysis but Wilson's book was not yet published. It seems, therefore, that Heaviside and Wilson independently introduced the misleading concept for the scalar and vector products between {{math|&amp;nabla;}} and a vector function. Both were, perhaps, induced by Gibbs' notations for the divergence and the curl. Heaviside did not even include the word 'formal' in his description of the products.&lt;ref&gt;[[#tai-95|Tai, 1995]], p.&amp;#8239;35.&lt;/ref&gt;
&lt;/blockquote&gt;
Whereas it was quite in character for Heaviside to treat an operator that way, the word &quot;independently&quot; would have surprised Wilson and is contradicted by Tai himself, who observes that Wilson's preface acknowledges Heaviside.&lt;ref&gt;[[#tai-95|Tai, 1995]], pp.&amp;#8239;25,&amp;#8239;29.&lt;/ref&gt; In Wilson's own words:
&lt;blockquote&gt;By far the greater part of the material used in the following pages has been taken from the course of lectures on Vector Analysis delivered annually at the University [Yale] by Professor Gibbs. Some use, however, has been made of the chapters on Vector Analysis in Mr. Oliver Heaviside's ''Electromagnetic Theory'' (Electrician Series, 1893) and in Professor F&amp;ouml;ppl's lectures on ''Die Maxwell'sche Theorie der Electricit&amp;auml;t'' (Teubner, 1894).&amp;#8239;....

Notwithstanding the efforts which have been made during more than half a century to introduce Quaternions into physics the fact remains that they have not found wide favor.{{efn|A ''quaternion''&amp;#8202; is a mathematical object invented by [[w:William Rowan Hamilton|William Rowan Hamilton]] in 1843, consisting of two parts which Hamilton later called the scalar part and the vector part. For most purposes the two parts were found to be more useful separately than together. By putting them together, however, Hamilton constructed a set which satisfied all the algebraic field axioms except commutativity of multiplication. This was, and is, considered a triumph.}} On the other hand there has been a growing tendency especially in the last decade toward the adoption of some form of Vector Analysis. The works of Heaviside and F&amp;ouml;ppl referred to before may be cited in evidence. As yet however no system of Vector Analysis which makes any claim to completeness has been published. In fact Heaviside says: &quot;I am in hopes that the chapter which I now finish may serve as a stopgap till regular vectorial treatises come to be written suitable for physicists, based upon the vectorial treatment of vectors&quot; (''Electromagnetic Theory'', Vol.&amp;#8239;{{serif|I}}., p.&amp;#8239;305). Elsewhere in the same chapter Heaviside has set forth the claims of vector analysis as against Quaternions, and others have expressed similar views.&lt;ref&gt;[[#wilson-1901|Wilson, 1901]], pp.&amp;#8239;ix,&amp;#8239;xi–xii.&lt;/ref&gt;
&lt;/blockquote&gt;

Most damaging to Tai's thesis, however, is Gibbs's original pamphlet, a copy of which Heaviside received from Gibbs himself in June 1888.&lt;ref&gt;[[#gibbs-1881-4|Gibbs, 1881–84]], privately printed version&amp;mdash;of which the scan in our bibliography is of the very copy that Gibbs sent to Heaviside, with annotations in Heaviside's hand. On the annotations see [[#rocci-20|Rocci, 2020]].&lt;/ref&gt; Sections 62 to 65 of the pamphlet appear under the heading
&lt;blockquote style=&quot;text-align: center&quot;&gt;{{math|&amp;nabla;,}} {{math|&amp;nabla;'''&amp;sdot;'''&amp;#8202;,}} ''and''&amp;#8202; {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8239; ''applied to Functions of Functions of Position''.
&lt;/blockquote&gt;In &amp;sect;&amp;#8239;62, Gibbs says that a constant scalar factor after such an operator may be placed before it (that is, taken outside the operator). {{nowrap|In &amp;sect;&amp;#8239;63}} he states our rule ({{EquationNote|73g}}) for the gradient of a function of a scalar field. His next section (in which I have bolded the vector field{{math| '''&amp;omega;'''}}) is worth quoting in full:
&lt;blockquote&gt;64.  If {{mvar|u}} or {{math|'''&amp;omega;'''}} is a function of several scalar or vector variables, which are themselves functions of the position of a single point, the value of&amp;#8202; {{math|&amp;nabla;''u''}} or {{math|&amp;nabla;'''&amp;sdot;&amp;#8239;&amp;omega;'''}}&amp;#8202; or {{math|&amp;nabla;&amp;#8202;&amp;times;&amp;#8202;'''&amp;omega;'''}}&amp;#8202; will be equal to the sum of the values obtained by making successively all but each one of these variables constant.
&lt;/blockquote&gt;
This proposition is a ''generalized product rule'' in the sense that the &quot;function of several scalar or vector variables&quot; may be, but is not restricted to, any sort of product of those variables. Gibbs continues:
&lt;blockquote&gt;65.  By the use of this principle, we easily derive the following identical equations:
&lt;/blockquote&gt;
Six &quot;equations&quot; follow. The first says that the gradient operation is distributive over addition, and the second says the same of the divergence and curl (on one line). The last four are our identities ({{EquationNote|69}}), ({{EquationNote|71d}}), ({{EquationNote|71c}}), and ({{EquationNote|67d}}), in that order (albeit with different symbols). Gibbs then remarks (with my italics):
&lt;blockquote&gt;The student will observe an analogy between these equations and the formul&amp;aelig; of ''multiplication''. (In the last four equations the analogy appears most distinctly when we regard all the factors but one as constant.) Some of the more curious features of this analogy are due to the fact that the {{math|&amp;nabla;}} contains implicitly the vectors {{math|'''i'''&amp;#8202;,}} {{math|'''j'''&amp;#8202;,}} and {{math|'''k'''&amp;#8202;,}} which are to be ''multiplied''&amp;#8202; into the following quantities.
&lt;/blockquote&gt;
Indeed, if the ''first''&amp;#8202; factor is constant, identities ({{EquationNote|69}}), ({{EquationNote|71d}}), ({{EquationNote|71c}}), and ({{EquationNote|67d}}) become
:&lt;math&gt;\begin{align}
    \nabla(p\varphi) &amp;= p\;\!\nabla\varphi \\
    \nabla\cdot p\mathbf{b} &amp;= p\,\nabla{\cdot}~\!\mathbf{b} \\
    \nabla\times p\mathbf{b} &amp;= p\,\nabla{\times}~\!\mathbf{b} \\
    \nabla\cdot(\mathbf{a}\!\times\!\mathbf{b})
    &amp;= -\mathbf{a}\cdot\nabla{\times}~\!\mathbf{b} \,,
  \end{align}&lt;/math&gt;
whereas if the ''second''&amp;#8202; factor is constant, they become respectively
:&lt;math&gt;\begin{align}
    \nabla(p\varphi) &amp;= \varphi\;\!\nabla p \\
    \nabla\cdot p\mathbf{b} &amp;= \nabla p \cdot \mathbf{b} \\
    \nabla\times p\mathbf{b} &amp;= \nabla p \times \mathbf{b} \\
    \nabla\cdot(\mathbf{a}\!\times\!\mathbf{b})
    &amp;= \nabla{\times}~\!\mathbf{a}\cdot\mathbf{b} \,.
  \end{align}&lt;/math&gt;
All eight equations look like rearrangements of ''products'' involving a vector{{math| &amp;nabla;}}.&amp;#8201; [Concerning the last ''three'' equations, we have made that observation before; see ({{EquationNote|15}}) above.]&amp;#8201; But only seven of the eight are explained by taking the constant outside the operator ({{nowrap|as in &amp;sect;&amp;#8239;62}}); the exception is the fourth, in which the minus sign is not explained by that step alone, but ''is'' explained by the change in the cyclic order of the formal triple product. And if we add the two right-hand sides corresponding to each of the four left-hand sides, we get the identities in which both factors are variable&amp;mdash;as claimed {{nowrap|in &amp;sect;&amp;#8239;64}}.

If &amp;sect;&amp;#8239;65 leaves any doubt that Gibbs approved of formal products with the symbolic vector{{math| &amp;nabla;}} (albeit without using those terms), this is dispelled {{nowrap|by &amp;sect;&amp;#8202;166}}, where he writes:
&lt;blockquote&gt;166.&amp;#8201; To the equations in No.&amp;#8239;65 may be added many others&amp;hellip;
&lt;/blockquote&gt;
followed by a list of seven identities terminated by &quot;etc.&quot; Six of the seven are beyond the scope of the present paper,{{efn|They involve ''dyadics'', i.e. 2nd-order tensors written in a vector-friendly notation. The fourth of the seven is
:{{math|&amp;nabla;('''&amp;tau;&amp;sdot;&amp;#8202;&amp;omega;''') {{=}} &amp;nabla;'''&amp;tau;&amp;#8239;&amp;sdot;&amp;#8201;&amp;omega;''' + &amp;nabla;'''&amp;omega;&amp;#8201;&amp;sdot;&amp;#8201;&amp;tau;'''&amp;#8239;,}}
which is our ({{EquationNote|68}}) expressed in terms of the dyadics {{math|&amp;nabla;'''&amp;tau;'''}} and{{math| &amp;nabla;'''&amp;omega;'''&amp;#8202;}}; the right-hand side is not to be confused with
:{{math|('''&amp;omega;&amp;#8202;&amp;sdot;'''&amp;nabla;)'''&amp;tau;''' + '''(&amp;tau;&amp;sdot;'''&amp;nabla;)'''&amp;omega;'''&amp;#8239;,}}
which would contradict our ({{EquationNote|68}}).}} while the third of the seven is our ({{EquationNote|67c}}). After that list comes the smoking gun ({{nowrap|&amp;sect;&amp;#8202;166, continued}}):
&lt;blockquote&gt;The principle in all these cases is that if we have one of the operators&amp;#8202; {{math|&amp;nabla;,}} {{math|&amp;nabla;'''&amp;sdot;'''&amp;#8202;,}} {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8239; prefixed to a ''product'' of any kind, and we make any transformation of the expression which would be allowable if the {{math|&amp;nabla;}} were a ''vector'', (viz: by changes in the order of the ''factors'', in the signs of ''multiplication'', in the parentheses written or implied, etc.,) by which changes the {{math|&amp;nabla;}} is brought into connection with one particular factor, the expression thus transformed will represent the part of the value of the original expression which results from the variation of that factor.
&lt;/blockquote&gt;
The italics are mine, but I have refrained from italicizing those instances of the word &quot;factor&quot; which are not applicable to{{math| &amp;nabla;}}. In particular, at the stage when &quot;the {{math|&amp;nabla;}} is brought into connection with one particular factor,&quot; the &quot;part of the value&amp;hellip; which results from the variation of that factor&quot; evidently means the term of the sum {{nowrap|in &amp;sect;&amp;#8239;64}}&amp;#8202;&amp;mdash;which, as we have noted, amounts to a generalized product rule. But, according to the stated &quot;principle', we reach that stage by treating{{math| &amp;nabla;}} as a factor. I rest my case.

&lt;br /&gt;

Wilson ([[#wilson-1901|1901]], p.&amp;#8239;157) gives a comprehensive list of sum and product rules for the gradient, divergence, and curl, and properly states (p.&amp;#8239;158) that the rules may be proven &quot;most naturally&quot; from Gibbs's definitions of the operators&amp;mdash;our equations ({{EquationNote|58g}}), ({{EquationNote|60d}}), and ({{EquationNote|59c}}). Understandably, Wilson uses a {{math|&amp;sum;}} sign rather than implicit summation. Less understandably, and less fortunately, he does not sum over a numerical index; e.g., he defines the curl operator as
:{{big|&lt;math&gt;\nabla\times
    \,=\, \textstyle\sum\,\mathbf{i}~\!\!\times\!\frac{\part}{\part x}
          \qquad\quad
  &lt;/math&gt;[sic]}}
and explains that &quot;The summation extends over {{math|''x'',&amp;#8239;''y'',&amp;#8239;''z''}}.&quot;&amp;#8201; With these definitions he proves our identities ({{EquationNote|71c}}) and ({{EquationNote|68}}) essentially as we have done, but inevitably with greater difficulty, which may explain why he then says &quot;The other formul&amp;aelig; are demonstrated in a similar manner&quot; before reverting to Gibbs's strategy of varying one factor at a time. He announces (p.&amp;#8239;159) that the variable held constant will be written as a subscript after the product, and he combines this notation with his {{math|&amp;sum;}} notation in a rigorous proof that varying one factor at a time is valid for our ({{EquationNote|68}}), i.e. the gradient of a dot-product. Noting that this result is analogous to
:&lt;math&gt;d(\mathbf{u}\cdot\mathbf{v})
    = \mathbf{u}\cdot d\mathbf{v} + d\mathbf{u}\cdot\mathbf{v} \,,
  &lt;/math&gt;
he then jumps to the conclusion that varying one factor at a time is valid for ''all''&amp;#8202; of his product rules&amp;mdash;notwithstanding that a small change in a vector is not related to its divergence or curl as a small change in a scalar is related to its gradient.

That ''per saltum''&amp;#8202; conclusion is his cue to go formal and symbolic. To obtain the curl of a cross-product [as in our ({{EquationNote|67c}})], he &quot;formally&quot; expands a vector triple product to obtain the curl when the first factor is constant, states the curl when the second factor is held constant, and adds the two partial curls ([[#wilson-1901|Wilson, 1901]], p.&amp;#8239;161). Next he gives various arrangements of our ({{EquationNote|8q}}), except that he presents the first vector not as strictly uniform, but as merely ''held'' constant for the gradient operation. He states in passing that a proof may be effected by &quot;expanding in terms of&amp;#8202; {{math|'''i'''&amp;#8202;,&amp;#8239;'''j''',&amp;#8239;'''k'''}}&quot;; but instead of such a proof, he offers a &quot;method of remembering the result&quot; by expanding the &quot;product&quot;&amp;#8239; {{math|'''u'''&amp;#8239;&amp;times;&amp;#8239;(&amp;nabla;&amp;#8239;&amp;times;&amp;#8239;'''v''')}}&amp;#8202; &quot;formally as if&amp;#8202; {{math|&amp;nabla;,&amp;#8239;'''u'''&amp;#8202;,&amp;#8202;'''v'''}}&amp;#8202; were all real vectors&quot; (pp.&amp;#8239;161–2). Concerning the curl of the gradient, and the divergence of the curl (pp.&amp;#8239;167,&amp;#8239;168), he recommends expanding in terms of&amp;#8202; {{math|'''i'''&amp;#8202;,&amp;#8239;'''j''',&amp;#8239;'''k'''&amp;#8202;,}}&amp;#8239; but does not elaborate. Concerning the curl of the curl, however, he shows what would happen if it were &quot;expanded formally according to the law of the triple vector product&quot; (p.&amp;#8239;169).

In defense of the &quot;formal product&quot; method, we should note that the operators {{math|''&amp;part;&lt;sub&gt;x&lt;/sub&gt;''&amp;#8202;,}} {{math|''&amp;part;&lt;sub&gt;y&lt;/sub&gt;''&amp;#8202;,}} and {{math|''&amp;part;&lt;sub&gt;z&lt;/sub&gt;''&amp;#8202;}} are ''linear'', so that they are distributive over addition and may be permuted with multiplication by a constant, as if the operators themselves were multipliers (like components of vectors). They may be similarly permuted with other like operators&amp;mdash;explaining why the formal-product method correctly deals with the curl of the gradient, the divergence of the curl, and the curl of the curl. But such an operator ''cannot'' be permuted with multiplication by a ''variable'', because then the product rule of differentiation applies, yielding an extra term. The formal-product system responds to this difficulty by generalizing the product rule as in &amp;sect;&amp;sect;&amp;#8239;64 &amp;amp;&amp;#8239;166 of Gibbs ([[#gibbs-1881-4|1881–84]]). As Borisenko &amp;amp; Tarapov put it ([[#borisenko-tarapov-68|1968]], p.&amp;#8239;169),
&lt;blockquote&gt;the operator {{math|&amp;nabla;}} acts on each factor separately with the other held fixed. Thus {{math|&amp;nabla;}} should be written after any factor regarded as a constant in a given term and before any factor regarded as variable.
&lt;/blockquote&gt;
In this they differ inconsequentially from Gibbs, who requires that the operator be &quot;brought into connection&quot; with the factor considered variable.

To illustrate, let us find the gradient of a dot-product, essentially in the manner of Borisenko &amp;amp; Tarapov ([[#borisenko-tarapov-68|1968]], p.&amp;#8239;180), quoted by Tai ([[#tai-95|1995]], p.&amp;#8239;46; the next five equation numbers are Tai's). In this case the generalized product rule gives
{{NumBlk|:|&lt;math&gt;\nabla(\mathbf{A ~\!\!\cdot B})
    = \nabla(\mathbf{A}_c {\cdot}~\!\mathbf{B}) +
      \nabla(\mathbf{A} ~\!\!\cdot \mathbf{B}_c) \,,
  &lt;/math&gt;|{{EquationRef|7.26}}}}
where the subscript {{mvar|c}} marks the factor held ''constant'' during the differentiation. In Wilson's notation, this equation would be written
:{{midsize|&lt;math&gt;\nabla(\mathbf{A ~\!\!\cdot B})
    = \nabla(\mathbf{A ~\!\!\cdot B})_{\mathbf{A}} +
      \nabla(\mathbf{A ~\!\!\cdot B})_{\mathbf{B}} \,,
  &lt;/math&gt;}}
where a trailing subscript indicates which factor is held constant.&amp;#8201; In the ''Feynman'' subscript notation, the subscript is attached to the {{math|&amp;nabla;}} operator and indicates which factor is allowed to ''vary'', so that the same equation would be written
:{{midsize|&lt;math&gt;\nabla(\mathbf{A ~\!\!\cdot B})
    = \nabla_{\mathbf{B}}(\mathbf{A ~\!\!\cdot B}) +
      \nabla_{\!\mathbf{A}}(\mathbf{A ~\!\!\cdot B}) \,.
  &lt;/math&gt;}}
But, as we are discussing Borisenko &amp;amp; Tarapov, we press on with ({{EquationNote|7.26}}).&amp;#8201; By the algebraic identity
{{NumBlk|:|&lt;math&gt;\mathbf{c}(\mathbf{a\cdot b})
    \,=\, (\mathbf{a\cdot c})\mathbf{b}
    \,-\, \mathbf{a}\times(\mathbf{b}\times\mathbf{c}) \,,
  &lt;/math&gt;|{{EquationRef|7.27}}}}
i.e.
:&lt;math&gt;\mathbf{c}(\mathbf{a\cdot b})
    \,=\, (\mathbf{a\cdot c})\mathbf{b}
    \,+\, \mathbf{a}\times(\mathbf{c}\times\mathbf{b}) \,,
  &lt;/math&gt;
we can say
{{NumBlk|:|&lt;math&gt;\nabla(\mathbf{A}_c {\cdot}~\!\mathbf{B})
    \,=\, (\mathbf{A}_c {\cdot}\nabla)\mathbf{B}
    \,+\, \mathbf{A}_c\times(\nabla\times\mathbf{B}) \,.
  &lt;/math&gt;|{{EquationRef|7.28}}}}
Similarly,&lt;ref&gt;In the next equation as printed in Borisenko &amp;amp; Tarapov ([[#borisenko-tarapov-68|1968]], p.&amp;#8239;180), the first cross should be &quot;&amp;equals;&quot;; Tai ([[#tai-95|1995]], p.&amp;#8239;46) corrects it.&lt;/ref&gt;
{{NumBlk|:|&lt;math&gt;\nabla(\mathbf{B}_c {\cdot}~\!\mathbf{A})
    \,=\, (\mathbf{B}_c {\cdot}\nabla)\mathbf{A}
    \,+\, \mathbf{B}_c\times(\nabla\times\mathbf{A}) \,.
  &lt;/math&gt;|{{EquationRef|7.29}}}}
Substituting ({{EquationNote|7.28}}) and ({{EquationNote|7.29}}) into ({{EquationNote|7.26}}), in which the order of the dot-products is immaterial, and dropping the {{mvar|c }}subscripts (because they are now outside the differentiations), we get the correct result
{{NumBlk|:|{{midsize|&lt;math&gt;\nabla(\mathbf{A{\cdot}B})
    =     (\mathbf{A}{\cdot}\nabla)\mathbf{B}
      +   (\mathbf{B}\;\!{\cdot}\nabla)\mathbf{A}
      +   \mathbf{A}{\times}(\nabla{\times}\mathbf{B})
      +   \mathbf{B}{\times}(\nabla{\times}\mathbf{A}) \,,
  &lt;/math&gt;}}|{{EquationRef|7.30}}}}
corresponding to our ({{EquationNote|68}}).

Tai ([[#tai-95|1995]], p.&amp;#8239;47) is unimpressed, asking why we cannot apply ({{EquationNote|7.27}}) directly to the left side of ({{EquationNote|7.26}}). The answer to that is obvious: on the left side, the {{math|&amp;nabla;}} operator is applied to a product of ''two variables'', and the variations of ''both'' must be taken into account. But there is a harder question which Tai does not ask: in ({{EquationNote|7.28}}), why can't we have {{math|&amp;nabla;'''&amp;sdot;A'''&lt;sub&gt;c&lt;/sub&gt;}} instead of{{math| '''A'''&lt;sub&gt;c&lt;/sub&gt;'''&amp;sdot;'''&amp;nabla;}}&amp;#8239;? (Or, in terms of Feynman subscripts, why can't we have {{math|&amp;nabla;'''&lt;sub&gt;B&lt;/sub&gt;&amp;#8202;&amp;sdot;&amp;#8202;A'''}} instead of{{math| '''A&amp;sdot;'''&amp;nabla;&lt;sub&gt;'''B'''&lt;/sub&gt;}}?) Because that would make the term vanish? Yes, it would; but, as there is only one variable factor on the left side, why do we need two terms on the right? Because the rule says {{math|&amp;nabla;}} should be written after the constant but before the variable? Yes, but that rule serves the purpose of varying ''each'' variable, whereas there is only one variable to vary on the left of ({{EquationNote|7.28}}). The same issue arises in ({{EquationNote|7.29}}). We cannot settle the question even by appealing to symmetry. Obviously the right side of ({{EquationNote|7.30}}), like the left, must be unchanged if we switch {{math|'''A'''}} and {{math|'''B'''}}; and indeed it is. But if the first term on the right of ({{EquationNote|7.28}}) and of ({{EquationNote|7.29}}) were to vanish, the necessary symmetry of ({{EquationNote|7.30}}) would be maintained. And unless I'm missing something, Tai's &quot;symbolic vector&quot; method does not circumvent the problem; Tai's &quot;Lemma 2&quot; ([[#tai-95|1995]], p.&amp;#8239;53) is the Gibbs&amp;#10744;Wilson method of &quot;varying one factor at a time&quot;, written with Feynman subscripts attached to the symbolic vector instead of the del operator.{{efn|I don't overlook the fact that Tai's symbolic vector, unlike the del operator, is subject to commutative and anticommutative laws. Neither do I see how it helps.}}

For another example of the same issue, consider the following two-liner offered by Panofsky &amp;amp; Phillips ([[#panofsky-phillips-62|1962]], pp.&amp;#8239;470–71) and rightly pilloried by Tai ([[#tai-95|1995]], pp.&amp;#8239;47–8):
:&lt;math&gt;\begin{align}
  &amp;   \nabla{\times}(\mathbf{A}{\times}\mathbf{B})
   =  (\nabla{\cdot}\;\!\mathbf{B})\mathbf{A}
    - (\nabla{\cdot}    \mathbf{A})\mathbf{B}               &amp;&amp;[\mathsf{sic}] \\
  &amp;=  (\nabla{\cdot}\;\!\mathbf{B}_c ~\!\!)\mathbf{A}
    + (\nabla{\cdot}\;\!\mathbf{B}~\!\!)\mathbf{A}_c ~\!\!
    - (\nabla\mathbf{\cdot A}_c ~\!\!)\mathbf{B}
    - (\nabla\mathbf{\cdot A}~\!\!)\mathbf{B}_c \!\!\!\!\!\!\!&amp;&amp;[\mathsf{sic}].
  \end{align}&lt;/math&gt;
If the first line were right, the authors would hardly bother to continue; but evidently it isn't, because it doesn't begin by &quot;varying one factor at a time&quot;. The second line does not follow from the first and includes divergences of constants, which ought to vanish but somehow apparently do not. Let's try again, this time sticking to the rules:
:&lt;math&gt;\begin{align}
  &amp; \nabla\!\times\!(\mathbf{A}\!\times\!\mathbf{B})
  \,=\,   \nabla\!\times\!(\mathbf{A}_c \!\times\!\mathbf{B})
    \,+\, \nabla\!\times\!(\mathbf{A}\!\times\!\mathbf{B}_c) \\
  &amp;~=\,   (\nabla{\cdot}\;\!\mathbf{B})\mathbf{A}_c
        - (\mathbf{A}_c {\cdot}\nabla)\mathbf{B}
    \,+\, (\mathbf{B}_c {\cdot}\nabla)\mathbf{A}
        - (\nabla{\cdot}    \mathbf{A})\mathbf{B}_c \\
  &amp;~=\,   \mathbf{A}(\nabla{\cdot}\;\!\mathbf{B})
        - \mathbf{B}(\nabla{\cdot}    \mathbf{A})
    \,+\, (\mathbf{B\;\!\cdot}\nabla)\mathbf{A}
        - (\mathbf{A    \cdot}\nabla)\mathbf{B} \,,
  \end{align}&lt;/math&gt;
in agreement with our ({{EquationNote|67c}}). Here the first line comes from the generalized product rule, and the third is obtained from the second by rearranging terms and dropping the (now redundant) subscripts. The interesting line is the second, which is obtained from the first by expanding the formal vector triple products. But again, why must we have {{math|'''A'''&lt;sub&gt;c&lt;/sub&gt;'''&amp;sdot;'''&amp;nabla;}} and {{math|'''B'''&lt;sub&gt;c&lt;/sub&gt;'''&amp;sdot;'''&amp;nabla;,}} instead of {{math|&amp;nabla;'''&amp;sdot;A'''&lt;sub&gt;c&lt;/sub&gt;}} and {{math|&amp;nabla;'''&amp;sdot;B'''&lt;sub&gt;c&lt;/sub&gt;&amp;#8202;,}} which would make the middle two terms vanish? Again symmetry does not give an answer. The right-hand side, like the left, must change sign if we switch {{math|'''A'''}} and {{math|'''B'''&amp;#8202;}}; but the disappearance of the {{math|'''A'''&lt;sub&gt;c&lt;/sub&gt;'''&amp;sdot;'''&amp;nabla;}} and {{math|'''B'''&lt;sub&gt;c&lt;/sub&gt;'''&amp;sdot;'''&amp;nabla;}} terms would maintain the required (anti)symmetry. Funnily enough, the result would then agree with the incorrect first line given by Panofsky &amp;amp; Phillips (above). But then how would we know that it is incorrect?

The foregoing examples show that &quot;formal product&quot; arguments can be tenuous, even on their own terms. Before these examples, we might have been troubled by the omission of a general proof of the &quot;generalized&quot; product rule. After them, we might wonder whether the rule is even well defined.

I submit, however, that none of this matters. I submit that the popularity of using &quot;formal products&quot; with the del operator, in derivations of vector-analytic identities, is a reaction to the failure of early writers to use indicial notation in the Cartesian definitions of differential operators.{{efn|Indicial notation is standard in higher-order tensor analysis, which however tends not to use unit vectors of coordinate systems, and therefore tends not to encourage the indexing of unit vectors in elementary vector analysis&amp;mdash;whereas in the present paper, I have unapologetically indexed the unit vectors.}} The ensuing proliferation of terms in coordinate-based derivations led authors to seek shortcuts through &quot;formal products&quot; when more rigorous but no-less convenient shortcuts could have been taken through indicial notation, especially in combination with implicit summation. Our derivation of the gradient of a dot-product ({{EquationNote|68}}) is shorter than that of Borisenko &amp;amp; Tarapov, and even uses the right-hand sides of their identities ({{EquationNote|7.28}}) and ({{EquationNote|7.29}}), but obtains them rigorously with no ambiguity and no {{mvar|c }}subscripts. Our derivation of the curl of a cross-product ({{EquationNote|67c}}) takes six lines with a single column of &quot;&amp;equals;&quot; signs. Our subsequent formal-product derivation (not to be confused with the attempt of Panofsky &amp;amp; Phillips) seems to take only three lines; but it is only through our earlier indicial derivation that we have any confidence in our result (not to be confused with the result of Panofsky &amp;amp; Phillips). Our other indicial derivations of identities are mostly shorter than the two just mentioned. Having amassed so comprehensive a collection of identities so rigorously with so little effort, I submit that the use of formal products, Wilson subscripts, {{mvar|c }}subscripts, and Feynman subscripts for this purpose is a historical aberration, to be deciphered in other people's writings but avoided in one's own.

That being said, it is one thing to conclude, as Tai duly does, that the del-cross and del-dot notations should not be interpreted as products in derivations and proofs, and another thing to allege, as Tai also does ([[#tai-95|1995]], p.&amp;#8239;22), that&amp;#8202; {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8202; and {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; are &quot;not compound operators&quot; but only &quot;assemblies&quot;, or in other words that &quot;{{math|&amp;#8202;&amp;nabla; }}is not a constituent of the divergence operator nor of the curl operator.&quot; Against the latter proposition, our equations ({{EquationNote|14}}), ({{EquationNote|61o}}), and ({{EquationNote|62o}}) have been ''derived'', not merely defined, and our derivation of ({{EquationNote|14}}) is as general as we could wish. Moreover, whereas ({{EquationNote|61o}}) and ({{EquationNote|62o}}) are for Cartesian coordinates, we shall see that they have counterparts in more general coordinates.

{{cob}}

== General coordinates ==

{{cot}}

From our initial definitions of the differential operators, we derived certain identities, from which we derived expressions for the operators in Cartesian coordinates, from which we derived a comprehensive collection of identities, two of which (the multivariate chain rule, and the curl of the product of a scalar and a vector) will now be useful for expressing the operators in other coordinate systems. Cartesian coordinates are traditionally called {{math|''x'',&amp;#8239;''y'',&amp;#8239;''z'',}}&amp;#8201; which we renamed {{mvar|x&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; where&amp;#8202; {{math|''i''&amp;#8201;{{=}}&amp;#8239;1,&amp;#8202;2,&amp;#8202;3&amp;#8202;,}}&amp;#8201; respectively. The best-known 3D ''non''&amp;#8202;-Cartesian coordinate systems are the cylindrical coordinates {{math|(''&amp;rho;'',&amp;#8239;''&amp;phi;'',&amp;#8239;''z'')}} and the spherical coordinates {{math|(''r'',&amp;#8239;''&amp;theta;'',&amp;#8239;''&amp;phi;'')}}; we have already seen {{mvar|r}}&amp;#8202; in the guise of the magnitude of the position vector{{math| '''r'''}}.&amp;#8201; But now we want our coordinate system to be as general as possible&amp;mdash;with the Cartesian, cylindrical, and spherical systems and many others, and even ''classes'' of systems, as special cases.

{{cob}}

=== Natural and dual basis vectors ===

{{cot}}

We shall call our general coordinates {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}}&amp;#8202; where&amp;#8202; {{math|''i''&amp;#8201;{{=}}&amp;#8239;1,&amp;#8202;2,&amp;#8202;3&amp;#8202;}};&amp;#8201; yes, for reasons which will emerge, we shall write the coordinate index as a {{nowrap|''super''&amp;#8202;script}}. But we shall write {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; for{{math| ''{{sfrac|&amp;part;|&amp;part;u&lt;sup&gt;i&lt;/sup&gt;&amp;#8202;}}''&amp;#8202;,}}&amp;#8202; relying on context to distinguish it from the special case{{mvar| {{sfrac|&amp;part;|&amp;part;x&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202;}}.&amp;#8201; By describing the {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}}&amp;#8202; as ''coordinates''&amp;#8239; we mean two things. First, for some domain of interest, the position vector is a smooth function
:&lt;math&gt;\mathbf{r} = \mathbf{r}(u^1,u^2,u^3) \,,&lt;/math&gt;
which possesses partial derivatives w.r.t. its arguments. Second, for every position vector in the resulting range, there is only one ordered triplet&amp;#8239; {{math|(''u&lt;sup&gt;i&lt;/sup&gt;''&amp;#8202;)&amp;#8201;{{=}}&amp;#8201;(''u''&amp;sup1;,&amp;#8239;''u''&amp;sup2;,&amp;#8239;''u''&amp;sup3;),}}&amp;#8239; so that we can think of each coordinate as
:{{big|&lt;math&gt;u^i = u^i(\mathbf{r}) \,;&lt;/math&gt;}}
&amp;mdash;that is, we can think of each {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}}&amp;#8202; as a scalar field, which possesses a gradient.{{efn|Hence we want each {{math|''u&lt;sup&gt;i&lt;/sup&gt;''('''r''')}} to be, as far as possible, a ''smooth'' function. This may require some tweaking of definitions. E.g., in cylindrical coordinates, the angular coordinate {{mvar|&amp;phi;}} must be confined to some 360&amp;deg; range in order to make it unique, and we don't want it jumping from the end of the range to the beginning within the region of interest.}} (I say &quot;think of&quot; because {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}}, being obviously dependent on a coordinate system, would not normally be considered a true scalar; but sometimes we need to treat the coordinate system itself as an object under study.)

These two properties of coordinates respectively suggest two simple ways of choosing basis vectors related to the coordinates: we shall define the '''natural basis''' vectors as
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_i := \part_i \mathbf{r} \,,
  \qquad&lt;/math&gt;}}|{{EquationRef|80a}}}}
and the '''dual basis''' vectors as
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^i := \nabla u^i .
  \qquad&lt;/math&gt;}}|{{EquationRef|80b}}}}
(We could ''normalize'' the natural basis vectors by dividing them by their magnitudes to obtain unit vectors; but, for the moment, we won't bother.) Just as we may think of each {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}} as a scalar field and inquire after its directional derivative or its gradient or its Laplacian, so we may think of each {{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt;}} or{{math| '''h'''&lt;sup&gt;''i''&lt;/sup&gt;}} as a vector field and inquire after its directional derivative or its curl or its divergence or its Laplacian. (That the curl of{{math|&amp;#8202; '''h'''&lt;sup&gt;''i''&lt;/sup&gt;}}&amp;#8202; is zero&amp;#8239; will be especially useful.)

In Cartesian coordinates,&amp;#8201; {{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt;}} and {{math|'''h'''&lt;sup&gt;''i''&lt;/sup&gt;}} are both equal to the unit vector{{math| '''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;}}; thus, in Cartesian coordinates, the natural basis vectors are their own duals.&amp;#8201; In ''general'' coordinates,&amp;#8201; {{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt;}} and {{math|'''h'''&lt;sup&gt;''i''&lt;/sup&gt;}} may differ in both direction and magnitude and are not generally unit vectors. Nevertheless, even in general coordinates, there is a simple relation between the natural and dual basis vectors. Consider the dot-product
:{{big|&lt;math&gt;
    \mathbf{h}_i \cdot \mathbf{h}^j
    = \part_i\mathbf{r} \cdot \nabla u^j \,.
  &lt;/math&gt;}}
If{{math|&amp;#8202; ''i&amp;#8239;&amp;ne;&amp;#8239;j''&amp;#8202;,}} then {{math|''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''r'''&amp;#8202;,}} being in a direction in which {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}} varies while each other {{mvar|u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;}} does not, is tangential to a surface of constant {{mvar|u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;}} and therefore normal to {{math|&amp;nabla;''u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;'',}} so that the dot-product is zero. But by ({{EquationNote|26g}}),
:{{big|&lt;math&gt;
    du^i = \nabla u^i \cdot d\mathbf{r} \,;
  &lt;/math&gt;}}
and if we vary {{math|'''r'''}} by varying {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}} while holding each other {{mvar|u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;}} constant, we can divide by {{mvar|du&lt;sup&gt;i&lt;/sup&gt;}} and obtain
{{NumBlk|:|{{big|&lt;math&gt;
    1 ~\!= \nabla u^i \cdot \part_i\mathbf{r}
      = \mathbf{h}^i \!\cdot \mathbf{h}_i
  \qquad&lt;/math&gt;}}[with no summation].|{{EquationRef|81i}}}}
Putting the two cases together, we have
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_i \cdot \mathbf{h}^j =~\! \delta_i^j
  &lt;/math&gt;}}|{{EquationRef|81}}}}
where the right-hand function, known as the '''Kronecker delta''' function, is defined by
{{NumBlk|:|{{big|&lt;math&gt;\delta_i^j = \delta_{ij} = \delta^{ij}
  =~&lt;/math&gt;}}&lt;math&gt;\begin{cases}
    0 &amp;\mathsf{if}~\, i \neq j \\
    1 &amp;\mathsf{if}~\, i = j \,.
  \end{cases}&lt;/math&gt;|{{EquationRef|82}}}}
Obviously the function is symmetric: the indices {{mvar|i }}and{{mvar| j}}&amp;#8202; can be interchanged. If two lists of vectors are related so that the dot-product of the {{mvar|i&amp;#8202;}}th vector in one list and the {{mvar|j&amp;#8202;}}th in the other is{{mvar| &amp;delta;&lt;sub&gt;ij&lt;/sub&gt;&amp;#8202;}}, the two lists are described as '''reciprocal'''. Thus the triplets {{math|('''h'''&lt;sub&gt;''i''&lt;/sub&gt;)}} and {{math|('''h'''&lt;sup&gt;''i''&lt;/sup&gt;)}} are '''reciprocal bases''': the dual basis is the reciprocal of the natural basis and vice versa. Hence, taking the natural basis as a reference, the dual basis is sometimes called &quot;the&quot; reciprocal basis.

In Cartesian coordinates, ({{EquationNote|81}}) becomes
:{{big|&lt;math&gt;
    \mathbf{e}_i ~\!\!\cdot \mathbf{e}_j =~\! \delta_{ij} \,.
  &lt;/math&gt;}}
So we have a relation for general coordinates ({{EquationNote|81}}) which is just as simple as its special case for Cartesian coordinates, ''provided that we use the natural basis for one factor and the dual basis for the other''. This will be a recurring pattern.

We have deduced the reciprocity relation ({{EquationNote|81}}) from prior definitions of the natural basis {{math|('''h'''&lt;sub&gt;''i''&lt;/sub&gt;)}} and the dual basis {{math|('''h'''&lt;sup&gt;''i''&lt;/sup&gt;)}}.&amp;#8201; This result has a partial converse, in that a reciprocity relation between bases is enough to define either basis in terms of the other&amp;mdash;as we shall see later. But first we proceed to components of vector fields.

{{cob}}

=== Contravariant and covariant components ===

{{cot}}

A '''coordinate grid''' is a set of intersecting curves such that on each curve, one coordinate varies while the others are constant. If we could embed such a grid in an elastic medium, and then stretch and rotate the medium, the natural basis vectors{{math| '''h'''&lt;sub&gt;''i''&lt;/sub&gt;}} given by ({{EquationNote|80a}}) would stretch and rotate ''with the medium'' and ''with the grid''. Accordingly, the ''natural'' basis is also called the '''covariant''' basis. But according to ({{EquationNote|81}}), the dot-product of a natural basis vector and a dual basis vector is '''invariant''' (independent of the coordinate system), so that the variation of one factor ''compensates''&amp;#8202; for the variation of the other. So, as the natural basis is &quot;covariant&quot; with the coordinate grid, we say that the dual basis is '''contravariant'''. Notice that the {{nowrap|''co''&amp;#8202;variant}} factor has a {{nowrap|''sub''&amp;#8202;script}} index (easily remembered because &quot;''co''&amp;#8202; rhymes with ''low''&amp;#8239;&quot;) whereas the {{nowrap|''contra''&amp;#8202;variant}} factor has a {{nowrap|''super''&amp;#8202;script}} index, and that one kind of variation must combine with the other in order to produce an {{nowrap|''in''&amp;#8202;variant}} result; these will be recurring patterns.

A vector field {{math|'''q'''}} may be expressed in components w.r.t. the natural (covariant) basis as
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q} = q^i \mathbf{h}_i
  \qquad&lt;/math&gt;}}|{{EquationRef|83a}}}}
with summation, or in components w.r.t. the dual (contravariant) basis as
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q} = q_i \mathbf{h}^i
  \qquad&lt;/math&gt;}}|{{EquationRef|83b}}}}
with summation. If{{math| '''q'''}} is to be invariant (a true vector, existing independently of the coordinate system), the components must be contravariant in the former case and covariant in the latter, and accordingly are written with superscripts and subscripts respectively. In Cartesian coordinates, the two bases are the same, so that the components w.r.t. the two bases are also the same; that's why, in the above section headed &quot;[[#Cartesian coordinates|Cartesian coordinates]]&quot;, we got away with writing component indices as subscripts. In ''general'' coordinates, however, the basis vectors have subscripts and the components have superscripts or vice versa, so that ''the index of implicit summation appears once as a superscript and once as a subscript''.

Taking dot-products of ({{EquationNote|83a}}) with{{math| '''h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;}}, applying ({{EquationNote|81}}), and noting that only one term on the right is non-zero, we obtain
{{NumBlk|:|{{big|&lt;math&gt;
     q^j =~\! \mathbf{q} \cdot \mathbf{h}^j .
  \qquad&lt;/math&gt;}}|{{EquationRef|83c}}}}
Similarly, taking dot-products of ({{EquationNote|83b}}) with{{math| '''h'''&lt;sub&gt;''j''&lt;/sub&gt;}} yields
{{NumBlk|:|{{big|&lt;math&gt;
     q_j =~\! \mathbf{q} \cdot \mathbf{h}_j \,.
  \qquad&lt;/math&gt;}}|{{EquationRef|83d}}}}
These results depend on the reciprocity relation ({{EquationNote|81}}) but not on the earlier definitions of the bases to which that relation applies. They say:
* to find the contravariant components of a vector, take its dot-products with the contravariant basis vectors, and
* to find the covariant components of a vector, take its dot-products with the covariant basis vectors;
''or'', in terms of the bases themselves:
* to find the components of a vector w.r.t. either basis, take dot-products of that vector with the ''other'' basis.

If a particular {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}} has a particular name, such as{{mvar| &amp;theta;}} or{{mvar| &amp;phi;}}, then, if we're not using indexed summation, we may find it convenient to write that name in place of the index{{mvar| i}}&amp;#8202; in the superscript or subscript.

At the present level of generality, the basis vectors {{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;,}} unlike their Cartesian counterparts {{math|'''e'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;,}} are ''not''&amp;#8202; assumed to be uniform (i.e., '''homogeneous'''). One consequence of this general non-uniformity (inhomogeneity) is that, although we can say&amp;#8239; {{math|'''r'''&amp;#8201;{{=}}&amp;#8201;''x&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''e'''&lt;sub&gt;''i''&lt;/sub&gt;}}&amp;#8239; in Cartesian coordinates and&amp;#8239; {{math|'''q'''&amp;#8201;{{=}}&amp;#8201;''q&lt;sup&gt;i&lt;/sup&gt;''&amp;#8239;'''h'''&lt;sub&gt;''i''&lt;/sub&gt;}}&amp;#8239; in general coordinates, we ''cannot'' say
:{{big|&lt;math&gt;\mathbf{r} = u^i \mathbf{h}_i \qquad&lt;/math&gt;[''sic!''&amp;#8239;]}}
in general coordinates. For example, we have seen that in spherical coordinates the position vector {{math|'''r'''}} is simply &lt;math&gt;r\mathbf{\hat{r}}&lt;/math&gt;, i.e.{{math| ''r''&amp;#8202;'''h'''&lt;sub&gt;''r''&lt;/sub&gt;&amp;#8239;}};&amp;#8202; it is ''not''&amp;#8201; {{math|''r''&amp;#8202;'''h'''&lt;sub&gt;''r''&lt;/sub&gt;&amp;#8202;+&amp;#8202;''&amp;theta;''&amp;#8202;'''h'''&lt;sub&gt;''&amp;theta;''&lt;/sub&gt;&amp;#8202;+&amp;#8202;''&amp;phi;''&amp;#8202;'''h'''&lt;sub&gt;''&amp;phi;''&lt;/sub&gt;&amp;#8202;,}} because {{mvar|&amp;theta;}} and {{mvar|&amp;phi;}} are encoded in the direction of{{math|&amp;#8202; '''h'''&lt;sub&gt;''r''&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; Similarly, in cylindrical coordinates the position vector {{math|'''r'''}} is {{math|''&amp;rho;''&amp;#8202;'''h'''&lt;sub&gt;''&amp;rho;''&lt;/sub&gt;&amp;#8202;+&amp;#8239;''z''&amp;#8202;'''h'''&lt;sub&gt;''z''&lt;/sub&gt;&amp;#8239;}};&amp;#8202; it is ''not''&amp;#8201; {{math|''&amp;rho;''&amp;#8202;'''h'''&lt;sub&gt;''&amp;rho;''&lt;/sub&gt;&amp;#8202;+&amp;#8202;''&amp;phi;''&amp;#8202;'''h'''&lt;sub&gt;''&amp;phi;''&lt;/sub&gt;&amp;#8202;+&amp;#8239;''z''&amp;#8202;'''h'''&lt;sub&gt;''z''&lt;/sub&gt;&amp;#8202;,}} because {{mvar|&amp;phi;}} is encoded in the direction of{{math|&amp;#8202; '''h'''&lt;sub&gt;''&amp;rho;''&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; In both examples, encoding one coordinate in the direction of another coordinate's unit vector is circular in that the said direction depends on the position vector, which is the very thing that we want to represent.

A non-uniform basis is not a ''global''&amp;#8202; basis. It cannot give a uniform representation of a uniform vector field, because the standard of representation changes; it is like having a compass whose orientation varies from place to place and&amp;#10744;or a measuring stick whose length varies from place to place. But it can serve as a '''local basis'''&amp;#8202;&amp;mdash;as in ({{EquationNote|83a}}) and ({{EquationNote|83b}}), each of which expresses a vector field at a given location in terms of a basis at that location, notwithstanding that the basis may be different at other locations. And although a local basis (as we have just seen) cannot generally represent the position vector in a non-circular manner, it ''can''&amp;#8202; represent a ''change''&amp;#8202; in the position vector. By the generality of the multivariate chain rule ({{EquationNote|75}}),
:{{big|&lt;math&gt;
    \part_t \mathbf{r} = \part_i \mathbf{r} \,\part_t u^i .
  &lt;/math&gt;}}
Multiplying by {{mvar|dt&amp;#8239;}} we get
{{NumBlk|:|{{big|&lt;math&gt;
    d\mathbf{r} = \part_i \mathbf{r} \,du^i
  &lt;/math&gt;}}|{{EquationRef|84}}}}
or, substituting from ({{EquationNote|80a}}),
{{NumBlk|:|{{big|&lt;math&gt;
    d\mathbf{r} = \mathbf{h}_i ~\!du^i .
  \qquad&lt;/math&gt;}}|{{EquationRef|85}}}}
Thus the small changes in the coordinates{{mvar| u&lt;sup&gt;i&lt;/sup&gt;}}&amp;#8202; are the components of the true vector{{math| ''d'''''r'''}} w.r.t. the ''covariant''&amp;#8202; basis. That means the changes in the coordinates must be ''contravariant''. Here at last is the explanation why we write general coordinates with superscript indices. And again the point is moot for Cartesian coordinates, for which the covariant basis is also contravariant.

Since {{mvar|du&lt;sup&gt;i&lt;/sup&gt;}}&amp;#8202; is contravariant,&amp;#8201; {{math|''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;'''r'''}}&amp;#8202; in ({{EquationNote|84}}) must be covariant in order to yield the true vector{{math| ''d'''''r'''}}. This vindicates our decision to write {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}} with a subscript. Recall, however, that {{mvar|&amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; means{{math| ''{{sfrac|&amp;part;|&amp;part;u&lt;sup&gt;i&lt;/sup&gt;&amp;#8202;}}''&amp;#8202;}}. Thus ''the derivative w.r.t. the contravariant quantity is covariant''&amp;#8202;&amp;mdash;wherefore it is said that ''a superscript in the denominator of a derivative counts as a subscript in the derivative as a whole''.

In ({{EquationNote|85}}), the general term&amp;#8239; {{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8201;''du&lt;sup&gt;i&lt;/sup&gt;''}} (not the sum) is the displacement of{{math|&amp;#8202; '''r'''}} due to the small change {{mvar|du&lt;sup&gt;i&lt;/sup&gt;}} in the coordinate {{mvar|u&lt;sup&gt;i&lt;/sup&gt;}}. The three such displacements of{{math|&amp;#8202; '''r'''}} make concurrent edges of a parallelepiped whose signed volume is
:&lt;math&gt;
  dV =~\! \mathbf{h}_1~\!du^1 \cdot~\!\mathbf{h}_2~\!du^2
          ~\!\!\times\mathbf{h}_3~\!du^3 \,;
  &lt;/math&gt;
that is,
{{NumBlk|:|&lt;math&gt;dV = J \,du^1 du^2 du^3&lt;/math&gt;|{{EquationRef|86}}}}
where
:&lt;math&gt;J := \mathbf{h}_1 ~\!\!\cdot \mathbf{h}_2 \!\times~\!\!\mathbf{h}_3&lt;/math&gt;
or, to use a standard abbreviation for the scalar triple product,
{{NumBlk|:|&lt;math&gt;
    J := [~\!\mathbf{h}_1 \mathbf{h}_2 \mathbf{h}_3] \,.
  &lt;/math&gt;|{{EquationRef|87}}}}
{{mvar|J}}&amp;#8202; is called the '''Jacobian''' of the natural (covariant) basis. We describe the basis and the associated coordinate system as '''right-handed''' if this Jacobian is ''positive'', and '''left-handed''' if this Jacobian is ''negative''. Thus the handedness depends on the standard order in which we write the vectors; e.g., the standard Cartesian basis is right-handed because we write it as{{math| ('''i''',&amp;#8202;'''j''','''k''')}} but would be left-handed if we wrote it as{{math| ('''i''','''k''',&amp;#8202;'''j''')}}.

If the covariant basis is indeed a basis, its member vectors must not be coplanar; that is, {{mvar|J }}must not be zero. Hence, if the covariant basis is to be a local basis in some region of interest, {{mvar|J }}must not vanish anywhere in that region, and therefore must have the same sign throughout the region; that is, the handedness of the coordinate system must be the same throughout the region.

{{cob}}

=== Properties of reciprocal bases ===

{{cot}}

We have noted that formulae ({{EquationNote|83c}}) and ({{EquationNote|83d}}), for the components of a vector w.r.t. the covariant and contravariant bases, depend only on the reciprocity relation ({{EquationNote|81}}) between the bases. Now, retaining the designations &quot;covariant&quot; and &quot;contravariant&quot; for convenience, let us see what else we can deduce from that relation.

Most obviously, the reciprocity relation leads to a simple component-based expression for the dot-product of two vector fields, say {{math|'''v'''}} and{{math| '''q'''&amp;#8202;,}} provided that we use the contravariant components and covariant basis ({{EquationNote|83a}}) for one vector, and the covariant components and contravariant basis ({{EquationNote|83b}}) for the other:
:{{big|&lt;math&gt;\mathbf{v} \cdot \mathbf{q}
    =~\! v^i ~\!\mathbf{h}_i \cdot q_j \mathbf{h}^j
    =~\! v^i \,\mathbf{h}_i \!\cdot\! \mathbf{h}^j \,q_j
    =~\! v^i ~\!\delta_i^j ~\!q_j \,,
  &lt;/math&gt;}}
whence selecting the non-zero terms gives
{{NumBlk|:|{{big|&lt;math&gt;\mathbf{v} \cdot \mathbf{q}
    =~\! v^i q_i \,.
  &lt;/math&gt;}}|{{EquationRef|88a}}}}
And the two vectors, being general, can swap roles in ({{EquationNote|83a}}) and ({{EquationNote|83b}}):
{{NumBlk|:|{{big|&lt;math&gt;\mathbf{v} \cdot \mathbf{q}
    =~\! v_i q^i \,.
  &lt;/math&gt;}}|{{EquationRef|88b}}}}

The cross-product needs a bit more preparation. First we define the '''permutation symbol''' {{mvar|&amp;epsiv;&lt;sub&gt;ijk&lt;/sub&gt;}} or{{mvar| &amp;epsiv;&lt;sup&gt;ijk&lt;/sup&gt;}} (also called the '''[[w:Tullio Levi-Civita|Levi-Civita]]''' symbol) as having the value&amp;#8202; {{math|+1}} if {{math|(''i'',&amp;#8201;''j'',&amp;#8239;''k'')}} is a permutation of{{math| (1,&amp;#8202;2,&amp;#8202;3)}} in the same cyclic order,&amp;#8201; {{math|&amp;minus;1 }}if {{math|(''i'',&amp;#8201;''j'',&amp;#8239;''k'')}} is a permutation of{{math| (1,&amp;#8202;2,&amp;#8202;3)}} in the reverse cyclic order, and {{math|0}} if {{math|(''i'',&amp;#8201;''j'',&amp;#8239;''k'')}} is not a permutation, i.e. if there is at least one repeated index. To put it more formally,
{{NumBlk|:|{{big|&lt;math&gt;\epsilon_{ijk\!} = \epsilon^{ijk\!} =&lt;/math&gt;}}
  {{resize|&lt;math&gt;\begin{cases}
    +1 &amp;\mathsf{if}\,\,(i,j,k)\in\big\{(1,2,3),~\!(2,3,1),~\!(3,1,2)\big\}\\
    -1 &amp;\mathsf{if}\,\,(i,j,k)\in\big\{(3,2,1),~\!(1,3,2),~\!(2,1,3)\big\}\\
    \phantom{-}0 &amp;\mathsf{otherwise}.
  \end{cases}&lt;/math&gt;}}|{{EquationRef|89}}}}
Note that because switching any two indices changes the cyclic order, ''switching any two indices changes the sign of the permutation symbol''. Now by ({{EquationNote|81}}),&amp;#8201; {{math|'''h'''&lt;sup&gt;1&lt;/sup&gt; }}is perpendicular to both {{math|'''h'''&lt;sub&gt;2&lt;/sub&gt; }}and{{math| '''h'''&lt;sub&gt;3&lt;/sub&gt;}}. So we can say
:&lt;math&gt;\mathbf{h}_2 \!\times\!\mathbf{h}_3 =~\! \alpha_1 ~\!\mathbf{h}^1&lt;/math&gt;
where {{math|''&amp;alpha;''&lt;sub&gt;1&lt;/sub&gt; }}is a real variable to be determined. Taking dot-products with{{math| '''h'''&lt;sub&gt;1&lt;/sub&gt;}} and applying ({{EquationNote|81}}) and ({{EquationNote|87}}), we find that&amp;#8202; {{math|''&amp;alpha;''&lt;sub&gt;1&lt;/sub&gt;&amp;#8239;{{=}}&amp;#8239;''J''&amp;#8202;,}} so that
{{NumBlk|:|&lt;math&gt;
    \mathbf{h}_2 \!\times\!\mathbf{h}_3 =~\! J \mathbf{h}^1 .
  &lt;/math&gt;|{{EquationRef|90.1}}}}
By the generality of the vectors we can rotate the three indices, but the sign of the left-hand side changes if we swap the two indices on the left. All six cases are covered by
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_i \!\times\!\mathbf{h}_j
    =~\! J \epsilon_{ijk\,} \mathbf{h}^k .
  &lt;/math&gt;}}|{{EquationRef|90a}}}}
Here we want only one term; but we need not specify &quot;no sum&quot;, because for given {{mvar|i&amp;#8202; }}and{{mvar| j}}&amp;#8201; the permutation symbol leaves only one non-zero term in the sum over{{mvar| k}}. In words, this result says that the cross-product of two covariant basis vectors, with their indices in the standard cyclic order, is the Jacobian times the contravariant basis vector with the omitted index. Similarly, or rather reciprocally,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^i \!\times\!\mathbf{h}^j
    =~\! J' \epsilon^{ijk\,} \mathbf{h}_k \,,
  &lt;/math&gt;}}|{{EquationRef|90b}}}}
where {{mvar|J&amp;prime;}}&amp;#8202; is the Jacobian ''of the contravariant basis''.

Equations ({{EquationNote|90a}}) and ({{EquationNote|90b}}), which we have obtained from the reciprocity relation ({{EquationNote|81}}), can be solved for {{math|'''h'''&lt;sup&gt;''k''&lt;/sup&gt; }}and{{math| '''h'''&lt;sub&gt;''k''&lt;/sub&gt;}} respectively; but now we ''do'' suppress the implicit sum, because {{mvar|k}}&amp;#8202; is &quot;given&quot; instead of {{mvar|i&amp;#8202; }}and{{mvar| j&amp;#8239;}}:
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^k = \tfrac{\,1\,}{J}~\! \mathbf{h}_i \!\times\!\mathbf{h}_j \quad
  &lt;/math&gt;}} [distinct {{math|''i'',&amp;#8201;''j'',&amp;#8239;''k''}}  in cyclic order];
  |{{EquationRef|90c}}}}
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_k = \tfrac{1}{\,J'}~\! \mathbf{h}^i \!\times\!\mathbf{h}^j \quad
  &lt;/math&gt;}}[distinct {{math|''i'',&amp;#8201;''j'',&amp;#8239;''k''}}  in cyclic order].
  |{{EquationRef|90d}}}}
Thus ''a reciprocity relation between bases is enough to define either basis in terms of the other''&amp;mdash;as claimed above.{{efn|Our ({{EquationNote|90c}}) corresponds to [[#stratton-41|Stratton, 1941]], p.&amp;#8239;39, eqs.&amp;#8239;(9). And our ({{EquationNote|90d}}) corresponds to Stratton's subsequent eqs.&amp;#8239;(11) except that Stratton has, in our notation, {{mvar|J}} instead of{{mvar| J&amp;prime;}}; the error is noted by Tai ([[#tai-95|1995]], p.&amp;#8239;59). See also our ({{EquationNote|92}}).}} If it is not convenient to suppress an implicit sum, the last two results can instead be written
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^k =
      \tfrac{1}{2J}~\!\epsilon^{ijk\,}\mathbf{h}_i {\times}~\!\mathbf{h}_j
  &lt;/math&gt;}}|{{EquationRef|90e}}}}
and
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_k =
      \tfrac{1}{2J'}~\!\epsilon_{ijk\,}\mathbf{h}^i {\times}~\!\mathbf{h}^j \,,
  &lt;/math&gt;}}|{{EquationRef|90f}}}}
where the factor 2 in each denominator is needed because the right-hand side has two equal non-zero terms&amp;mdash;the sign of the permutation symbol compensating for the order of the cross-product.

Now we're ready to consider the cross-product of two vector fields. In terms of the covariant basis,
:{{big|&lt;math&gt;\begin{align}\mathbf{v} \!\times\! \mathbf{q}
     =~\! v^i \mathbf{h}_i ~\!\!\times q^j \mathbf{h}_j
    &amp;=~\! v^i \,\mathbf{h}_i {\times}~\! \mathbf{h}_j \,q^j \\
    &amp;=~\! v^i J \epsilon_{ijk~\!} \mathbf{h}^k \;\!q^j \,;
  \end{align}&lt;/math&gt;}}
i.e.,
{{NumBlk|:|{{big|&lt;math&gt;\mathbf{v} \!\times\! \mathbf{q}
    =~\! J \epsilon_{ijk\,} v^i q^j \mathbf{h}^k .
  &lt;/math&gt;}}|{{EquationRef|91a}}}}
On the right, the two components and the basis vector are contravariant, but invariance is achieved by multiplying by the covariant Jacobian (which has three covariant factors). Similarly,
{{NumBlk|:|{{big|&lt;math&gt;\mathbf{v} \!\times\! \mathbf{q}
    =~\! J' \epsilon^{ijk} v_i q_j \;\!\mathbf{h}_k .
  &lt;/math&gt;}}|{{EquationRef|91b}}}}
On the right of ({{EquationNote|91a}}) or ({{EquationNote|91b}}), the implicit triple summation has 27 terms, of which only six&amp;mdash;corresponding to the six possible permutations of the three possible indices&amp;mdash;can be non-zero. Thus the factor following the Jacobian can be recognized as the familiar determinant whose columns (or rows), in cyclic order, are the components of{{math| '''v'''&amp;#8202;,}} the components of{{math| '''q'''&amp;#8202;,}} and the three basis vectors. In Cartesian coordinates, in which the Jacobians are equal to{{math| 1}} and we don't need the co&amp;#10744;contra distinction, both equations reduce to
:{{big|&lt;math&gt;\mathbf{v} \!\times\! \mathbf{q}
    =~\! \epsilon_{ijk\,} v_i q_j \;\!\mathbf{e}_k
  &lt;/math&gt;}}
&amp;mdash;a familiar result written in a possibly unfamiliar way.

The Jacobian of the contravariant basis is
:&lt;math&gt;J' =~\! \mathbf{h}^1 \cdot \mathbf{h}^2 \!\times\!\mathbf{h}^3&lt;/math&gt;
or, if we substitute from ({{EquationNote|90c}}),
:&lt;math&gt;\begin{align}J'
  &amp;= \frac{(\mathbf{h}_2 \!\times\!\mathbf{h}_3)}{J} \cdot
     \frac{(\mathbf{h}_3 \!\times\!\mathbf{h}_1)}{J} \times
     \frac{(\mathbf{h}_1 \!\times\!\mathbf{h}_2)}{J} \\[.5ex]
  &amp;= \frac{(\mathbf{h}_2 \!\times\!\mathbf{h}_3) \cdot
           (\mathbf{h}_3 \!\times\!\mathbf{h}_1) \times
           (\mathbf{h}_1 \!\times\!\mathbf{h}_2)}
          {J^3} \,.
  \end{align}&lt;/math&gt;
In the numerator, the cross-product of cross-products can be read as a vector triple product in which the first factor is a cross-product. Expanding that triple product and noting that one term is a scalar triple product with a repeated factor, we get
:&lt;math&gt;
    J' = \frac{(\mathbf{h}_2 \!\times\!\mathbf{h}_3) \cdot J\mathbf{h}_1}{J^3}
       = \frac{\,J^2}{~J^3 ~\!} = \frac{\,1\,}{J} \,,
  &lt;/math&gt;
so that we may write
{{NumBlk|:|&lt;math&gt;J' =~\! J^{-1} &lt;/math&gt;|{{EquationRef|92}}}}
in ({{EquationNote|90b}}), ({{EquationNote|90d}}), ({{EquationNote|90f}}), and ({{EquationNote|91b}}). In words, ''the Jacobian of the reciprocal basis is the reciprocal of the Jacobian'' of the original basis. Therefore the two Jacobians have the same sign. Therefore ''a basis is right-handed if and only if its reciprocal is right-handed''. Thus the natural and dual bases of a coordinate system have the same handedness, and the handedness of either may be identified with the handedness of the coordinate system.

{{cob}}

=== The gradient, del, and advection operators ===

{{cot}}

Let {{mvar|p}}&amp;#8202; be a scalar field, and let{{mvar| s}}&amp;#8202; be arc length in the direction of the unit vector{{math| '''s&amp;#770;'''}}. By the multivariate chain rule ({{EquationNote|75}}),
:{{big|&lt;math&gt;\begin{align}\part_s p
    &amp;= \part_i p \;\part_s u^i \\
    &amp;= \part_i p \;\mathbf{\hat{s}} \cdot \nabla u^i \\
    &amp;= \part_i p \;\mathbf{\hat{s}} \cdot \mathbf{h}^i \\
    &amp;=\mathbf{\hat{s}} \cdot \mathbf{h}^i \part_i p \,.
  \end{align}&lt;/math&gt;}}
So&amp;#8202; {{math|'''h'''&lt;sup&gt;''i''&lt;/sup&gt;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;p''}}&amp;#8239; is the vector whose (invariant) scalar component in the direction of any{{math| '''s&amp;#770;'''}} is the directional derivative of{{mvar| p}} in that direction; that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla p = \mathbf{h}^i \part_i p \,,
  &lt;/math&gt;}}|{{EquationRef|93g}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla =~\! \mathbf{h}^i \part_i \,.
  &lt;/math&gt;}}|{{EquationRef|93o}}}}
Apart from the need to pair a superscript with a subscript, these two results look as simple as their Cartesian special cases ({{EquationNote|58g}}) and ({{EquationNote|58o}}).

If {{mvar|&amp;psi;}}&amp;#8202; is a generic field and {{math|'''q'''}} is a general vector in the direction of the same{{math| '''s'''&amp;#8202;}} then by definition ({{EquationNote|11}}),
:{{big|&lt;math&gt;\begin{align}
    \mathbf{q}\;\!{\cdot}\nabla\,\psi
    &amp;= |\mathbf{q}| \,\part_s \psi \\
    &amp;= |\mathbf{q}| \,\part_i \psi \,\part_s u^i \\
    &amp;= \part_i \psi \;|\mathbf{q}| ~\!\part_s u^i \\
    &amp;= \part_i \psi \;\mathbf{q} \cdot \nabla u^i \\
    &amp;= \part_i \psi \;\mathbf{q} \cdot \mathbf{h}^i \\
    &amp;= \part_i \psi \;q^j \mathbf{h}_j \cdot \mathbf{h}^i \\
    &amp;= \part_i \psi \;q^j \epsilon_j^i \\
    &amp;= \part_i \psi \,q^i \,;
  \end{align}&lt;/math&gt;}}
that is,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla\,\psi = q^i \part_i \psi \,,
  &lt;/math&gt;}}|{{EquationRef|94}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla =~\! q^i \part_i \,.
  &lt;/math&gt;}}|{{EquationRef|94o}}}}
These results likewise look as simple as their Cartesian special cases ({{EquationNote|64}}) and ({{EquationNote|64o}}).&amp;#8201; And by ({{EquationNote|88a}}), the {{math|'''q&amp;sdot;'''&amp;nabla;}} operator again turns out to be the formal dot-product of&amp;#8202; {{math|'''q'''}} and{{math| &amp;nabla;}}.

{{cob}}

=== The curl and divergence operators ===

{{cot}}

To express the curl of a vector field{{math| '''q'''&amp;#8202;,}} we choose the contravariant basis ({{EquationNote|83b}}) and apply identity ({{EquationNote|71c}}):
:{{big|&lt;math&gt;\begin{align}\operatorname{curl}\mathbf{q}
    &amp;= \operatorname{curl} q_j \mathbf{h}^j \\
    &amp;= q_j \operatorname{curl}\mathbf{h}^j + \nabla q_j \times \mathbf{h}^j .
  \end{align}&lt;/math&gt;}}
On the right, the first term vanishes because {{math|'''h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;}}&amp;#8202; is {{math|&amp;nabla;''u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;''}} (and the curl of a gradient is zero). Substituting from ({{EquationNote|93o}}) in the second term, we obtain
:{{big|&lt;math&gt;\operatorname{curl}\mathbf{q}
    = \mathbf{h}^i \part_i q_j \times \mathbf{h}^j
    = \mathbf{h}^i {\times}~\! \mathbf{h}^j ~\!\part_i q_j
  &lt;/math&gt;}}
or, using ({{EquationNote|90b}}),
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{curl}\mathbf{q}
    = J' \epsilon^{ijk\,} \mathbf{h}_k \part_i q_j
  &lt;/math&gt;}}|{{EquationRef|95c}}}}
or, in a more familiar form,
:&lt;math&gt;
    \operatorname{curl}\mathbf{q} \,=\, J'\,
      \begin{vmatrix}
        \mathbf{h}_1 &amp; \part_1 &amp; q_1 \\
        \mathbf{h}_2 &amp; \part_2 &amp; q_2 \\
        \mathbf{h}_3 &amp; \part_3 &amp; q_3
      \end{vmatrix} \,.
  &lt;/math&gt;
Formula ({{EquationNote|95c}}) agrees with a result obtained by Tai with his &quot;symbolic vector&quot; method.&lt;ref&gt;[[#tai-95|Tai, 1995]], p.&amp;#8239;66, eq.&amp;#8239;(9.41).&lt;/ref&gt; It is also what we would get by naively using ({{EquationNote|91b}}) to evaluate {{math|&amp;nabla;&amp;#8202;&amp;times;&amp;#8202;'''q'''&amp;#8202;;}}&amp;#8201; it comes out so simply because each contravariant basis vector{{math| '''h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;}}&amp;#8202; is the actual gradient of{{mvar| u&amp;#8239;&lt;sup&gt;j&lt;/sup&gt;}} and not (e.g.) merely a unit vector in the same direction (remember that {{mvar|q&lt;sub&gt;j&lt;/sub&gt;}} is the component w.r.t.{{math| '''h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;&amp;#8202;,}}&amp;#8202; not{{math| '''h'''&lt;sub&gt;''j''&lt;/sub&gt;}}).

But ({{EquationNote|95c}}) does not end in a subexpression for the operand{{math| '''q'''}}&amp;#8202; and therefore does not directly yield an expression for the curl ''operator''. To find this operator and the divergence operator, we return to the original definitions ({{EquationNote|4g}}), ({{EquationNote|4c}}), and ({{EquationNote|4d}}), noting that they can be combined as
{{NumBlk|:|&lt;math&gt;\nabla ~\!\!* \psi
    \,=\, \tfrac{1}{dV}\!\iint_{\delta S} (\mathbf{\hat{n}}~\!dS*\psi) \,,
  &lt;/math&gt;|{{EquationRef|96}}}}
where {{math|&amp;lowast;}} may be a null for the gradient, a cross for the curl, or a dot for the divergence.{{efn|But not dot-del for the Laplacian, as in ({{EquationNote|19}}), because we want to use an elementary product rule inside the integral.}} Recalling that the value of this expression does not depend on the shape of{{mvar| dS&amp;#8202;}}, let{{mvar| dS&amp;#8202;}} be the parallelepiped defined by the six equicoordinate surfaces at {{mvar|u&lt;sup&gt;i&lt;/sup&gt; }}and{{mvar| u&lt;sup&gt;i&lt;/sup&gt;+du&lt;sup&gt;i&lt;/sup&gt;}}, so that {{mvar|dV}}&amp;#8202; is given by ({{EquationNote|86}}). Then the contribution to the integral from the face at{{math| ''u''&amp;sup1;+''du''&amp;sup1; &amp;#8202;}}is
:&lt;math&gt;\Big[
    \big(\mathbf{h}_2 du^2 \!\times\! \mathbf{h}_3 du^3\big) * \psi
  \Big]_{u^1 + du^1}&lt;/math&gt;
where the square brackets and subscripting mean &quot;evaluated at&quot;. This can be written
:&lt;math&gt;du^2 du^3 \Big[
    (\mathbf{h}_2 {\times}~\! \mathbf{h}_3) * \psi
  \Big]_{u^1 + du^1}&lt;/math&gt;
or, by ({{EquationNote|90.1}}),
:&lt;math&gt;du^2 du^3 \big[J \mathbf{h}^1 ~\!\!* \psi \big]_{u^1 + du^1} \,.&lt;/math&gt;
Similarly, the contribution from the face at{{math| ''u''&amp;sup1;}} (where {{math|'''h'''&lt;sub&gt;1&lt;/sub&gt;}} points inward instead of outward) is
:&lt;math&gt;-du^2 du^3 \big[J \mathbf{h}^1 ~\!\!* \psi \big]_{u^1} \,.&lt;/math&gt;
The sum of the contributions from the two opposite faces can then be written
:&lt;math&gt;du^1 du^2 du^3 ~\!\part_1 \big(J \mathbf{h}^1 ~\!\!* \psi\big) ~,&lt;/math&gt;
so that when we add in the contributions from the other two pairs of opposite faces, the entire integral becomes
:{{big|&lt;math&gt;
    du^1 du^2 du^3 ~\!\part_i \big(J \mathbf{h}^i ~\!\!* \psi\big)
  &lt;/math&gt;}}
(with implicit summation over{{mvar| i}}). Substituting this and ({{EquationNote|86}}) into ({{EquationNote|96}}), we get
{{NumBlk|:|{{big|&lt;math&gt;\nabla ~\!\!* \psi
    = \tfrac{\,1\,}{J} ~\!\part_i \big(J \mathbf{h}^i ~\!\!* \psi\big) \,.
  &lt;/math&gt;}}|{{EquationRef|97}}}}
Now applying the product rule gives
{{NumBlk|:|{{big|&lt;math&gt;\nabla ~\!\!* \psi
    = \mathbf{h}^i ~\!\!* \part_i \psi
      + \tfrac{\,1\,}{J} ~\!\part_i(J \mathbf{h}^i) * \psi \,.
  &lt;/math&gt;}}|{{EquationRef|98}}}}
Here the left-hand side is {{math|&amp;nabla;&amp;#8202;&amp;lowast;''&amp;psi;''}}&amp;#8239; according to our original volume-based definition ({{EquationNote|4g}}) of the {{math|&amp;nabla; }}operator&amp;mdash;which is known to yield the curl or the divergence if&amp;#8202; {{math|&amp;lowast;}} is a cross or a dot, respectively&amp;mdash;whereas the first term on the right is what we would get for {{math|&amp;nabla;&amp;#8202;&amp;lowast;''&amp;psi;''}}&amp;#8239; by using our latest definition ({{EquationNote|93o}}) of the {{math|&amp;nabla; }}operator and allowing{{mvar| &amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; to &quot;pass by&quot; the star in the Wilsonian manner. So, if we can show that the second term on the right is zero, we shall have established the precise sense in which the del-cross and del-dot notations are valid in general coordinates. In that second term, by ({{EquationNote|90e}}),
:{{big|&lt;math&gt;\begin{align}
    \part_i(J \mathbf{h}^i)
    &amp;= \part_i \big(
         \tfrac{\,1\,}{2}~\!\epsilon^{jki}\mathbf{h}_j {\times}~\!\mathbf{h}_k
       \big) \\
    &amp;= \tfrac{\,1\,}{2}~\!\epsilon^{jki}
       \part_i \big(\mathbf{h}_j {\times}~\!\mathbf{h}_k \big) \\[.5ex]
    &amp;= \tfrac{\,1\,}{2}~\!\epsilon^{jki}
       \big(\part_i \mathbf{h}_j \!\times~\!\!\mathbf{h}_k +
            \mathbf{h}_j \!\times~\!\!\part_i \mathbf{h}_k \big) \\[.5ex]
    &amp;= \tfrac{\,1\,}{2}~\!\epsilon^{jki}
       \big(\part_i \part_j \mathbf{r} ~\!\!\times\! \mathbf{h}_k +
            \mathbf{h}_j {\times}~\! \part_i \part_k \mathbf{r} \big) \,,
  \end{align}&lt;/math&gt;}}
where the last line follows by ({{EquationNote|80a}}). But the order of partial differentiation can be switched. So, in the sum over the permutations, for each term in{{math| ''&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;&amp;part;&lt;sub&gt;j&lt;/sub&gt;''&amp;#8239;'''r'''}}&amp;#8239; there is an equal term in{{math| ''&amp;part;&lt;sub&gt;j&lt;/sub&gt;&amp;#8202;&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''r'''}}&amp;#8239; to which the permutation symbol attaches the opposite sign, so that the terms in{{math| ''&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;&amp;part;&lt;sub&gt;j&lt;/sub&gt;''&amp;#8239;'''r'''}}&amp;#8202; cancel. Similarly the terms in{{math| ''&amp;part;&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;&amp;part;&lt;sub&gt;k&lt;/sub&gt;''&amp;#8239;'''r'''}}&amp;#8202; cancel. Thus, as anticipated, the second term in ({{EquationNote|98}}) is zero and we have
{{NumBlk|:|{{big|&lt;math&gt;\nabla ~\!\!* \psi
    = \mathbf{h}^i ~\!\!* \part_i \psi \,.
  &lt;/math&gt;}}|{{EquationRef|99}}}}

If&amp;#8202; {{math|&amp;lowast;}} is a null and {{mvar|&amp;psi;}} is a scalar field {{math|''p''&amp;#8202;,}} then ({{EquationNote|99}}) becomes ({{EquationNote|93g}}) and thus (fortunately!) confirms ({{EquationNote|93o}}) as the form of the del operator in general coordinates.

Now let {{mvar|&amp;psi;}} be a ''vector'' field {{math|'''q'''&amp;#8202;}}.&amp;#8201; If{{math|&amp;#8202; &amp;lowast;}} is a cross, then ({{EquationNote|99}}) becomes
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{curl}\mathbf{q}
    = \mathbf{h}^i ~\!\!\times \part_i \mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|100c}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{curl}
    = \mathbf{h}^i ~\!\!\times \part_i \,.
  &lt;/math&gt;}}|{{EquationRef|100o}}}}
If instead {{math|&amp;lowast;}} is a dot, ({{EquationNote|99}}) becomes
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{div}\mathbf{q}
    = \mathbf{h}^i ~\!\!\cdot \part_i \mathbf{q}
  &lt;/math&gt;}}|{{EquationRef|101d}}}}
or, in operational terms,
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{div}
    = \mathbf{h}^i ~\!\!\cdot \part_i \,.
  &lt;/math&gt;}}|{{EquationRef|101o}}}}
But if we take the {{math|&amp;nabla;}} operator as given by ({{EquationNote|93o}}) and try to construct the curl and divergence operators (in the ''same''&amp;#8202; coordinates) as {{math|&amp;nabla;&amp;#8202;&amp;times;}}&amp;#8202; and {{math|&amp;nabla;'''&amp;sdot;'''}}&amp;#8239; respectively, we get&amp;#8202; {{math|'''h'''&lt;sup&gt;''i''&lt;/sup&gt;&amp;#8239;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;&amp;times;}}&amp;#8239; and&amp;#8239; {{math|'''h'''&lt;sup&gt;''i''&lt;/sup&gt;&amp;#8239;''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''&amp;sdot;'''}}&amp;#8201; respectively [compare ({{EquationNote|61o}}) and ({{EquationNote|62o}})]; and if we then let{{mvar| &amp;part;&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; &quot;pass by&quot; the cross and the dot, we get ({{EquationNote|100o}}) and ({{EquationNote|101o}}), or ({{EquationNote|100c}}) and ({{EquationNote|101d}}) if we include the operand{{math| '''q'''&amp;#8202;}}. Thus ''the del-cross and del-dot notations work in general coordinates''.

Equations ({{EquationNote|100c}}) to ({{EquationNote|101o}}) are apparently due to Tai ([[#tai-95|1995]], eqs.&amp;#8239;9.39, 9.40, 9.34, &amp;amp; 9.35, and text on p.&amp;#8239;66), who derives them, along with the corresponding form of the del operator (his eq.&amp;#8239;9.33), from volume-based definitions expressed in his &quot;symbolic vector&quot; notation. But he does not point out that the curl and divergence operators are obtainable from that del operator, as del-cross and del-dot, via the same &quot;pass by&quot; step that he condemns in the Cartesian context. Speaking of which, we should note that our equations ({{EquationNote|100c}}) to ({{EquationNote|101o}}), apart from the need to pair a superscript with a subscript, are as simple as their Cartesian special cases ({{EquationNote|59c}}), ({{EquationNote|59o}}), ({{EquationNote|60d}}), and ({{EquationNote|60o}}).

In ({{EquationNote|100c}}) and ({{EquationNote|101d}}), it goes without saying that&amp;#8202; {{math|''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''q'''}}&amp;#8202; must be evaluated correctly&amp;mdash;in particular, that if the operand is expressed in terms of non-uniform basis vectors, the non-uniformity must be taken into account. Formula ({{EquationNote|95c}}), for the curl, does not suffer from this complication, because it is already expressed in components w.r.t. the contravariant basis (whose non-uniformity has already been taken into account). To obtain a similarly convenient formula for the divergence, we use components w.r.t. the {{nowrap|''co''&amp;#8202;variant}} basis (i.e., contravariant components): in ({{EquationNote|97}}), if{{math|&amp;#8202; &amp;lowast;}} is a dot and {{mvar|&amp;psi;}} is a vector field{{math| '''q'''&amp;#8202;,}} we have
:{{big|&lt;math&gt;\operatorname{div}\mathbf{q}
    = \tfrac{\,1\,}{J} ~\!\part_i \big(J \mathbf{h}^i {\cdot}\, \mathbf{q}\big)
  &lt;/math&gt;}}
or, by ({{EquationNote|83c}}),
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{div}\mathbf{q}
    = \tfrac{\,1\,}{J} ~\!\part_i \big(J q^i \big) \,.
  &lt;/math&gt;}}|{{EquationRef|102d}}}}
This too agrees with Tai ([[#tai-95|1995]], p.&amp;#8239;65, eq.&amp;#8239;9.37).

{{cob}}

=== The Laplacian ===

{{cot}}

For a scalar operand, applying ({{EquationNote|101o}}) and reversing the &quot;pass by&quot;, we find that the Laplacian operator is
:{{big|&lt;math&gt;\operatorname{div}\nabla
    = \mathbf{h}^i \!\cdot \part_i \nabla
    = \mathbf{h}^i \part_i ~\!\!\cdot \nabla
    = \nabla {\cdot} \nabla = \nabla^2 .
&lt;/math&gt;}}
And by the linearity of the Laplacian, the{{math| &amp;nabla;&lt;sup&gt;2&lt;/sup&gt;}} formulation remains valid if the operand is a fixed linear combination of scalars&amp;mdash;including a vector field, because that is expressible (even if not actually expressed) w.r.t. a uniform basis. (And if it is expressed in terms of a non-uniform basis, the non-uniformity must be taken into account in differentiations.)

In what follows, however, we shall find it convenient to take a different approach. If{{mvar| &amp;psi;}}&amp;#8202; is a scalar field, its gradient as given by ({{EquationNote|93g}}) is&amp;#8202; {{math| '''h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;''&amp;part;&lt;sub&gt;j&lt;/sub&gt;&amp;#8239;&amp;psi;''&amp;#8202;,}} of which the {{mvar|i&amp;#8202;}}th contravariant component is{{math| '''h'''&lt;sup&gt;''i''&lt;/sup&gt;'''&amp;sdot;&amp;#8239;h'''&amp;#8239;&lt;sup&gt;''j''&lt;/sup&gt;''&amp;part;&lt;sub&gt;j&lt;/sub&gt;&amp;#8239;&amp;psi;''&amp;#8202;,}} which takes the place of{{mvar| q&lt;sup&gt;i&lt;/sup&gt;}} in ({{EquationNote|102d}}), so that the divergence of the gradient of{{mvar| &amp;psi; &amp;#8202;}}is
{{NumBlk|:|{{big|&lt;math&gt;\triangle\psi
    = \tfrac{\,1\,}{J} ~\!\part_i
        \big(J \mathbf{h}^i {\cdot}~\! \mathbf{h}^j \part_j \psi \big) \,.
  &lt;/math&gt;}}|{{EquationRef|103L}}}}
This remains well-defined if{{mvar| &amp;psi;}}&amp;#8202; is a generic field (although we still need to deal with any non-uniformity of the basis in which{{mvar| &amp;psi;}}&amp;#8202; might be expressed).

{{cob}}

=== Affine coordinates ===

{{cot}}

If a basis is ''uniform'' (homogeneous), so is its Jacobian. Hence, by ({{EquationNote|90c}}) and ({{EquationNote|90d}}), the dual (contravariant) basis is uniform if and only if the natural (covariant) basis is uniform. A coordinate system in which these bases are uniform is described as '''affine'''. In affine coordinates,
* by ({{EquationNote|80a}}), {{math|''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''r''' }}is uniform, so that the curves on which only one coordinate varies are straight parallel lines; and
* by ({{EquationNote|80b}}), {{math|&amp;nabla;''u&lt;sup&gt;i&lt;/sup&gt;'' }}is uniform, so that the level surfaces of each coordinate (being perpendicular to {{math|&amp;nabla;''u&lt;sup&gt;i&lt;/sup&gt;''}}) are parallel planes.
Obviously Cartesian coordinates are affine; but one can also construct affine coordinate systems in which the three vectors of each basis are not mutually perpendicular and&amp;#10744;or the coordinates have different scales or different units.

We have noted above that the correct application of the del-cross, del-dot, and del-squared notations must allow for non-uniformity of the basis vectors. Obviously this issue does not arise in affine coordinates, including Cartesian coordinates. Hence, while these notations are not (as is sometimes alleged) invalid in other coordinate systems, it would be fair to say that they are safer and more convenient in affine coordinates, including Cartesian coordinates.

{{cob}}

=== Orthogonal coordinates ===

{{cot}}

We know, e.g. from ({{EquationNote|90a}}) and ({{EquationNote|90b}}), that if two bases are reciprocal, the cross-product of the {{mvar|i&amp;#8202;}}th and {{mvar|j&amp;#8202;}}th members of one basis is collinear with the {{mvar|k&amp;#8202;}}th member of the other, if {{math|''i'',&amp;#8201;''j'',&amp;#8239;''k''}}&amp;#8239; are distinct. But if the first basis is ''orthogonal'' (that is, if its three member vectors are mutually orthogonal), the same cross-product is also collinear with the {{mvar|k&amp;#8202;}}th member of the ''same'' basis, so that ''corresponding members of the two bases are collinear''. It follows that ''the natural basis of a coordinate system is orthogonal if and only if the dual basis is orthogonal''. And if the bases are orthogonal, the coordinate system itself is said to be '''orthogonal'''.

Cartesian coordinates are obviously both affine and orthogonal, and we have already implied that there is a class of coordinate systems that are affine but not orthogonal. The most widely-used class of non-Cartesian systems, however, contains the systems that are orthogonal but not affine; this class, of which the cylindrical and spherical systems are the best-known members, is the class of '''curvilinear orthogonal coordinates'''. But we shall drop the word ''curvilinear''&amp;#8202; in order to include Cartesian coordinates as a special case.

In orthogonal coordinates, expressing a member of one basis in terms of its reciprocal basis is especially simple because corresponding members of the two bases are collinear, wherefore we can say
:{{big|&lt;math&gt;
    \mathbf{h}^i =~\! \beta_i \mathbf{h}_i \,,
  &lt;/math&gt;}}
where {{mvar|&amp;beta;&lt;sub&gt;i&lt;/sub&gt;}} is a real variable to be determined (and the single index on the left-hand side means ''no summation''). Substituting this into ({{EquationNote|81i}}) gives
:{{big|&lt;math&gt;
    \beta ~\!= 1/h_i^{~2}
  &lt;/math&gt;}}
where
{{NumBlk|:|{{big|&lt;math&gt;
    h_i = \big|\mathbf{h}_i \big| \,,
  &lt;/math&gt;}}|{{EquationRef|104}}}}
so that
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^i =~\! \mathbf{h}_i \big/ h_i^{~2} \,.
  &lt;/math&gt;}}|{{EquationRef|105}}}}
And substituting that into ({{EquationNote|83c}}), and comparing the result with ({{EquationNote|83d}}), we get
{{NumBlk|:|{{big|&lt;math&gt;
    q^i =~\! q_i \big/ h_i^{~2} \,.
  &lt;/math&gt;}}|{{EquationRef|106}}}}
Comparing ({{EquationNote|104}}) with definition ({{EquationNote|80a}}), we see that {{mvar|h&lt;sub&gt;i&lt;/sub&gt;}} is the magnitude of{{math| ''&amp;part;&lt;sub&gt;i&lt;/sub&gt;''&amp;#8239;'''r'''}}. Accordingly {{mvar|h&lt;sub&gt;i&lt;/sub&gt;}} is called the '''scale factor''' associated with the coordinate{{mvar| u&lt;sup&gt;i&lt;/sup&gt;&amp;#8202;}}; it is the factor by which we multiply a small change in{{mvar| u&lt;sup&gt;i&lt;/sup&gt;}} to obtain the magnitude of the consequent change in position.{{efn|Hsu ([[#hsu-84|1984]], p.&amp;#8239;171) implies that the scale factors are also called &quot;metric coefficients&quot;, and Tai ([[#tai-94|1994]], [[#tai-95|1995]]) prefers the latter term. This is loose terminology because, in general, the ''metric coefficient''&amp;#8202; is defined as
:{{math|''g&lt;sub&gt;ij&lt;/sub&gt;'' {{=}} '''h'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8239;'''&amp;sdot;&amp;#8201;h'''&lt;sub&gt;''j''&lt;/sub&gt;&amp;#8201;}}.
Hence, in the special case of orthogonal coordinates, we have {{math|''g&lt;sub&gt;ij&lt;/sub&gt;''&amp;#8201;{{=}}&amp;#8201;0}}&amp;#8202; for {{math|''i&amp;#8239;&amp;ne;&amp;#8239;j''&amp;#8202;,}}&amp;#8239; and&amp;#8202; {{math|''g&lt;sub&gt;ii&lt;/sub&gt;&amp;#8239;{{=}}&amp;#8201;h&lt;sub&gt;i&lt;/sub&gt;''&lt;sup&gt;2&lt;/sup&gt;}}&amp;#8202; [no sum]. Thus the scale factors are not special cases of the metric coefficients, but the ''square roots''&amp;#8202; of special cases of the metric coefficients (''cf''. [[#tai-95|Tai, 1995]], p.&amp;#8239;43, line 3).}}

If we now define
{{NumBlk|:|&lt;math&gt;\varsigma \,=~\! \begin{cases}
    +1 &amp;\mathsf{for~a~right{\operatorname{-}}handed~system} \\[.5ex]
    -1 &amp;\mathsf{for~a~left{\operatorname{-}}handed~system} ~,
  \end{cases}&lt;/math&gt;|{{EquationRef|107}}}}
then, due to the orthogonality,&amp;#8201; ({{EquationNote|87}}) and ({{EquationNote|92}}) are respectively reduced to
{{NumBlk|:|&lt;math&gt;
    J = \varsigma\, h_1 h_2 h_3
  &lt;/math&gt;|{{EquationRef|108}}}}
and
{{NumBlk|:|&lt;math&gt;
    J' = \frac{\,1\,}{J} = \frac{1}{\varsigma\, h_1 h_2 h_3}
       = \frac{\varsigma}{h_1 h_2 h_3}
  &lt;/math&gt;|{{EquationRef|109}}}}
&amp;mdash;although, for brevity, we shall sometimes leave things in terms of{{mvar| J}}.

At this point, we ''could''&amp;#8202; substitute ({{EquationNote|105}}) and ({{EquationNote|106}}) into earlier equations and obtain a suite of formulae for the differential operators in terms of the covariant basis and {{nowrap|''co''&amp;#8202;variant}} components! But we can avoid this confusing breach of convention by ''normalizing'' the basis vectors.

An '''orthonormal''' basis is one whose members are mutually orthogonal ''unit'' vectors. The assumption of unit vectors is introduced so late because it is more useful with orthogonality than without. If one basis consisted of unit vectors that were not all orthogonal, then the reciprocal basis vectors given by ({{EquationNote|90c}}) or ({{EquationNote|90d}}) would not all be unit vectors.{{efn|Outline of proof: If the reciprocal vectors were unit vectors, then the angles between the ''original''&amp;#8202; unit vectors would need to be equal, in order that their cross products have the same magnitude as their scalar triple product (Jacobian); and the latter condition requires the common angle to be 90&amp;deg;.}} But if the basis{{math| ('''h'''&lt;sub&gt;''i''&lt;/sub&gt;)}} consists of orthogonal unit vectors, equation ({{EquationNote|105}}) implies that the reciprocal basis consists of the ''same'' vectors; and the converse is also true, by the symmetry of the reciprocity relations. Thus ''an orthonormal basis is its own reciprocal''. Hence, if we choose an orthonormal basis, we do not need superscripts to distinguish the reciprocal basis from the original, or to distinguish components w.r.t. the latter basis from those w.r.t. the former.

An orthonormal basis is not generally covariant, because it doesn't stretch with the coordinate grid (although it does rotate with the grid). Neither is it generally contravariant, because its reciprocal (i.e. itelf) is not generally covariant. Hence, if a ''non''&amp;#8202;-orthonormal natural or dual basis of an orthogonal coordinate system is ''normalized'' (replaced by unit vectors in the same directions), the resulting orthonormal basis is not covariant or contravariant, and components with respect thereto are not contravariant or covariant, and the new basis vectors are not given in terms of the coordinates by ({{EquationNote|80a}}) or ({{EquationNote|80b}}); the basis is therefore described as a '''non-coordinate basis'''. By default, the indices of the orthonormal basis vectors and associated components are written as subscripts, but these are not indicative of covariance. The coordinates themselves remain contravariant (e.g., if the grid dilates, the same movement in space corresponds to ''smaller'' changes in the coordinates); but, for want of covariant basis vectors to pair them with, we tend to write the coordinates with subscripts when the basis is orthonormal.

Nevertheless, it is convenient to have one basis instead of two. Moreover, the components of a vector w.r.t. an orthonormal basis are '''physical components''': they have the same dimension (same units) as the represented vector, and they are the components that we would have in mind if we wanted to ''measure'' the &quot;components&quot; in the directions of the basis vectors. Hence an orthonormal basis is called a '''physical basis'''. Accordingly, it is indeed common practice to normalize the basis vectors of orthogonal coordinate systems. This together with the prevalence of such coordinate systems helps to account for the familiarity of subscripts as indices, and for the jarring unfamiliarity of superscript indices when general (possibly non-orthogonal) coordinates are encountered for the first time.

To normalize the covariant basis, let{{math| '''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;}} (as usual) be the unit vector in the direction of{{math|&amp;#8202; '''h'''&lt;sub&gt;''i''&lt;/sub&gt;&amp;#8202;.}} Then, by ({{EquationNote|104}}),
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}_i =~\! h_i \mathbf{\hat{h}}_i
  &lt;/math&gt;}}|{{EquationRef|110}}}}
(again with no summation, due to the single index on the left). Hence ({{EquationNote|105}}) becomes:
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{h}^i =~\! \mathbf{\hat{h}}_i \big/ h_i \,.
  &lt;/math&gt;}}|{{EquationRef|111}}}}
A vector field {{math|'''q'''}} is expressed in components w.r.t. the basis{{math| ('''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;)}} as
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q} = \hat{q_i} \;\!\mathbf{\hat{h}}_i
  \qquad&lt;/math&gt;}}|{{EquationRef|112}}}}
(with summation), where
{{NumBlk|:|{{big|&lt;math&gt;
    \hat{q_i} =~\! \mathbf{q} \!\cdot\! \mathbf{\hat{h}}_i \,.
  \qquad&lt;/math&gt;}}|{{EquationRef|113}}}}
(Here the hat on{{math| ''q''&amp;#770;&lt;sub&gt;''i''&lt;/sub&gt;}} is needed to distinguish the coefficient of{{math| '''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;}} from the coefficient of{{math| '''h'''&lt;sup&gt;''i''&lt;/sup&gt;,}} and indicates that{{math| ''q''&amp;#770;&lt;sub&gt;''i''&lt;/sub&gt;}} is the coefficient of a unit vector&amp;mdash;''not'' that{{math| ''q''&amp;#770;&lt;sub&gt;''i''&lt;/sub&gt;}} has unit magnitude.) Taking{{mvar| q&lt;sub&gt;i&lt;/sub&gt;}} as given by ({{EquationNote|83d}}) and applying ({{EquationNote|110}}) and ({{EquationNote|113}}), we get
{{NumBlk|:|{{big|&lt;math&gt;
    q_i =~\! h_i \hat{q_i} \,,
  \qquad&lt;/math&gt;}}|{{EquationRef|114}}}}
whence ({{EquationNote|106}}) gives
{{NumBlk|:|{{big|&lt;math&gt;
    q^i =~\! \hat{q_i} \big/ h_i \,.
  \qquad&lt;/math&gt;}}|{{EquationRef|115}}}}

Equation ({{EquationNote|110}}) quantifies the non-covariance of the orthonormal basis; substituting ({{EquationNote|110}}) into ({{EquationNote|85}}), we find that the components of{{math| ''d'''''r'''}} with respect to{{math| '''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;}}&amp;#8202; are not simply{{math| ''du&lt;sup&gt;i&lt;/sup&gt;'',}} but{{math| ''h&lt;sub&gt;i&lt;/sub&gt;&amp;#8239;du&lt;sup&gt;i&lt;/sup&gt;''}}&amp;#8202; [no sum]. So, as the coordinates{{mvar| u&lt;sup&gt;i&lt;/sup&gt;}} are still contravariant, the orthonormal basis vectors{{math| '''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;}} are not covariant unless the scale factors{{mvar| h&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; are equal to{{math| 1}}&amp;#8202;&amp;mdash;&amp;#8239;that is, unless the coordinates are Cartesian (except possibly for the handedness). And in Cartesian coordinates we can use subscripts throughout. This is another reason why, when using an orthonormal basis, we might as well write the coordinates as{{math| ''u&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;}}.

We can now re-express dot- and cross-products w.r.t. the orthonormal basis{{math| ('''h&amp;#770;'''&lt;sub&gt;''i''&lt;/sub&gt;)}}. If we apply ({{EquationNote|114}}) and ({{EquationNote|115}}) in ({{EquationNote|88a}}) or ({{EquationNote|88b}}), the scale factors cancel and we are left with
{{NumBlk|:|{{big|&lt;math&gt;\mathbf{v} \cdot \mathbf{q}
    =~\! \hat{v_i} \hat{q_i} \,,
  &lt;/math&gt;}}|{{EquationRef|116}}}}
as if the coordinates were Cartesian. And if we apply ({{EquationNote|108}}), ({{EquationNote|115}}), and ({{EquationNote|111}}) in ({{EquationNote|91a}}), the product of the scale factors cancels and we are left with
:{{big|&lt;math&gt;\mathbf{v} \!\times\! \mathbf{q} =~\!
    \varsigma~\!\epsilon_{ijk\,} \hat{v_i} \hat{q_j} \;\!\mathbf{\hat{h}}_k \,,
  &lt;/math&gt;}}
again as if the coordinates were Cartesian, except that the handedness symbol {{mvar|&amp;varsigma;}}&amp;#8202; gives a change of sign for left-handed coordinates. The last result is confirmed by applying ({{EquationNote|109}}), ({{EquationNote|114}}), and ({{EquationNote|110}}) in ({{EquationNote|91b}}). It can also be written
{{NumBlk|:|&lt;math&gt;\mathbf{v} \!\times\! \mathbf{q} \,=\,
    \varsigma\, \begin{vmatrix}
      \hat{v_1} &amp; \hat{q_1} &amp; \mathbf{\hat{h}}_1 \\
      \hat{v_2} &amp; \hat{q_2} &amp; \mathbf{\hat{h}}_2 \\
      \hat{v_3} &amp; \hat{q_3} &amp; \mathbf{\hat{h}}_3
    \end{vmatrix} \,.
  &lt;/math&gt;|{{EquationRef|117}}}}

We can similarly re-express the first-order differential operators. Applying ({{EquationNote|111}}) in ({{EquationNote|93o}}), ({{EquationNote|101o}}), and ({{EquationNote|100o}}) gives respectively
{{NumBlk|:|{{big|&lt;math&gt;
    \nabla = \tfrac{1}{\,h_{\scriptstyle i}}\;\!\mathbf{\hat{h}}_i \part_i \,,
  &lt;/math&gt;}}|{{EquationRef|118}}}}
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{div}
      = \tfrac{1}{\,h_{\scriptstyle i}}\;\!\mathbf{\hat{h}}_i ~\!\!\cdot \part_i \,,
  &lt;/math&gt;}}|{{EquationRef|119}}}}
and
{{NumBlk|:|{{big|&lt;math&gt;
    \operatorname{curl}
      = \tfrac{1}{\,h_{\scriptstyle i}}\;\!\mathbf{\hat{h}}_i ~\!\!\times \part_i \,.
  &lt;/math&gt;}}|{{EquationRef|120}}}}
And applying ({{EquationNote|115}}) in ({{EquationNote|94o}}) and ({{EquationNote|102d}}) gives
{{NumBlk|:|{{big|&lt;math&gt;
    \mathbf{q}\;\!{\cdot}\nabla
      = \tfrac{1}{\,h_{\scriptstyle i}} \;\!\hat{q_i} \part_i
  &lt;/math&gt;}}|{{EquationRef|121}}}}
and
{{NumBlk|:|{{big|&lt;math&gt;\operatorname{div}\mathbf{q}
    = \tfrac{\,1\,}{J} ~\!\part_i \Big(\tfrac{J}{\,h_{\scriptstyle i}} ~\!\hat{q_i} \Big) \,.
  &lt;/math&gt;}}|{{EquationRef|122}}}}
And applying ({{EquationNote|109}}), ({{EquationNote|110}}), and ({{EquationNote|114}}) in ({{EquationNote|95c}}) gives
:{{big|&lt;math&gt;\operatorname{curl}\mathbf{q}
    = \tfrac{\,1\,}{J} ~\!\epsilon_{ijk\,}
      h_k \mathbf{\hat{h}}_k ~\!\part_i \big(h_j \hat{q_j} \big)
  &lt;/math&gt;}}
or, in determinant form,
{{NumBlk|:|&lt;math&gt;
    \operatorname{curl}\mathbf{q} \,=\, \frac{\varsigma}{h_1 h_2 h_3}\,
      \begin{vmatrix}
        h_1 \mathbf{\hat{h}}_1 &amp; \part_1 &amp; h_1 \hat{q_1} \\
        h_2 \mathbf{\hat{h}}_2 &amp; \part_2 &amp; h_2 \hat{q_2} \\
        h_3 \mathbf{\hat{h}}_3 &amp; \part_3 &amp; h_3 \hat{q_3}
      \end{vmatrix} \,.
  &lt;/math&gt;|{{EquationRef|123}}}}

For the Laplacian, applying ({{EquationNote|111}}) twice in ({{EquationNote|103L}}) gives
:{{big|&lt;math&gt;\triangle\psi =
    \tfrac{\,1\,}{J} ~\!\part_i \Big(
      \tfrac{J}{h_i h_j} (\mathbf{\hat{h}}_i {\cdot}~\! \mathbf{\hat{h}}_j)
      ~\!\part_j \psi
    \Big) \,,
  &lt;/math&gt;}}
where the parenthesized dot-product is simply{{mvar| &amp;delta;&lt;sub&gt;ij&lt;/sub&gt;&amp;#8202;}}.&amp;#8201; Selecting the non-zero terms, we are left with
{{NumBlk|:|{{big|&lt;math&gt;\triangle\psi =
    \tfrac{\,1\,}{J} ~\!\part_i \Big(
      \tfrac{J}{\,h_{\scriptstyle i}^{~2}} ~\!\part_i \psi
    \Big) \,.
  &lt;/math&gt;}}|{{EquationRef|124}}}}

Working entirely within the coordinates{{math| ''u&lt;sub&gt;i&lt;/sub&gt;''&amp;#8202;,}} we can use equations ({{EquationNote|118}}), ({{EquationNote|121}}) to ({{EquationNote|123}}), and ({{EquationNote|124}}) for scalar{{math| ''&amp;psi;''&amp;#8202;,}} provided that we know the scale factors in terms of{{mvar| u&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;}}.{{efn|In ({{EquationNote|124}}), if {{mvar|&amp;psi;}}&amp;#8202; is a ''vector'' field, we also need the derivatives of its basis vectors w.r.t.{{mvar| u&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;}} in terms of{{mvar| u&lt;sub&gt;i&lt;/sub&gt;&amp;#8202;}}.}}&amp;#8201; And we can find the scale factors in terms of{{mvar| u&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; if we know the Cartesian coordinates{{mvar| x&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; in terms of{{mvar| u&lt;sub&gt;i&lt;/sub&gt;}}.&amp;#8201; For then the position vector can be written
:{{big|{{math|'''r''' {{=}} ''x&lt;sub&gt;j&lt;/sub&gt;''&amp;#8239;'''e'''&lt;sub&gt;''j''&lt;/sub&gt; ,}}}}
whence
:{{big|{{math|'''h'''&lt;sub&gt;''i''&lt;/sub&gt; {{=}} ''&amp;part;&lt;sub&gt;u&lt;sub&gt;i&lt;/sub&gt;&lt;/sub&gt;''&amp;#8239;'''r''' {{=}} ''&amp;part;&lt;sub&gt;u&lt;sub&gt;i&lt;/sub&gt;&lt;/sub&gt;&amp;#8201;x&lt;sub&gt;j&lt;/sub&gt;''&amp;#8201;'''e'''&lt;sub&gt;''j''&lt;/sub&gt; ,}}}}
so that the scale factors can be found from
:{{big|{{math|''h&lt;sub&gt;i&lt;/sub&gt;''&lt;sup&gt;2&lt;/sup&gt; {{=}} &amp;sum;&lt;sub&gt;&amp;#8202;''j'' &lt;/sub&gt;(''&amp;part;&lt;sub&gt;u&lt;sub&gt;i&lt;/sub&gt;&lt;/sub&gt;&amp;#8201;x&lt;sub&gt;j&lt;/sub&gt;'')&lt;sup&gt;2&lt;/sup&gt;.}}}}

In ({{EquationNote|121}}) to ({{EquationNote|123}}), the hat on{{math| ''q''&amp;#770;&lt;sub&gt;''i''&lt;/sub&gt;}} was needed because we treated the orthonormal basis as a special case, having used a hatless{{mvar| q&lt;sub&gt;i&lt;/sub&gt;}}&amp;#8202; in less special cases; the hat would not have been needed if we had assumed an orthonormal basis at the outset. In more elementary introductions to curvilinear orthogonal coordinates, the basis vectors are indeed chosen as unit vectors and consequently as orthonormal vectors. Hence, if the coordinates are called&lt;math&gt;~u,v,w\,&lt;/math&gt; and the respective basis vectors are called&lt;math&gt;~\mathbf{e}_u , \mathbf{e}_v , \mathbf{e}_w&lt;/math&gt; (understood to be unit vectors), the components of the vector{{math| '''q'''}} w.r.t. that basis are called&lt;math&gt;~q_u , q_v , q_w ~\!,\,&lt;/math&gt; with no hats. In this notation, in which sums are written out longhand without numerical indices, it is convenient also to write out the Jacobian in full, in order to exploit cancellations of scale factors. If the Jacobian appears in both a numerator inside parentheses and a denominator outside, the handedness symbol{{mvar| &amp;varsigma;}}&amp;#8202; also cancels. Thus the equations numbered ({{EquationNote|116}}) to ({{EquationNote|124}}) can be rewritten as, respectively,
{{NumBlk||&lt;math&gt;\begin{align}
    \mathbf{f} \cdot \mathbf{q} ~\!&amp;=~\!
      f_u q_u + f_v q_v + f_w q_w \\[1ex]
    \mathbf{f} ~\!\!\times\! \mathbf{q} ~\!&amp;=~\!
      \varsigma\, \begin{vmatrix}
        f_u &amp; q_u &amp; \mathbf{e}_u \\
        f_v &amp; q_v &amp; \mathbf{e}_v \\
        f_w &amp; q_w &amp; \mathbf{e}_w
      \end{vmatrix} \\[1ex]
    \nabla &amp;=
      \tfrac{1}{~\!h_{\scriptstyle u}}\;\!\mathbf{e}_u \part_u +
      \tfrac{1}{~\!h_{\scriptstyle v}}\;\!\mathbf{e}_v \part_v +
      \tfrac{1}{~\!h_{\scriptstyle w}}\;\!\mathbf{e}_w \part_w \\[.5ex]
    \operatorname{div} &amp;=
      \tfrac{1}{~\!h_{\scriptstyle u}} \mathbf{e}_u ~\!\!\cdot \part_u +
      \tfrac{1}{~\!h_{\scriptstyle v}} \mathbf{e}_v ~\!\!\cdot \part_v +
      \tfrac{1}{~\!h_{\scriptstyle w}} \mathbf{e}_w ~\!\!\cdot \part_w \\[.5ex]
    \operatorname{curl} ~\!&amp;=
      \tfrac{1}{~\!h_{\scriptstyle u}} \mathbf{e}_u ~\!\!\times \part_u +
      \tfrac{1}{~\!h_{\scriptstyle v}} \mathbf{e}_v ~\!\!\times \part_v +
      \tfrac{1}{~\!h_{\scriptstyle w}} \mathbf{e}_w ~\!\!\times \part_w\\[.5ex]
    \mathbf{q}\;\!{\cdot}\nabla &amp;=
      \tfrac{1}{~\!h_{\scriptstyle u}} \;\!q_u \part_u +
      \tfrac{1}{~\!h_{\scriptstyle v}} \;\!q_v \part_v +
      \tfrac{1}{~\!h_{\scriptstyle w}} \;\!q_w \part_w \\[.5ex]
    \operatorname{div}\mathbf{q} ~\!&amp;= \tfrac{1}{h_u h_v h_w} \Big(
        \part_u (h_v h_w q_u) + \part_v (h_w h_u q_v) + \part_w (h_u h_v q_w)
      \Big) \\[.5ex]
    \operatorname{curl}\mathbf{q} ~\!&amp;=~\! \frac{\varsigma}{h_u h_v h_w}\,
      \begin{vmatrix}
        h_u \mathbf{e}_u &amp; \part_u &amp; h_u q_u \\
        h_v \mathbf{e}_v &amp; \part_v &amp; h_v q_v \\
        h_w \mathbf{e}_w &amp; \part_w &amp; h_w q_w
      \end{vmatrix} \\[.5ex]
    \triangle\psi ~\!&amp;=
      \tfrac{1}{h_{\scriptstyle u\;\!}h_{\scriptstyle v\;\!}h_{\scriptstyle w}\!}
      \bigg\{\!
        \tfrac{\part}{\part u\!}
          \Big(\!\tfrac{h_v h_w}{h_u\;\!}\tfrac{\part\psi}{\part u}\!\Big)
        \!+~\!\! \tfrac{\part}{\part v\!}
          \Big(\!\tfrac{h_w h_u}{\;\!h_v}\tfrac{\part\psi}{\part v}\!\Big)
        \!+~\!\! \tfrac{\part}{\part w\!}
          \Big(\!\tfrac{h_u h_v}{\;\!h_w}\tfrac{\part\psi}{\part w}\!\Big)
      \!\bigg\} .
  \end{align}&lt;/math&gt;|{{EquationRef|125}}}}
Only in the cross-product and the curl does the handedness factor{{mvar| &amp;varsigma;}}&amp;#8202; make any difference. If the system is right-handed&amp;mdash;as is also often assumed at the outset&amp;mdash;this factor is replaced by{{math| 1}}.

For some readers, equation group ({{EquationNote|125}}) will announce a return to familiar territory. For the writer, it offers a convenient place to stop.

{{cob}}

== Appendix: Mathematizing Huygens' principle ==

{{cot}}

If a wavelike disturbance originating ''outside''&amp;#8202; a region{{math| ''V'',}} bounded by a surface{{math| ''S''&amp;#8202;,}} enters the region, it must do so through the surface{{mvar| S}}.&amp;#8201; Unless we believe in &quot;action at a distance&quot;, we must conclude that ''the behavior of the wave function throughout the region is fully determined by its behavior on the bounding surface''. That reasoning, being qualitative, does not tell us precisely what aspects of the behavior at the boundary determine the behavior throughout the region, or how. In this appendix, we shall answer these questions using tools of vector analysis. The aim is to express the wave function in the region{{mvar| V}} as a surface integral, over the bounding surface{{math| ''S''&amp;#8202;,}} of an integrand related to the wave function incident at a general point on that surface.

'''[[w:Huygens' principle|Huygens' principle]]''' asserts not only that the behavior of the wave function throughout the region (containing no sources) is determined by the behavior at the boundary, but also that the behavior at the boundary is equivalent to a distribution of sources over the boundary, so that the wave function throughout the region is ''as if''&amp;#8202; the original sources outside the region ('''primary sources''') were ''replaced''&amp;#8202; by sources distributed over the boundary ('''secondary sources''').{{efn|Notice that the desired secondary sources are ''not''&amp;#8202; segments of the moving wavefronts, but segments of a stationary surface influenced by the passing waves. Compare Huygens' original statement: &quot;that ''each particle of matter''&amp;#8202; in which a wave spreads, ought not to communicate its motion only to the next particle which is in the straight line drawn from the luminous point, but that it also imparts some of it necessarily to all the others which touch it and which oppose themselves to its movement. So it arises that around each particle there is made a wave of which ''that particle''&amp;#8202; is the centre&quot; ([[#huygens-1690-thompson|Huygens, 1690, tr.&amp;#8239;Thompson]], p.&amp;#8239;19; my emphasis). Huygens chooses secondary sources on the same primary wavefront at the same time for the purpose of constructing the &quot;continuation&quot; of the wavefront (the same wavefront at a later time) in the same medium (''ibid.'', pp.&amp;#8239;19,&amp;#8239;50–51), but ''not''&amp;#8202; for the purpose of constructing a wavefront reflected or refracted at an interface between two media; for the latter purpose, he chooses secondary sources at various points on the reflecting or refracting surface, although the primary wavefront reaches those points at various times (''ibid.'', pp.&amp;#8239;23–4,&amp;#8239;35–7,&amp;#8239;etc.).}} We shall find that by appropriately arranging the integrand for the wave function inside the region, we can indeed recognize the distribution of boundary sources that would generate the wave function.

{{cob}}

=== Hints ===

{{cot}}

Let {{math|''&amp;psi;''('''r''',&amp;#8239;''t'')}} be the primary wave function, and let {{math|'''r&amp;prime;'''}} be the position of the observation point ('''field point''') at a distance{{mvar| s}}&amp;#8202; from position{{math| '''r'''}}. If the surface integrand is the wave function at{{math| '''r&amp;prime;'''}} due to a secondary source-strength density, it will not only be related to the primary wave function{{mvar| &amp;psi;}} at a general point{{math| '''r'''}} on{{mvar| S}}, but will also be delayed by the propagation time from {{math|'''r'''}} to{{math| '''r&amp;prime;'''}}, and attenuated in accordance with the propagation distance{{mvar| s}}. Hence the integrand (or at least the dominant term thereof) will be proportional to
{{NumBlk|:|{{big|{{math|{{sfrac|&amp;#8239;''s''&amp;#8239;}}&amp;#8202;''&amp;psi;''('''r''', ''t&amp;#8201;&amp;minus;&amp;#8201;s''&amp;#10744;''c'') .}}}}|{{EquationRef|126}}}}
But the necessary operations may cause{{mvar| &amp;psi;}} to be replaced by, e.g., one of its derivatives or a linear combination of its derivatives; such a replacement would be &quot;proportional&quot; to{{mvar| &amp;psi;}} in the requisite sense.

Of course we would like our distribution of secondary sources to be valid for an arbitrarily shaped boundary{{mvar| S}}. This preference will be easier to satisfy if the distribution of secondary sources, by itself, produces a zero wave function outside{{mvar| V}}&amp;#8202;&amp;mdash;in other words, ''no backward secondary waves''&amp;#8202;&amp;mdash;because in that case, even if {{mvar|S}}&amp;#8202; is concave outward, the wave function inside{{mvar| V}}&amp;#8239; will not be complicated by &quot;backward&quot; waves generated at one point on{{mvar| ''S''}}&amp;#8202; and entering{{mvar| V}}&amp;#8202; through another point on{{mvar| S}}.&amp;#8201; Accordingly, we would like our surface integral to be equal to the ''volume''&amp;#8202; integral over{{mvar| V}}&amp;#8239; of
{{NumBlk|:|{{big|{{math|''&amp;psi;''('''r''',&amp;#8239;''t'')&amp;#8239;''&amp;delta;''('''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;''') ,}}}}|{{EquationRef|127}}}}
because that volume integral will be {{math| ''&amp;psi;''('''r&amp;prime;''',&amp;#8239;''t'')}}&amp;#8202; if {{math|'''r&amp;prime;'''}} is inside{{mvar| V}} (where the secondary waves are forward), but zero if it is outside (where any secondary waves are backward).

Relating the volume integral of ({{EquationNote|127}}) to the surface integral of ({{EquationNote|126}}) would seem to require a surface-to-volume '''integral identity''' involving two different fields. Some promising identities are available; but, as we shall see, they tend to treat the two fields symmetrically, and they get simpler if the two fields have more properties in common. We might therefore seek fields with more in common than ({{EquationNote|126}}) and ({{EquationNote|127}}). In the first factor in ({{EquationNote|127}}),&amp;#8201; {{mvar|t}} can be replaced by&amp;#8202; {{math|''t&amp;#8202;&amp;minus;&amp;#8202;s''&amp;#10744;''c''}}&amp;#8202; because the second factor is zero for non-zero{{mvar| s}}. Thus the second factor in ({{EquationNote|127}}), by selecting the time, makes the primary wave function{{math| ''&amp;psi;''('''r''',&amp;#8239;''t'')}} equivalent to the second factor in ({{EquationNote|126}}). That primary wave function is of course a solution of the wave equation in{{mvar| V}}. So, if the factor
:{{big|{{math|''&amp;delta;''('''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;''')}}}}
in ({{EquationNote|127}}) can be replaced by solution of the wave equation with an equivalent &quot;selecting&quot; property, and especially if that solution includes the factor {{math|1&amp;#10744;''s''}} in ({{EquationNote|126}}), perhaps we can pick that solution and the primary wave function as the two fields to substitute into the integral identity. The &quot;solution&quot; that suggests itself is
{{NumBlk|:|{{big|{{math|{{sfrac|&amp;#8239;''s''&amp;#8239;}}&amp;#8202;''&amp;delta;''(''t&amp;#8201;+&amp;#8201;s''&amp;#10744;''c'') ,}}}}|{{EquationRef|128}}}}
where the defining properties of {{math|''&amp;delta;''(''t'')}} are that its integral over all ''time'' is zero and, ideally, that it is zero except at {{math|''t''&amp;#8239;{{=}}&amp;#8202;0}}; but, to ensure that {{math|''&amp;delta;''(''t'')}} is sufficiently differentiable for our purposes, we shall allow it to be a smooth function which is zero except within a negligibly short interval around{{math| ''t''&amp;#8239;{{=}}&amp;#8202;0}}. Solution ({{EquationNote|128}}) describes an ''incoming'' spherical wave converging on{{math| '''r&amp;prime;'''}} [recall the discussion of ({{EquationNote|54a}}) above], which is appropriate because, for an observer at{{math| '''r&amp;prime;'''}}, the secondary waves are incoming; to put it more precisely, the delta function selects the time{{math| ''t&amp;#8239;{{=}}&amp;#8202;&amp;minus;s''&amp;#10744;''c''}}, which is the time of emission of the secondary waves that affect the wave function at{{math| '''r&amp;prime;'''}} at{{math| ''t''&amp;#8239;{{=}}&amp;#8202;0}} (which is a general time, because the origin of{{mvar| t}} is arbitrary). Moreover, as{{math| ''t''&amp;rightarrow;&amp;#8202;0&lt;sup&gt;&amp;minus;&lt;/sup&gt;}}, the temporal delta function in ({{EquationNote|128}}) looks like the spatial delta function in ({{EquationNote|127}}).

So let us tentatively pick the primary wave function {{math| ''&amp;psi;''('''r''',&amp;#8239;''t'')}} and the auxiliary wave function ({{EquationNote|128}}) as the two fields to be related by the integral identity&amp;mdash;which we must now choose.

{{cob}}

=== Green's identities ===

{{cot}}

If&lt;math&gt;~u&lt;/math&gt; and&lt;math&gt;~v&lt;/math&gt; are scalar fields, then by identity ({{EquationNote|71d}}),
:&lt;math&gt;\mathrm{div}(u~\!\nabla v)
    \equiv u~\!\triangle v + \nabla u \cdot~\!\! \nabla v \,.
  &lt;/math&gt;
Integrating both sides over a volume{{math| ''V''}} enclosed by a surface{{math| ''S''&amp;#8202;,}} and applying the divergence theorem on the left, we get
:&lt;math&gt;\iint_S u~\!\nabla v \cdot \mathbf{\hat{n}}~\!dS \,\equiv
    \iiint_V \big(u~\!\triangle v + \nabla u \cdot~\!\! \nabla v \big)~\!dV \,,
  &lt;/math&gt;
where &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is the unit normal to{{mvar| S}}&amp;#8202; pointing out of{{mvar| V}}. This integral equation is called '''[[w:George Green (mathematician)|Green]]'s first identity'''. Switching the roles of&lt;math&gt;~u&lt;/math&gt; and&lt;math&gt;~v&lt;/math&gt; yields a second integral equation, which can be subtracted from the first to obtain
:&lt;math&gt;
    \iint_S \!\big(u~\!\nabla v - v~\!\nabla u \big) \cdot \mathbf{\hat{n}}~\!dS
    \,\equiv \iiint_V \!\big(u~\!\triangle v - v~\!\triangle u \big)~\!dV \,;
  &lt;/math&gt;
this is '''Green's second identity'''. If {{mvar|n}} is the normal distance from{{mvar| S}} (positive outside{{math| ''V'',}} negative inside), then, by relation ({{EquationNote|9g}}) between the gradient and the directional derivative, we can rewrite Green's second identity in the alternative form
{{NumBlk|:|&lt;math&gt;
    \iint_S \!\big(u~\!\part_n v - v~\!\part_n u\big)~\!dS
    \,\equiv \iiint_V \!\big(u~\!\triangle v - v~\!\triangle u\big)~\!dV \,,
  &lt;/math&gt;|{{EquationRef|129}}}}
which remains meaningful if one of the two operands is a ''generic'' field. And indeed, by the linearity of the various operators, the identity remains valid in that case (which is not always pointed out).

{{cob}}

=== Kirchhoff's integral theorem ===

{{cot}}

Now, as foreshadowed above,&lt;ref&gt;The following demonstration of the Kirchhoff integral theorem is indebted to Stratton ([[#stratton-41|1941]], pp.&amp;#8239;424–8), especially as regards the choice of the &quot;auxiliary&quot; wave function &lt;math&gt;v&lt;/math&gt; (my nomenclature) and the insight that the time origin is arbitrary (p.&amp;#8239;427). However, Stratton's treatment does not consider the case with {{math|'''r&amp;prime;'''}} outside{{mvar| V}}, handles the &quot;inside{{mvar|&amp;#8201;V&amp;#8239;}}&quot; case differently, and takes a less heuristic approach, without introductory &quot;hints&quot;.&lt;/ref&gt; let us see what happens if we put
{{NumBlk|:|&lt;math&gt;u = \psi(\mathbf{r},t)&lt;/math&gt;|{{EquationRef|130u}}}}
and
{{NumBlk|:|&lt;math&gt;v = \tfrac{\,1\,}{s}~\!\delta\big(t+s/c\big)&lt;/math&gt;|{{EquationRef|130v}}}}
in ({{EquationNote|129}}). As &lt;math&gt;u&lt;/math&gt; satisfies the wave equation in{{mvar| V}}, we have
{{NumBlk|:|&lt;math&gt;\triangle u
  = \tfrac{1}{c^2} \ddot{u} \,.&lt;/math&gt;|{{EquationRef|131u}}}}
With &lt;math&gt;v&lt;/math&gt;&amp;#8201; we need to be more careful, because &lt;math&gt;v&lt;/math&gt; is undefined at{{math| '''r&amp;prime;'''}}, which may be inside{{mvar| V}}.  By rule ({{EquationNote|54}}), the D'Alembertian of &lt;math&gt;v&lt;/math&gt; (with the {{math|&amp;#9744;}} operator written out in full) is
:&lt;math&gt;\triangle v - \tfrac{1}{c^2} \ddot{v}
  = -4\pi \delta(t)\,\delta(\mathbf{r}{-}\mathbf{r}') \,,&lt;/math&gt;
whence
{{NumBlk|:|&lt;math&gt;\triangle v = \tfrac{1}{c^2} \ddot{v}
  - 4\pi \delta(t)\,\delta(\mathbf{r}{-}\mathbf{r}') \,.
  &lt;/math&gt;|{{EquationRef|131v}}}}
Substituting ({{EquationNote|131u}}) and ({{EquationNote|131v}}) into ({{EquationNote|129}}) gives
:&lt;math&gt;\begin{align}
  &amp;\iint_S \big(u~\!\part_n v - v~\!\part_n u\big)~\!dS \\
  &amp;= \tfrac{1}{c^2}\!\iiint_V (u\ddot{v} - v\ddot{u})~\!dV
     - \!\iiint_V\!4\pi u~\!\delta(t)~\!\delta(\mathbf{r}{-}\mathbf{r}')~\!dV \\
  &amp;= \tfrac{1}{c^2}\!
     \iiint_V \tfrac{\part}{\part t} \big(u\dot{v} - v\dot{u}\big) ~\!dV
     - \!\iiint_V \!4\pi\psi(\mathbf{r},t)~\!\delta(t)
                  ~\!\delta(\mathbf{r}{-}\mathbf{r}') ~\!dV \,,
  \end{align}&lt;/math&gt;
using ({{EquationNote|130u}}) in the last term. In that term we may now set{{math| '''r'''}} to{{math| '''r&amp;prime;'''}} [because {{math|''&amp;delta;''('''r'''&amp;#8202;&amp;minus;&amp;#8202;'''r&amp;prime;''')}} is zero elsewhere] and then take the {{math|'''r'''}}-independent factor outside the volume integral, obtaining
:&lt;math&gt;\begin{align}
  \iint_S &amp;\big(u~\!\part_n v - v~\!\part_n u\big)~\!dS \\
  &amp;= \tfrac{1}{c^2}\!
     \iiint_V \tfrac{\part}{\part t} \big(u\dot{v} - v\dot{u}\big) ~\!dV
     - \,4\pi\psi(\mathbf{r}',t)~\!\delta(t)
       ~\!\operatorname{if}(\mathbf{r}'{\in}~\!V) \,,
  \end{align}&lt;/math&gt;
where {{math|if()}} is an ad-hoc function taking the value{{math|&amp;#8201;1}} if its argument is true ({{math|'''r&amp;prime;''' }}is in{{mvar| V&amp;#8202;}}), and{{math| 0}}&amp;#8202; if its argument is false. Then, to eliminate{{math| ''&amp;delta;''(''t'')}}, we integrate w.r.t.{{mvar| t}}&amp;#8202; over all time, obtaining
{{NumBlk|:|&lt;math&gt;\begin{align}
    \iint\limits_{S\;} &amp;\int_{-\infty}^{\infty}
      \!\!\big(u~\!\part_n v - v~\!\part_n u\big)
    dt\;dS \\
    &amp;= \tfrac{1}{c^2}\!\iiint_V\!(u\dot{v}-v\dot{u})\Big|_{-\infty}^{\infty} dV
       - \,4\pi\psi(\mathbf{r}',0)~\!\operatorname{if}(\mathbf{r}'{\in}~\!V)\,.
  \end{align}&lt;/math&gt;|{{EquationRef|132}}}}
In the remaining volume integral, substituting for &lt;math&gt;v&lt;/math&gt; from ({{EquationNote|130v}}), we have
{{NumBlk|:|&lt;math&gt;\begin{align}(u\dot{v}-v\dot{u})\Big|_{-\infty}^{\infty}
    &amp;= \Big(\tfrac{\,u\,}{s}~\!\delta'\!(t+s/c)
            - \tfrac{\,\dot{u}\,}{s}~\!\delta(t+s/c)\Big)
       \bigg|_{t\to-\infty}^{t\to\infty} \\
    &amp;= ~\!0
  \end{align}&lt;/math&gt;|{{EquationRef|133}}}}
because the expression in the big parentheses is zero except where {{math|''t&amp;#8201;&amp;#8776;&amp;#8201;&amp;minus;s''&amp;#10744;''c''}}, and{{mvar| s}} is finite in{{mvar| V}}.  So the volume integral in ({{EquationNote|132}}) vanishes, and what remains is
{{NumBlk|:|&lt;math&gt;
    \iint_{\!S} \!\textstyle\Big\{\!
      \int_{-\infty}^{\infty} \!u~\!\part_n v \,dt
      - \!\int_{-\infty}^{\infty} \!v~\!\part_n u \,dt
    \Big\} ~\!dS
    = - 4\pi\psi(\mathbf{r}'\!,0) ~\!\operatorname{if}(\mathbf{r}'{\in}~\!V)
  &lt;/math&gt;|{{EquationRef|134}}}}
&amp;mdash;which is what we wanted: a surface integral over{{mvar| S}}, equal (up to a scale factor) to the primary wave function at{{math| '''r&amp;prime;'''}} if {{math|'''r&amp;prime;'''}} is inside{{mvar| V}}, but zero if it is outside.

It remains to put the surface integral into a more convenient form, by substituting from ({{EquationNote|130u}}) and ({{EquationNote|130v}}) and simplifying. The second inner time-integral is
:&lt;math&gt;\begin{align}
    \int_{-\infty}^{\infty} \!v~\!\part_n u \,dt
    &amp;= \!\int_{-\infty}^{\infty}\!
         \tfrac{\,1\,}{s}~\!\delta\big(t+s/c\big)~\!\part_n\psi(\mathbf{r},t)
       \,dt \\[.5ex]
    &amp;= \!\int_{-\infty}^{\infty}\!
         \tfrac{\,1\,}{s}~\!\delta\big(t+s/c\big)~\!\part_n\psi(\mathbf{r},-s/c)
       \,dt \\
    &amp;= \tfrac{\,1\,}{s}~\!\part_n\psi(\mathbf{r},-s/c)\!\int_{-\infty}^{\infty}
         \!\delta\big(t+s/c\big)
       \,dt \,,
  \end{align}&lt;/math&gt;
i.e.
{{NumBlk|:|&lt;math&gt;\textstyle
    \int_{-\infty}^{\infty} \!v~\!\part_n u \,dt
    = \frac{\,1\,}{s}~\!\frac{\part\psi}{\part n}\big(\mathbf{r},-s/c\big) \,,
  &lt;/math&gt;|{{EquationRef|135}}}}
where the differentiation w.r.t.{{mvar| n}} does ''not'' account for the variation of{{mvar| s}} with{{mvar| n}}, because the {{mvar|s}}-dependence arises from selecting the time ''after'' the spatial differentiation. For the other time-integral in ({{EquationNote|134}}), however, it's the other way around: we differentiate a function of{{mvar| s}}, treating {{mvar|s}} as a function of{{mvar| n}} (and of two other coordinates which are also parameters of the surface{{mvar| S}}), using the chain rule and the product rule:
:&lt;math&gt;\begin{align}
    \int_{-\infty}^{\infty} \!u~\!\part_n v \,dt
    &amp;= \!\int_{-\infty}^{\infty}\!
         \psi(\mathbf{r},t)\,
         \part_n\!\Big(\!\tfrac{\,1\,}{s}~\!\delta(t+s/c)\Big)
       \,dt \\[.5ex]
    &amp;= \!\int_{-\infty}^{\infty}\!
         \psi(\mathbf{r},t)\,
         \part_s\!\Big(\!\tfrac{\,1\,}{s}~\!\delta(t+s/c)\Big)~\!
         \tfrac{\part s}{\part n}
       \,dt \\[.5ex]
    &amp;= \!\int_{-\infty}^{\infty}\!
         \psi(\mathbf{r},t)
         \Big(\tfrac{1}{cs}~\!\delta'\!(t+s/c)
              -\tfrac{1}{s^2}~\!\delta(t+s/c)\Big)
         \tfrac{\part s}{\part n}
       \,dt \\[.5ex]
    &amp;= \!\int_{-\infty}^{\infty}\!
         \tfrac{1}{cs}~\!\tfrac{\part s}{\part n}~\!\psi(\mathbf{r},t)
         ~\delta'\!(t+s/c)
       \,dt \\[.5ex]
    &amp;~~~~~- \int_{-\infty}^{\infty}\!
              \tfrac{1}{s^2}~\!\tfrac{\part s}{\part n}~\!\psi(\mathbf{r},t)
              \,\delta(t+s/c)
            \,dt \,.
  \end{align}&lt;/math&gt;
Expanding the first integral by parts, and processing the delta function in the second integral in the usual manner, we get
:&lt;math&gt;\begin{align}
    \int_{-\infty}^{\infty} \!u~\!\part_n v \,dt =\;
    &amp;\Big(\tfrac{1}{cs}~\!\tfrac{\part s}{\part n}~\!\psi(\mathbf{r},t)
         ~\delta(t+s/c)\Big)\bigg|_{-\infty}^{\infty} \\
    &amp; - \!\int_{-\infty}^{\infty}\!
          \tfrac{1}{cs}~\!\tfrac{\part s}{\part n}~\!\dot{\psi}(\mathbf{r},t)
          ~\delta(t+s/c)
        \,dt \\[.5ex]
    &amp; - \tfrac{1}{s^2}~\!\tfrac{\part s}{\part n}~\!\psi(\mathbf{r},-s/c) \,,
  \end{align}&lt;/math&gt;
in which we can now process the remaining delta functions to obtain
:&lt;math&gt;\textstyle\int_{-\infty}^{\infty} \!u~\!\part_n v \,dt \,=\,
    0 - \tfrac{1}{cs}~\!\tfrac{\part s}{\part n}~\!\dot{\psi}(\mathbf{r},-s/c)
      - \tfrac{1}{s^2}~\!\tfrac{\part s}{\part n}~\!\psi(\mathbf{r},-s/c) \,.
  &lt;/math&gt;
Substituting this and ({{EquationNote|135}}) into ({{EquationNote|134}}), renaming the (arbitrary) time origin as time{{mvar| t}}, and multiplying through by{{math| &amp;minus;1}}, we get the desired result:
{{NumBlk|:|&lt;math&gt;\begin{align}
    \iint_{S} \!\Big\{\!
    &amp; \tfrac{1}{cs} \tfrac{\part s}{\part n}
      ~\!\dot{\psi}\big(\mathbf{r},t{-}\tfrac{s}{c}\big)
      + \tfrac{1}{s^2} \tfrac{\part s}{\part n}
        ~\!\psi\big(\mathbf{r},t{-}\tfrac{s}{c}\big)
      + \tfrac{\,1\,}{s}~\!
        \tfrac{\part\psi}{\part n}\big(\mathbf{r},t{-}\tfrac{s}{c}\big)
    \Big\} ~\!dS \\
    &amp;=\, 4\pi\psi(\mathbf{r}',t) ~\!\operatorname{if}(\mathbf{r}'{\in}~\!V)\,.
  \end{align}&lt;/math&gt;|{{EquationRef|136}}}}
This is more usually written
:&lt;math&gt;
    \iint_{S} \!\Big\{\!
      \tfrac{1}{cs} \tfrac{\part s}{\part n} ~\!\big[\dot{\psi}\big]
      + \tfrac{1}{s^2} \tfrac{\part s}{\part n} ~\![\psi]
      + \tfrac{\,1\,}{s} \Big[\tfrac{\part\psi}{\part n}\Big]
    \Big\} ~\!dS
    \,=\, 4\pi\psi(\mathbf{r}',t) ~\!\operatorname{if}(\mathbf{r}'{\in}~\!V)\,,
  &lt;/math&gt;
where the square brackets indicate that the contents are to be ''delayed'' (or, in older literature, &quot;retarded&quot;) by the propagation time from {{math|'''r'''}} to{{math| '''r&amp;prime;'''}}&amp;mdash;that is, delayed by{{math| ''s''&amp;#10744;''c''}}&amp;#8202; relative to the default arguments{{math| ('''r''',&amp;#8239;''t'')}}.

It is common to write &lt;math&gt;-\tfrac{\part}{\part n}\big(\!\tfrac{\,1\,}{s}\!\big)&lt;/math&gt;&amp;#8202; instead of&amp;#8201; &lt;math&gt;\tfrac{1}{s^2}\tfrac{\part s}{\part n}&lt;/math&gt; (reversing the chain rule), so that the last result becomes
{{NumBlk|:|&lt;math&gt;
    \iint_{S} \!\Big\{\!
      \tfrac{1}{cs} \tfrac{\part s}{\part n} ~\!\big[\dot{\psi}\big]
      - [\psi]~\!\tfrac{\part}{\part n}\big(\!\tfrac{\,1\,}{s}\!\big)
      + \tfrac{\,1\,}{s} \Big[\tfrac{\part\psi}{\part n}\Big]
    \Big\} ~\!dS
    \,=\, 4\pi\psi(\mathbf{r}',t) ~\!\operatorname{if}(\mathbf{r}'{\in}~\!V)\,.
  &lt;/math&gt;|{{EquationRef|137}}}}

Although we derived ({{EquationNote|137}}) by supposing that {{mvar|V &amp;#8202;}}is a ''finite''&amp;#8202; region, we can extend the result to an infinite region by adding another sheet to the bounding surface{{mvar| S}}&amp;#8202; in such a way that (i) the region becomes finite, but (ii) the additional sheet makes no contribution to the surface integral. The simplest way to do this is to suppose that the additional sheet is at such a large distance that the disturbance has not reached it yet! Alternatively, we can consider how the wave function decays with distance.&lt;ref&gt;[[#baker-copson-39|Baker &amp;amp; Copson, 1939]], pp.&amp;#8239;37–8.&lt;/ref&gt; By such methods we can apply ({{EquationNote|137}}) not only to the region inside a closed surface, but also (e.g.) to the region outside a closed surface, or the region on one side of an infinite open surface.

Although we derived ({{EquationNote|137}}) by supposing, as usual in this paper, that&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; points out of{{mvar| V}}&amp;#8239; and that {{mvar|n &amp;#8202;}}is measured out of{{mvar| V}}, this has the arguably counterintuitive implication that&amp;#8202; &lt;math&gt;\mathbf{\hat{n}}&lt;/math&gt; is typically against the direction of propagation&amp;mdash;''directly'' against it in the simplest case, in which {{mvar|V}}&amp;#8202; is the exterior of a sphere with a monopole source at its center.

So, in the following formal statement of our result, let us drop the symbol {{mvar|V}}&amp;#8202; and define {{mvar|n}}&amp;#8202; as being measured out of the region containing the sources, and consequently ''into'' the region that satisfies the homogeneous wave equation, ''changing the signs''&amp;#8202; on the left side of ({{EquationNote|137}}).

'''[[w:Gustav Kirchhoff|Kirchhoff]]'s integral theorem''':  If
* the wave function {{mvar|&amp;psi;}}&amp;#8202; satisfies the wave equation (with speed{{mvar| c}}) in a region{{mvar| R}}&amp;#8202; bounded by a surface{{mvar| S}}&amp;#8202; (with all sources consequently on the other side of{{mvar| S&amp;#8202;}}), and
* {{mvar|s}}&amp;#8202; is the distance of the general point at position{{math| '''r'''}}&amp;#8202; from the observation point at position{{math| '''r&amp;prime;''',&amp;#8202;}} and
* quantities in square brackets are to be delayed by{{math| ''s''&amp;#10744;''c''&amp;#8202;,}} and
* {{mvar|n}}&amp;#8202; is the normal coordinate measured from the general point on{{mvar| S}}&amp;#8202; ''into''{{mvar| R}}&amp;#8202; [contrary to the usual direction for a named region, and contrary to the convention we have used above!],
then the expression
{{NumBlk|:|&lt;math&gt;
    \tfrac{1}{4\pi} \!\iint_{S} \!\Big\{\!
      [\psi]~\!\tfrac{\part}{\part n}\big(\!\tfrac{\,1\,}{s}\!\big)
      - \tfrac{1}{cs} \tfrac{\part s}{\part n} ~\!\big[\dot{\psi}\big]
      - \tfrac{\,1\,}{s} \Big[\tfrac{\part\psi}{\part n}\Big]
    \Big\} ~\!dS
  &lt;/math&gt;|{{EquationRef|138}}}}
is equal to the wave function at{{math| '''r&amp;prime;'''}}&amp;#8202; if{{math| '''r&amp;prime;'''}}&amp;#8202; is inside{{math| ''R''&amp;#8202;,}} but zero if it is outside.&lt;ref&gt;[[#born-wolf-02|Born &amp;amp; Wolf, 2002]], pp.&amp;#8239;420–21, eq.&amp;#8239;(13).&amp;#8201; ''Cf''. Baker &amp;amp; Copson ([[#baker-copson-39|1939]], p.&amp;#8239;37) and Miller ([[#miller-91|1991]], eq.&amp;#8239;2), who use {{mvar|r}}&amp;#8202; instead of{{mvar| s}}&amp;#8202; (among other notational differences). Baker &amp;amp; Copson, in their last equation on p.&amp;#8239;40, give the opposite sign because on this occasion they measure the normal coordinate&lt;math&gt;~\nu&lt;/math&gt; ''out'' of the region.&lt;/ref&gt;

The above derivation does not assume sinusoidal time-dependence at any stage. An alternative approach&lt;ref&gt;E.g., [[#baker-copson-39|Baker &amp;amp; Copson, 1939]], pp.&amp;#8239;36–7; [[#born-wolf-02|Born &amp;amp; Wolf, 2002]], pp.&amp;#8239;420–21.&lt;/ref&gt; is to derive the special case for sinusoidal time-dependence (due to [[w:Hermann von Helmholtz|Helmholtz]]) from Green's identities, and then generalize the time-dependence; this method has the advantage of being more readily applicable to ''dispersive''&amp;#8202; media (in which {{mvar|c &amp;#8202;}}is frequency-dependent), but the disadvantages of depending on complex numbers and on the premise that a general function of time can be expressed as a sum of sinusoids. Helmholtz's integrand is a sinusoidal version of our expression ({{EquationNote|139}}) below. That expression, and thence the Kirchhoff integral, can be obtained in a far more elementary manner, albeit with some loss of rigor, by ''assuming'' (instead of justifying) the form of the wave function due to a monopole source. From this, together with considerations of causality and superposition, we can work out the required distribution of secondary sources and then expresses the wave function as a surface integral.&lt;ref&gt;[[#putland-22|Putland, 2022&amp;ndash;]].&lt;/ref&gt; In the present paper, however, we argue in the other direction: from the integral to the secondary sources.

{{cob}}

=== Monopole and dipole secondary sources ===

{{cot}}

If the dependence on{{math| '''r'''}} is taken as implicit, the integrand inside the braces in ({{EquationNote|138}}) can be written out as
:&lt;math&gt;\begin{align}
    \psi&amp;\big(t\!-\!s/c\big)\,\part_n \tfrac{\,1\,}{s}
    - \tfrac{1}{cs}~\!\part_n s \,\psi'\big(t\!-\!s/c\big)
    - \tfrac{\,1\,}{s}~\!\tfrac{\part\psi}{\part n}\big(t\!-\!s/c\big) \\
    &amp;= \psi\big(t\!-\!s/c\big)\,\part_n \tfrac{\,1\,}{s}
       + \tfrac{\,1\,}{s}~\!\psi'\big(t\!-\!s/c\big)~\!
         \big({-}1/c\big)~\!\part_n s
       - \tfrac{\,1\,}{s}~\!\tfrac{\part\psi}{\part n}\big(t\!-\!s/c\big) \\
    &amp;= \psi\big(t\!-\!s/c\big)\,\part_n \tfrac{\,1\,}{s}
       + \tfrac{\,1\,}{s}~\!\part_n \psi\big(t\!-\!s/c\big)
       - \tfrac{\,1\,}{s}~\!\tfrac{\part\psi}{\part n}\big(t\!-\!s/c\big)
  \end{align}&lt;/math&gt;
or, recognizing the first two terms as the derivative of a product,
{{NumBlk|:|&lt;math&gt;
    \part_n \Big(\tfrac{\,1\,}{s}~\!\psi\big(t\!-\!s/c\big)\Big)
    - \tfrac{\,1\,}{s}~\!\tfrac{\part\psi}{\part n}\big(t\!-\!s/c\big) \,,
  &lt;/math&gt;|{{EquationRef|139}}}}
where {{mvar|&amp;part;&lt;sub&gt;n&lt;/sub&gt;}} accounts for the variation of {{mvar|s}}&amp;#8202; through{{mvar| n}}, but {{mvar|{{sfrac|&amp;part;&amp;psi;|&amp;part;n}}}} does not [see remarks after ({{EquationNote|135}}) above]. If{{mvar| h}} is a ''small''&amp;#8202; change in{{math| ''n''&amp;#8202;,}} from&amp;#8201; {{mvar|n&amp;#8201;{{=}}&amp;#8201;&amp;minus;h}}&amp;#8201; to&amp;#8201; {{math|''n''&amp;#8201;{{=}}&amp;#8201;0&amp;#8202;,}} the integrand can be written
{{NumBlk|:|&lt;math&gt;
    h~\!\part_n \bigg(\frac{\,1\,}{s}~\!\frac{\psi\big(t\!-\!s/c\big)}{h}\bigg)
    - \tfrac{\,1\,}{s}~\!\tfrac{\part\psi}{\part n}\big(t\!-\!s/c\big) \,.
  &lt;/math&gt;|{{EquationRef|140}}}}
The second term (including the minus sign) is recognizable as the contribution to the wave function from a monopole source with strength{{mvar| &amp;minus;{{sfrac|&amp;part;&amp;psi;|&amp;part;n}}&amp;#8202;}}.&lt;ref&gt;''Reminder&amp;#8202;:''&amp;#8201; There are rival definitions of the &quot;strength&quot; of a monopole source; see the text and footnote under equation ({{EquationNote|54}}) above.&lt;/ref&gt; Similarly, in the first term, the expression in the big parentheses is the contribution from a monopole source with strength{{math| ''&amp;psi;''&amp;#10744;''h''&amp;#8202;}}; and the operator {{mvar|h&amp;#8202;&amp;part;&lt;sub&gt;n&lt;/sub&gt;}} gives the change in that contribution due to{{mvar| n}}&amp;#8202; increasing from {{mvar|&amp;minus;h}}&amp;#8202; to{{math| 0&amp;#8202;,}}&amp;#8201; i.e. the change in that contribution due to moving the said monopole from&amp;#8201; {{mvar|n&amp;#8201;{{=}}&amp;#8201;&amp;minus;h}}&amp;#8201; to&amp;#8201; {{math|''n''&amp;#8201;{{=}}&amp;#8201;0&amp;#8202;,}}&amp;#8201; i.e. the whole contribution due to the combination of a monopole with strength{{math| &amp;minus;''&amp;psi;''&amp;#10744;''h''}}&amp;#8202; at&amp;#8201; {{mvar|n&amp;#8201;{{=}}&amp;#8201;&amp;minus;h}}&amp;#8201; and a monopole with strength{{math| ''&amp;psi;''&amp;#10744;''h''}}&amp;#8202; at&amp;#8201; {{math|''n''&amp;#8201;{{=}}&amp;#8201;0}}. This combination is called a '''dipole''' (or ''doublet'')&lt;ref&gt;The term ''doublet'', which seems to be older, is used by Baker &amp;amp; Copson ([[#baker-copson-39|1939]]), Born &amp;amp; Wolf ([[#born-wolf-02|2002, p.&amp;#8239;421]]), and Larmor ([[#larmor-1904|1904]]).&lt;/ref&gt; with strength{{mvar| &amp;psi;}}&amp;#8202; in the normal ({{mvar|n}}) direction.

According to ({{EquationNote|138}}), the expression ({{EquationNote|140}}) is to be scaled by {{math|{{sfrac|4''&amp;pi;''}}}}&amp;#8202; and integrated over the surface{{mvar| S}}. Thus the secondary source distribution can be described as a monopole distribution of strength density {{math|&amp;minus;{{sfrac|4''&amp;pi;''}}{{sfrac|''&amp;part;&amp;psi;''|''&amp;part;n''}}}}&amp;#8239; plus a normal dipole distribution of strength density{{math| {{sfrac|''&amp;psi;''|4''&amp;pi;''}}&amp;#8202;,}} where &quot;strength density&quot; means strength per unit area. This description is well known.&lt;ref&gt;E.g., [[#born-wolf-02|Born &amp;amp; Wolf, 2002]], p.&amp;#8239;421.&lt;/ref&gt;

The implication is not that the specified secondary sources really exist, or even that they ''could''&amp;#8202; exist, but only that the wave function in the region{{mvar| R}}&amp;#8202; is ''as if''&amp;#8202; it had been generated by the specified secondary sources (which would also give a null wave function outside the region). We should note, however, that a monopole contribution of the form ({{EquationNote|48}}) ''can''&amp;#8202; really exist, even for a vector wave function, notwithstanding that it requires not only the magnitude but also the direction of the vector to be independent of the direction of propagation. That requirement might seem to exclude electromagnetic waves, for which the electric and magnetic fields are transverse to the direction of propagation and therefore not independent of it. But it is possible to describe such waves in terms of an electric scalar potential and a magnetic vector potential, such that the contribution to the latter from a current element has the same direction as the current element for all directions of propagation.&lt;ref&gt;[[#stratton-41|Stratton, 1941]], pp.&amp;#8239;428–30.&lt;/ref&gt;

{{cob}}

=== Spatiotemporal-dipole secondary sources ===

{{cot}}

The &quot;dipole&quot; discussed so far is a ''spatial''&amp;#8202; dipole, in which the constituent monopoles differ only in sign and by a small spatial displacement. In the Helmholtz–Kirchhoff integrand ({{EquationNote|139}}), the second term (including the sign) represents a monopole strength density{{mvar| &amp;minus;{{sfrac|&amp;part;&amp;psi;|&amp;part;n}}}}&amp;#8202; and the first term represents a spatial dipole strength density{{mvar| &amp;psi;}}&amp;#8202; in the {{mvar|n }}direction; the dipole source per unit area of{{mvar| S}}&amp;#8202; comprises a monopole with strength{{math| &amp;minus;''&amp;psi;''&amp;#10744;''h''}}&amp;#8202; at&amp;#8201; {{mvar|n&amp;#8201;{{=}}&amp;#8201;&amp;minus;h}}&amp;#8202; (the ''inverted''&amp;#8202; monopole), and a monopole with strength{{math| ''&amp;psi;''&amp;#10744;''h''}}&amp;#8202; at&amp;#8201; {{math|''n''&amp;#8201;{{=}}&amp;#8201;0}} (the ''uninverted''&amp;#8202; monopole), where {{mvar|h}}&amp;#8202; is small (and the indicated strength densities are eventually to be divided by{{math| 4''&amp;pi;''}}).

If there is only a '''single monopole primary source''', this combination of a monopole and a spatial dipole is exactly equivalent to a modified dipole in which the inverted monopole has a certain fixed delay, and a certain fixed attenuation, relative to the uninverted monopole.&lt;ref&gt;The derivation of this &quot;generalized spatiotemporal dipole&quot; (GSTD) was first given in ver.&amp;#8239;0.3 of [[#putland-22|Putland, 2022&amp;ndash;]] (&amp;sect;&amp;#8239;3.7). It was included in earlier versions of the present paper, but is now more conveniently available in a much smaller document ([[#putland-25|Putland, 2025]]).&lt;/ref&gt;

If, in addition, the surface {{mvar|S}}&amp;#8202; coincides with a primary wavefront, the required &quot;fixed delay&quot; is simply{{math| ''h''&amp;#10744;''c''&amp;#8202;}}, i.e. the propagation time from the uninverted monopole to the inverted one. If the primary wavefronts are plane (for a general{{mvar| S&amp;#8202;}}), the inverted monopole should be unattenuated. If {{mvar|S}}&amp;#8202; coincides with a primary wavefront ''and''&amp;#8202; is plane (a large-{{mvar|r}} approximation), the modified dipole reduces to what D.A.B.&amp;#8239;Miller called a '''spatiotemporal dipole''',&lt;ref&gt;[[#miller-91|Miller, 1991]].&lt;/ref&gt; in which the only modification of the spatial dipole is the delay{{math| ''h''&amp;#10744;''c''}}.

{{cob}}

=== Application to diffraction by an aperture ===

{{cot}}

Suppose that the primary sources are partly obstructed by an opaque baffle with an aperture in it. What is the wave function that propagates beyond the baffle? Let us choose a surface{{mvar| S}}&amp;#8202; consisting of two segments, namely {{mvar|S&lt;sub&gt;a&lt;/sub&gt; }}spanning the aperture, and {{mvar|S&lt;sub&gt;b&lt;/sub&gt; }}on the side of the baffle facing away from the sources (the dark side or quiet side of the baffle). The obvious way to proceed is to suppose that the baffle simply eliminates the secondary sources on{{mvar| S&lt;sub&gt;b&lt;/sub&gt;}} while leaving the secondary sources on{{mvar| S&lt;sub&gt;a&lt;/sub&gt;}} unchanged (as if the baffle were not there). The result, as far as the wave function in{{mvar| R}} (beyond the baffle) is concerned, is simply that the integral is taken over {{mvar|S&lt;sub&gt;a&lt;/sub&gt; }}only.

Integrating over the aperture alone is indeed the standard answer, but there are various other ways of explaining it. Some explanations, including the famously inconsistent one offered by Kirchhoff himself, are discussed in [[#putland-22|Putland, 2022&amp;ndash;]] (&amp;sect;&amp;#8239;2.2 and Appendices A &amp;amp; B), and the references therein.

{{cob}}

== Acknowledgment ==

This learning resource uses images from ''Wikimedia Commons''.

== Notes ==

{{cot}}

{{notelist|30em}}

{{cob}}

== Citations ==

{{cot}}

{{reflist|19em}}

{{cob}}

== References ==

{{cot}}

&lt;div style=&quot;font-size: 111%&quot;&gt;

{{refbegin|indent=yes}}

*&lt;span id=&quot;axler-95&quot;&gt;S.J. Axler, 1995, &quot;Down with Determinants!&quot;&amp;#8201; ''American Mathematical Monthly'', vol.&amp;#8239;102, no.&amp;#8239;2 (Feb.&amp;#8239;1995), pp.&amp;#8239;139–54; [https://www.jstor.org/stable/2975348 jstor.org/stable/2975348].&amp;#8201; (Author's preprint, with different pagination: [https://www.researchgate.net/publication/265273063_Down_with_Determinants researchgate.net/publication/265273063_Down_with_Determinants].)&lt;/span&gt;
*&lt;span id=&quot;axler-23-&quot;&gt;S.J. Axler, 2023–, ''Linear Algebra Done Right'', 4th Ed., Springer; [https://linear.axler.net/ linear.axler.net] (open access).&lt;/span&gt;
*&lt;span id=&quot;baker-copson-39&quot;&gt;B.B. Baker and E.T. Copson, 1939, ''The Mathematical Theory of&amp;#8202; Huygens' Principle'', Oxford; 3rd Ed.&amp;#8201;(same pagination, with addenda), New York: Chelsea, 1987, [https://archive.org/details/mathematicaltheo0000bake archive.org/details/mathematicaltheo0000bake].&lt;/span&gt;
*&lt;span id=&quot;borisenko-tarapov-68&quot;&gt;A.I. Borisenko and I.E.&amp;#8239;Tarapov (tr.&amp;#8239;&amp;amp; ed. R.A.&amp;#8239;Silverman), 1968, ''Vector and Tensor Analysis with Applications'', Prentice-Hall; reprinted New York: Dover, 1979, [https://archive.org/details/vectortensoranal0000bori archive.org/details/vectortensoranal0000bori].&lt;!-- Typo on p.180: First cross in equation before (4.93) should be &quot;=&quot;. --&gt;&lt;/span&gt;
*&lt;span id=&quot;born-wolf-02&quot;&gt;M.&amp;#8201;Born and E.&amp;#8239;Wolf, 2002, ''Principles of Optics'', 7th Ed., Cambridge, 1999 (reprinted with corrections, 2002).&lt;/span&gt;
*&lt;span id=&quot;broyden-75&quot;&gt;C.G. Broyden, 1975, ''Basic Matrices'', London: Macmillan.&lt;/span&gt;
*&lt;span id=&quot;feynman-63&quot;&gt;R.P. Feynman, R.B. Leighton, &amp;amp; M.&amp;#8239;Sands, 1963 etc., ''The Feynman Lectures on Physics'', California Institute of Technology; [http://www.feynmanlectures.caltech.edu/ feynmanlectures.caltech.edu].&lt;/span&gt;
*&lt;span id=&quot;fletcher-74&quot;&gt;N.H. Fletcher, 1974, &quot;Adiabatic assumption for wave propagation&quot;, ''American Journal of Physics'', vol.&amp;#8239;42, no.&amp;#8239;6 (June 1974), pp.&amp;#8239;487–9; [https://doi.org/10.1119/1.1987757 doi.org/10.1119/1.1987757].&lt;/span&gt;
*&lt;span id=&quot;gibbs-1881-4&quot;&gt;J.W. Gibbs, 1881–84, &quot;Elements of Vector Analysis&quot;, privately printed New Haven: Tuttle, Morehouse &amp;amp; Taylor, 1881 (&amp;sect;&amp;sect;&amp;#8239;1–101), 1884 (&amp;sect;&amp;sect;&amp;#8239;102–189, etc.), [https://archive.org/details/elementsvectora00gibb archive.org/details/elementsvectora00gibb]; published in ''The Scientific Papers of J.&amp;#8239;Willard Gibbs'' (ed. H.A.&amp;#8239;Bumstead &amp;amp; R.G.&amp;#8239;Van Name), New York: Longmans, Green, &amp;amp; Co., 1906, vol.&amp;#8239;2, [https://archive.org/details/scientificpapers02gibbuoft archive.org/details/scientificpapers02gibbuoft], pp.&amp;#8239;17–90.&lt;/span&gt;
*&lt;span id=&quot;hsu-84&quot;&gt;H.P. Hsu, 1984, ''Applied Vector Analysis'', Harcourt Brace Jovanovich; [https://archive.org/details/appliedvectorana00hsuh archive.org/details/appliedvectorana00hsuh].&lt;/span&gt;
*&lt;span id=&quot;huygens-1690-thompson&quot;&gt;C. Huygens, 1690, tr. S.P.&amp;#8239;Thompson, ''Treatise on Light'', University of Chicago Press, 1912 / [https://gutenberg.org/files/14725/14725-h/14725-h.htm gutenberg.org/files/14725/14725-h/14725-h.htm], 2005. (See also &quot;Errata in various editions of Huygens' ''Treatise on Light''&amp;#8239;&quot;, ''www.grputland.com'' or ''grputland.blogspot.com'', June 2016.)&lt;/span&gt;
*&lt;span id=&quot;katz-79&quot;&gt;V.J. Katz, 1979, &quot;The history of Stokes' theorem&quot;, ''Mathematics Magazine'', vol.&amp;#8239;52, no.&amp;#8239;3 (May 1979), pp.&amp;#8239;146–56; [https://www.jstor.org/stable/2690275 jstor.org/stable/2690275].&lt;/span&gt;
*&lt;span id=&quot;kemin-et-al-00&quot;&gt;S. Kemin, X.&amp;#8239;Zhenting, T.&amp;#8239;Jinsheng, &amp;amp; H.&amp;#8239;Xuemei, 2000, &quot;The comprehension, some problems and suggestions to symbolic vector method and some defenses for Gibbs' symbol&quot;, ''Applied Mathematics and Mechanics'' (English Ed.), vol.&amp;#8239;21, no.&amp;#8239;5 (May 2000), pp.&amp;#8239;603–6; [https://doi.org/10.1007/BF02459044 doi.org/10.1007/BF02459044].&lt;/span&gt;
*&lt;span id=&quot;kemmer-77&quot;&gt;N. Kemmer, 1977, ''Vector Analysis: A physicist's guide to the mathematics of fields in three dimensions'', Cambridge; [https://archive.org/details/isbn_0521211581 archive.org/details/isbn_0521211581].&lt;/span&gt;
*&lt;span id=&quot;kreyszig-62-&quot;&gt;E. Kreyszig, 1962 etc., ''Advanced Engineering Mathematics'', New York: Wiley;&amp;#8201; 5th Ed., 1983;&amp;#8201; 6th Ed., 1988;&amp;#8201; 9th Ed., 2006;&amp;#8201; 10th Ed., 2011.&lt;/span&gt;
*&lt;span id=&quot;larmor-1904&quot;&gt;J. Larmor, 1904, &quot;On the mathematical expression of the principle of&amp;#8202; Huygens&quot; (read 8 Jan.&amp;#8239;1903), ''Proceedings of the London Mathematical Society'', Ser.&amp;#8239;2, vol.&amp;#8239;1 (1904), pp.&amp;#8239;1–13.&lt;!-- Listed as &quot;Issue 1&quot;; only issue for that volume. --&gt;&lt;/span&gt;
*&lt;span id=&quot;miller-91&quot;&gt;D.A.B. Miller, 1991, &quot;Huygens's wave propagation principle corrected&quot;, ''Optics Letters'', vol.&amp;#8239;16, no.&amp;#8239;18 (15 Sep.&amp;#8239;1991), pp.&amp;#8239;1370–72; [http://ee.stanford.edu/~dabm/146.pdf stanford.edu/~dabm/146.pdf].&lt;/span&gt;
*&lt;span id=&quot;moon-spencer-65&quot;&gt;P.H. Moon and D.E.&amp;#8239;Spencer, 1965, ''Vectors'', Princeton, NJ: Van Nostrand.&lt;/span&gt;
*&lt;span id=&quot;panofsky-phillips-62&quot;&gt;W.K.H. Panofsky and M.&amp;#8239;Phillips, 1962, ''Classical Electricity and Magnetism'', 2nd Ed., Addison-Wesley; reprinted Mineola, NY: Dover, 2005.&lt;/span&gt;
*&lt;span id=&quot;putland-22&quot;&gt;G.R. Putland, 2022&amp;ndash;, &quot;Consistent derivation of Kirchhoff's integral theorem and diffraction formula and the Maggi-Rubinowicz transformation using high-school math&quot; (working paper), [https://doi.org/10.5281/zenodo.7205781 doi.org/10.5281/zenodo.7205781] (Creative Commons).&lt;/span&gt;
*&lt;span id=&quot;putland-25&quot;&gt;G.R. Putland, 2025, &quot;Exact formulation of Huygens' principle in terms of generalized spatiotemporal-dipole secondary sources&quot;, [https://doi.org/10.48550/arXiv.2510.20825 doi.org/10.48550/arXiv.2510.20825] (Creative Commons).&lt;/span&gt;
*&lt;span id=&quot;rocci-20&quot;&gt;A. Rocci, 2020, &quot;Back to the roots of vector and tensor calculus: Heaviside versus Gibbs&quot; (online 10 Nov.&amp;#8239;2020), ''Archive for History of Exact Sciences'', vol.&amp;#8239;75, no.&amp;#8239;4 (July 2021), pp.&amp;#8239;369–413. (Author's preprint, with different pagination: [https://arxiv.org/abs/2010.09679 arxiv.org/abs/2010.09679].)&lt;/span&gt;
*&lt;span id=stratton-41&gt;J.A. Stratton, 1941, ''Electromagnetic Theory'', New York: McGraw-Hill; [https://archive.org/details/electromagnetict0000juli archive.org/details/electromagnetict0000juli].&lt;/span&gt;
*&lt;span id=&quot;tai-94&quot;&gt;C.-T. Tai, 1994, &quot;A survey of the improper use of &amp;nabla; in vector analysis&quot; (Technical Report RL&amp;#8239;909), Dept.&amp;#8201;of Electrical Engineering &amp;amp; Computer Science, University of Michigan; [https://deepblue.lib.umich.edu/handle/2027.42/7869 hdl.handle.net/2027.42/7869].&lt;/span&gt;
*&lt;span id=&quot;tai-95&quot;&gt;C.-T. Tai, 1995, &quot;A historical study of vector analysis&quot; (Technical Report RL&amp;#8239;915), Dept.&amp;#8201;of Electrical Engineering &amp;amp; Computer Science, University of Michigan; [https://deepblue.lib.umich.edu/handle/2027.42/7868 hdl.handle.net/2027.42/7868].&lt;/span&gt;
*&lt;span id=&quot;tai-fang-91&quot;&gt;C.-T. Tai and N.&amp;#8239;Fang, 1991, &quot;A systematic treatment of vector analysis&quot;, ''{{serif|IEEE}} Transactions on Education'', vol.&amp;#8239;34, no.&amp;#8239;2 (May 1991), pp.&amp;#8239;167–74; [https://doi.org/10.1109/13.81596 doi.org/10.1109/13.81596].&lt;/span&gt;
*&lt;span id=&quot;wilson-1901&quot;&gt;E.B.&amp;#8201;Wilson, 1901, ''Vector Analysis: A text-book for the use of students of mathematics and physics'' (&quot;Founded upon the lectures of J.&amp;#8239;Willard Gibbs&amp;hellip;&quot;), New York: Charles Scribner's Sons; 12th printing, Yale University Press, 1958, [https://archive.org/details/vectoranalysiste0000gibb archive.org/details/vectoranalysiste0000gibb].&lt;/span&gt;
*&lt;span id=&quot;wrede-spiegel-10&quot;&gt;R.C.&amp;#8201;Wrede and M.R.&amp;#8239;Spiegel, 2010, ''Advanced Calculus'', 3rd Ed., New York: McGraw-Hill (Schaum's Outlines); [https://archive.org/details/schaumsoutlinesa0000wred archive.org/details/schaumsoutlinesa0000wred].&lt;/span&gt;

{{refend}}

&lt;/div&gt;

{{cob}}

== Further reading ==

{{cot}}

M.J. Crowe, &quot;A History of Vector Analysis&quot; (address at the University of Louisville, Autumn term, 2002), [https://www.researchgate.net/publication/244957729_A_History_of_Vector_Analysis researchgate.net/publication/244957729_A_History_of_Vector_Analysis] (including much discussion of quaternions).

P. Lynch, &quot;Matthew O'Brien: An inventor of vector analysis&quot;, ''Bulletin of the Irish Mathematical Society'', No.&amp;#8239;74 (Winter 2014), pp.&amp;#8239;81–8; [https://doi.org/10.33232/BIMS.0074.81.88 doi.org/10.33232/BIMS.0074.81.88].

{{cob}}

[[Category:Mathematics]]
[[Category:Calculus]]
[[Category:Vectors]]
[[Category:Vector calculus]]
[[Category:Multivariable calculus]]
[[Category:Applied mathematics]]
[[Category:Mathematical physics]]
[[Category:Waves]]
[[Category:Coordinate systems]]</text>
      <sha1>9v5g8sjkrtrux8zloz23s8ww8tm7g9m</sha1>
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  <page>
    <title>GNU Octave</title>
    <ns>0</ns>
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        <id>3705</id>
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      <text bytes="3001" sha1="kdcp3832u0sgh0055p06x4azhzkss2h" xml:space="preserve">GNU Octave is a free-as-in-freedom mathematical software package and a programming language, with considerable compatibility with MATLAB, a proprietary package. This page complements Wikipedia and Wikibooks. It should help readers who want to learn GNU Octave rather than just read about it. For a start, the page collects links to learning materials outside of Wikiversity.

==Getting started==
To get started, one can use octave-online.net, avoiding the need to install Octave. Using that environment, one can peruse [https://wiki.octave.org/Using_Octave Using Octave] from octave.org, try the examples and modify them. One can then move to the Octave Programming Tutorial in Wikibooks and solve the exercises that it has.

A quick glance, to be expanded:
* sin(3)
** Evaluates an expression and outputs the result.
* x = 0:0.1:7; plot (x, sin (x));
** Plots sine function on the given range, with the given step of 0.1.
* plot(x, sin(x), x, cos(x))
** Building on the above, plots multiple functions.
* m = [1, 1, 2; 3, 5, 8; 13, 21, 34]
** Creates a matrix.

Links:
* [https://octave-online.net/ Try Octave online], octave-online.net

==Compatibility with MATLAB==
As per octave.org, &quot;The Octave syntax is largely compatible with Matlab.&quot;&lt;ref&gt;[https://octave.org/ octave.org]&lt;/ref&gt; Moreover, differences between Octave and MATLAB are usually considered to be bugs. &lt;ref&gt;[https://wiki.octave.org/Differences_between_Octave_and_Matlab Differences between Octave and Matlab], wiki.octave.org&lt;/ref&gt;

Links:
* {{W|GNU Octave#MATLAB compatibility}}, wikipedia.org

==References==
&lt;references/&gt;

==Further reading==
Wikipedia and Wikibooks:
* {{W|GNU Octave}}, wikipedia.org
* [[B:Octave Programming Tutorial|Octave Programming Tutorial]], wikibooks.org

Official pages and documentation from octave.org:
* [https://www.octave.org/ GNU Octave], octave.org
* [https://wiki.octave.org/GNU_Octave_Wiki Octave], wiki.octave.org -- has links to different kinds of Octave documentation
* [https://wiki.octave.org/Using_Octave Using Octave], wiki.octave.org -- a quick introduction
* [https://wiki.octave.org/Octave_Basics Octave Basics], wiki.octave.org -- something like a reference card
* [https://docs.octave.org/latest/ Top (GNU Octave (version 8.4.0))], docs.octave.org -- the complete documentation

YouTube:
* [https://www.youtube.com/watch?v=woiU5PRVm7M GNU Octave - Full Tutorial For Beginners], MCC Py Tutorials, youtube.com
* [https://www.youtube.com/watch?v=LhPZwdhutgU Octave/MATLAB® for Beginners, Part 1: Starting from Scratch], MIT OpenCourseWare, youtube.com

Other:
* [https://octave-online.net/ Try Octave online], octave-online.net
* [http://www-h.eng.cam.ac.uk/help/programs/octave/tutorial/ CUED - Introduction to Octave], www-h.eng.cam.ac.uk -- a single-page introduction with about 15 000 words
* [https://learnxinyminutes.com/docs/matlab/ Learn MATLAB in Y Minutes], learnxinyminutes.com -- although for MATLAB, will work to a considerable extent for Octave

[[Category:GNU Octave|*]]</text>
      <sha1>kdcp3832u0sgh0055p06x4azhzkss2h</sha1>
    </revision>
    <revision>
      <id>2818437</id>
      <parentid>2818436</parentid>
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      <contributor>
        <username>Mu301</username>
        <id>3705</id>
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      <text bytes="3087" sha1="kbmke871z0sxhghpbejm5isvj5mjz0x" xml:space="preserve">[[File:Octave-11.1.0.png|thumb|right|The GNU Octave GUI workspace running on Linux.]]
GNU Octave is a free-as-in-freedom mathematical software package and a programming language, with considerable compatibility with MATLAB, a proprietary package. This page complements Wikipedia and Wikibooks. It should help readers who want to learn GNU Octave rather than just read about it. For a start, the page collects links to learning materials outside of Wikiversity.

==Getting started==
To get started, one can use octave-online.net, avoiding the need to install Octave. Using that environment, one can peruse [https://wiki.octave.org/Using_Octave Using Octave] from octave.org, try the examples and modify them. One can then move to the Octave Programming Tutorial in Wikibooks and solve the exercises that it has.

A quick glance, to be expanded:
* sin(3)
** Evaluates an expression and outputs the result.
* x = 0:0.1:7; plot (x, sin (x));
** Plots sine function on the given range, with the given step of 0.1.
* plot(x, sin(x), x, cos(x))
** Building on the above, plots multiple functions.
* m = [1, 1, 2; 3, 5, 8; 13, 21, 34]
** Creates a matrix.

Links:
* [https://octave-online.net/ Try Octave online], octave-online.net

==Compatibility with MATLAB==
As per octave.org, &quot;The Octave syntax is largely compatible with Matlab.&quot;&lt;ref&gt;[https://octave.org/ octave.org]&lt;/ref&gt; Moreover, differences between Octave and MATLAB are usually considered to be bugs. &lt;ref&gt;[https://wiki.octave.org/Differences_between_Octave_and_Matlab Differences between Octave and Matlab], wiki.octave.org&lt;/ref&gt;

Links:
* {{W|GNU Octave#MATLAB compatibility}}, wikipedia.org

==References==
&lt;references/&gt;

==Further reading==
Wikipedia and Wikibooks:
* {{W|GNU Octave}}, wikipedia.org
* [[B:Octave Programming Tutorial|Octave Programming Tutorial]], wikibooks.org

Official pages and documentation from octave.org:
* [https://www.octave.org/ GNU Octave], octave.org
* [https://wiki.octave.org/GNU_Octave_Wiki Octave], wiki.octave.org -- has links to different kinds of Octave documentation
* [https://wiki.octave.org/Using_Octave Using Octave], wiki.octave.org -- a quick introduction
* [https://wiki.octave.org/Octave_Basics Octave Basics], wiki.octave.org -- something like a reference card
* [https://docs.octave.org/latest/ Top (GNU Octave (version 8.4.0))], docs.octave.org -- the complete documentation

YouTube:
* [https://www.youtube.com/watch?v=woiU5PRVm7M GNU Octave - Full Tutorial For Beginners], MCC Py Tutorials, youtube.com
* [https://www.youtube.com/watch?v=LhPZwdhutgU Octave/MATLAB® for Beginners, Part 1: Starting from Scratch], MIT OpenCourseWare, youtube.com

Other:
* [https://octave-online.net/ Try Octave online], octave-online.net
* [http://www-h.eng.cam.ac.uk/help/programs/octave/tutorial/ CUED - Introduction to Octave], www-h.eng.cam.ac.uk -- a single-page introduction with about 15 000 words
* [https://learnxinyminutes.com/docs/matlab/ Learn MATLAB in Y Minutes], learnxinyminutes.com -- although for MATLAB, will work to a considerable extent for Octave

[[Category:GNU Octave|*]]</text>
      <sha1>kbmke871z0sxhghpbejm5isvj5mjz0x</sha1>
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  <page>
    <title>WikiJournal Preprints/Mental health in Sri Lanka</title>
    <ns>0</ns>
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      <text bytes="94164" sha1="j33joznds4505qwervyjankosfoevww" xml:space="preserve">{{Article info
| journal = WikiJournal of Medicine      &lt;!-- WikiJournal of Medicine, Science, or Humanities --&gt;
| last1 = Azeez
| orcid1 = 0009-0007-9202-4614
| first1 = Aaqib
| last2 = 
| first2 = 
| last3 = 
| first3 = 
| last4 = 
| first4 = &lt;!-- up to 9 authors can be added in this above format --&gt;
| et_al = &lt;!-- if there are &gt;9 authors, hyperlink to the list here --&gt;
| affiliation1 = Old Dominion University
| correspondence1 = aaqib.azeez@yahoo.com
| affiliations = institutes / affiliations
| correspondence = email@address.com
| keywords = &lt;!-- up to 6 keywords --&gt;
| license = &lt;!-- default is CC-BY --&gt;
| abstract = Mental health issues continue to be a significant problem in Sri Lanka, with 2022 suicide rates in the country reporting 15 suicides per 100,000 people, above the global average of 10.5 suicides per 100,000 people. The barriers to mental healthcare on the island are multi-faceted and are best understood with historical context. This narrative review covers the historical developments of mental healthcare,  mental health impacts of historical events within the last 100 years, current challenges affecting mental health outcomes, the role of the island's major religions in mental health and mental healthcare, and recommendations for improving future mental healthcare. The author uses peer-reviewed journal articles, relevant books, historical documents, and governmental/non-governmental reports to support clinical and historical claims, though non-peer-reviewed sources were used to contextualize historical and non-clinical claims. The narrative review concludes that outdated legislation, impacts from recent conflicts or disasters, stigma surrounding mental health, and economic vulnerability contribute to mental health issues and the inefficiency of mental healthcare services. The author recommends updating legal frameworks, expanding services, and raising awareness to mitigate social stigma.
}}

== Introduction ==
Mental health continues to be a critically relevant topic as the island nation has experienced decades of [[w:Black_July|violent ethnic conflict]], terrorist attacks, alleged war crimes, and economic disruptions. Sri Lanka continues to recover from a [[w:Sri_Lankan_economic_crisis_(2019–2024)|severe economic crisis (2019 - 2024)]], a [[w:Sri_Lankan_civil_war|nearly 30-year civil war ending in 2009]], a [[w:2019_Sri_Lanka_Easter_bombings|2019 terrorist attack]], and the [[w:2004_Boxing_Day_tsunami|2004 Boxing Day tsunami]]. The exact effect these major events have had on mental health in the country is &quot;unknown&quot;, but the statistics remain concerning despite a declining trend in the overall suicide rate.

Suicide rates in the country during the mid-1990s were the second-highest in the world, with ingesting toxic products being the main suicide method. Despite the decline in suicide numbers since then—possibly attributed to Sri Lanka's ban on toxic products—evidence from a 2023 study reports an upward trend in suicide through hanging from 2016 to 2021—independent of the [[w:COVID-19_pandemic_in_Sri_Lanka|COVID-19 pandemic]]. Several risk factors for suicide, such as poverty and economic instability, are still prevalent and even increasing in the country&lt;ref&gt;{{Cite journal|last=Rajapakse|first=Thilini|last2=Silva|first2=Tharuka|last3=Hettiarachchi|first3=Nirosha Madhuwanthi|last4=Gunnell|first4=David|last5=Metcalfe|first5=Chris|last6=Spittal|first6=Matthew J.|last7=Knipe|first7=Duleeka|date=2023-01-19|title=The Impact of the COVID-19 Pandemic and Lockdowns on Self-Poisoning and Suicide in Sri Lanka: An Interrupted Time Series Analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC9914278/|journal=International Journal of Environmental Research and Public Health|volume=20|issue=3|pages=1833|doi=10.3390/ijerph20031833|issn=1660-4601|pmc=9914278|pmid=36767200}}&lt;/ref&gt;.

== Methods ==
A narrative review was conducted on mental health in Sri Lanka. Sources used included peer-reviewed journal articles, relevant books, historical documents, and governmental/non-governmental reports. These sources were found on Google Scholar, PubMed/PMC, Sri Lankan journals, and official Sri Lankan governmental websites showing relevant statistics/reports. Keywords used to conduct searches included, but were not limited to: &quot;Sri Lanka mental health&quot;, &quot;Sri Lanka civil war trauma&quot;, &quot;Sri Lanka suicide&quot;, &quot;Sri Lanka mental health ordinances&quot;, &quot;Sri Lanka religion and mental health&quot;, &quot;Sri Lanka public mental healthcare&quot;, and &quot;Sri Lanka poverty/economic crisis mental health impact.&quot; Studies that were included were relevant to the topic (Sri Lanka, South Asian mental health law, suicide, public mental health, conflict/disaster trauma, or cultural/religious practice), had full text available, and were in the English language. Non-peer-reviewed sources were primarily used to explain historical claims or contextualize non-clinical claims. 

==Historical Development of Mental Health Services==

Records attest to the care of the mentally ill through established hospitals in the island since the 4th century.&lt;ref name=&quot;:17&quot; /&gt; Prior to the incarceration of the mentally ill by the European colonizing forces, the mentally ill were regarded as ''Pissowetitch'', or people who had &quot;the spirit of the Gods within him&quot; and &quot;whatsoever he pronounceth, is looked upon as spoken by God himself, and the people will speak to him, as if it were the very person of God&quot;&lt;ref&gt;{{Cite web|url=https://www.gutenberg.org/files/14346/14346-h/14346-h.htm|title=An Historical Relation Of the Island Ceylon, in the East-Indies: Together, With an Account of the Detaining in Captivity the Author and divers other Englishmen now Living there, and of the Author’s Miraculous Escape.|last=Knox|first=Robert|website=www.gutenberg.org|language=en-us|access-date=2026-06-29}}&lt;/ref&gt;. With this religious understanding, Lucien de Alwis reasoned that the mentally ill in Sri Lanka were &quot;placed... at a higher social status than the mentally ill in the Western world&quot;, with this understanding correlating with the unsurprising absence of evidence of any &quot;large scale segregation[s] of [the] mentally ill from society&quot;&lt;ref name=&quot;:17&quot; /&gt;. 

In the 1800s, established care for mental health began shifting primarily from indigenous practices, mainly derived from [[w:Ayurveda|Ayurveda medicine]], [[w:Siddha_medicine|Siddha medicine]], and [[w:Unani_medicine|Unani medicine]], to a Western model by the British&lt;ref name=&quot;:17&quot; /&gt;&lt;ref name=&quot;:0&quot;&gt;Gambheera, H. (2011). [https://www.saarcpsychiatry.com/viewText?chapter=c6 The evolution of psychiatric services in Sri Lanka]. South Asian Journal of Psychiatry, 2(1), 25–27.&lt;/ref&gt;&lt;ref name=&quot;:15&quot;&gt;{{Cite book|url=https://doi.org/10.1007/978-981-96-8078-8_7|title=Social Psychiatry in Sri Lanka|last=Baminiwatta|first=Anuradha|last2=Williams|first2=Shehan|date=2025|publisher=Springer Nature|isbn=978-981-96-8078-8|editor-last=Arafat|editor-first=S. M. Yasir|location=Singapore|pages=141–158|language=en|doi=10.1007/978-981-96-8078-8_7|editor-last2=Singh|editor-first2=Amit|editor-last3=Kar|editor-first3=Sujita Kumar}}&lt;/ref&gt;. 

=== Adoption of a Western-based mental healthcare model and ordinances ===
In 1839, [[w:James_Alexander_Stewart-Mackenzie|James Alexander Stewart-Mackenzie]], the 7th Governor of British Ceylon, released the Lunacy Ordinance, authorizing municipal authorities to create lunatic asylums for the mentally ill&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=article&amp;id=6&amp;Itemid=125&amp;lang=en|title=History - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-10}}&lt;/ref&gt;. The ordinance was concerned with the legal frameworks of detaining individuals considered dangerous to others or individuals falsely presenting themselves as mentally ill, and not on medical treatments to alleviate the conditions of detained individuals. UK psychiatrist [[w:Edward_Mapother|Edward Mapother]] critiqued the ordinance during his 1937 inspection of British Ceylon's mental health institutions in a series of reports titled ''A Disgrace to a Civilised Community'', remarking that the ordinance &quot;[did] not seem to have contemplated treatment as a contingency to be considered&quot;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|title=Permeable walls: historical perspectives on hospital and asylum visiting|date=2009|publisher=Rodopi|isbn=978-90-420-2599-8|editor-last=Mooney|editor-first=Graham|series=Clio medica|location=Amsterdam New York, NY|editor-last2=Reinarz|editor-first2=Jonathan}}&lt;/ref&gt;.

The 1839 Ordinance was repealed and replaced by the 1840 Ordinance, which removed two requirements from the previous Ordinance: the requirement for official medical diagnoses of the mentally ill and the mandate to maintain adequate staff-to-patient ratios within lunatic asylums&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Alwis|first=L. A. P. de|last2=Seneviratne|first2=V. L.|last3=Mendis|first3=T. S. S.|last4=Abhayanayaka|first4=C.|date=2024-12-31|title=The development of laws related to the disposal of forensic patients in Sri Lanka: A historical review|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v15i2.8569|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=15|issue=2|doi=10.4038/sljpsyc.v15i2.8569|issn=2012-6883}}&lt;/ref&gt;.

In 1873, a third Ordinance was released. It included linguistic changes, where the term, &quot;insane&quot;, was replaced with &quot;of unsound mind&quot;. The Ordinance also gave more power to medical professionals in determining insanity diagnoses, and more power to detainees in appealing their commitment to the mental asylum. Despite the increased granted authority, the legal frameworks behind the detainment of the criminally insane were left identical to previous ordinances&lt;ref name=&quot;:3&quot; /&gt;.

=== Development of mental asylums ===
At the time the 1839 ordinance was released, mentally ill patients were placed either in prisons throughout the country or leprosy hospitals, such as the [[w:Hendala_Leprosy_Hospital|Hendala Leprosy Hospital]] in the Gampaha district&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;. After the creation of the first mental asylum in Borella in 1846, patients from the Hendala Leprosy Hospital were transferred to Borella. Overcrowding soon became an issue, which led to patients being sent to prisons. [[File:Edward Mapother.jpg|thumb|A portrait taken of Edward Mapother during his time working at [[w:Maudsley_Hospital|Maudsley Hospital]] in London.
]]

As medical institutions were being made to house the mentally ill, another mental asylum was created in the [[w:Cinnamon_Gardens|Cinnamon Gardens]] area of Colombo in 1884, though this mental asylum faced overcrowding issues in just one year&lt;ref name=&quot;:0&quot; /&gt;. Treatment in these asylums was limited to occupational and protection therapy, failing to provide treatment for the root causes.

In 1926, the Angoda Mental Hospital was established, marginally alleviating the severe overcrowding issues that were plaguing the preceding mental asylums. Despite the addition of 1,700 beds to the facility, treatment was still vastly limited and the patients were left in significantly poor conditions. 

=== Edward Mapother and his 1937 inspection of British Ceylon ===
Edward Mapother was born in Dublin, Ireland, on July 12, 1881 and moved to London when he was 7 years old&lt;ref&gt;{{Cite book|title=Madness to mental illness: a history of the Royal College of Psychiatrists|last=Bewley|first=Thomas|date=2008|publisher=RCPsych Publications ; Distributed in North America by Balogh International|isbn=978-1-904671-35-0|location=London : [S.l.]}}&lt;/ref&gt;. Mapother attained his M.D. in 1908. While Mapother was the Medical Superintendent of Maudsley Hospital in London, England, he was invited to inspect British Ceylon's mental health institutions by Dr S. T. Gunasekara, the first Medical Director of British Ceylon&lt;ref name=&quot;:1&quot; /&gt;.

In Mapother's visit, he commented that the Angoda Mental Hospital had the atmosphere of &quot;a prison that is neglected and dilapidated&quot;&lt;ref name=&quot;:1&quot; /&gt;. Overcrowding was still a major issue, with the institute hosting 3,000 patients—more than double the intended capacity. Patients were sleeping on mats and were clearly out of reach of adequate treatment. Mapother also noted that only 4% of public health expenditure in the country was being set for hospitals, drawing a stark comparison to London's 25%&lt;ref name=&quot;:1&quot; /&gt;. Mapother offered a vivid and grim account of the hospital in his reports:
&lt;blockquote&gt;
The floor, roof and walls of each cell consist alike of drab cement without any attempt at colouring or decoration. High up in one wall is a small window with stout iron bars. In the floor is a large hole into which the patient may pass his motion and urine. These cells are incompletely divided from one another by a partition which does not reach the roof so that the noise and stink from any one cell may reach at least all the others of the same row. Into these empty cells I was informed that the most noisy and troublesome patients in the hospital; were turned at night completely naked. The doors of the cell contain no observation window, and considering the violent character of many of these patients there is every ground for believing that the doors are rarely opened in the night by the solitary attendant on duty. It needs little imagination to picture the suffering of any patient in an early stage of bodily illness passing a night under such conditions, a situation which must frequently arise. I am told that the noise proceeding from this building is like that on a bad night in a menagerie&lt;ref name=&quot;:0&quot; /&gt;.&lt;/blockquote&gt;Mapother proposed a series of reinforcements to the legal, institutional, and medical frameworks of mental health care in British Ceylon. This included the decentralization of the psychiatric services, a reworking of the Lunacy Ordinance to incorporate treatment into the legal framework, and the establishment of a separate service of medical professionals dedicated to psychiatry. Mapother's recommendations led to several of the best local medical professionals to be sent to London for extensive training in psychiatry, while nurses from England were sent to British Ceylon to supervise hospital operations and train local staff&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;.

On August 25, 1938, the Executive Committee of Health approved the strategies proposed by Mapother, though the Government was unable to fully implement all of Mapother's interventions due to the 'heavy cost'. In fact, the Government decided to forego one of his proposals at the beheast of the &quot;Visiting Committee&quot;, a committee that was tasked to &quot;meet at the hospital, carry out inspections, and make recommendations&quot; to the Executive Committee of Health&lt;ref name=&quot;:1&quot; /&gt;. The Government believed that deficiencies in their mental healthcare system could prove to be &quot;costly&quot; for their reputation, which enraged Maptoher. Mapother intended to contact the Secretary of State regarding the &quot;distortion&quot; of his plans, but was interrupted by events preceding [[w:World_War_II|World War II]]&lt;ref name=&quot;:1&quot; /&gt;. Mapother passed away on March 20, 1940, without materializing his follow-up plans.

=== Post-Mapother developments and further innovations ===
[[File:Sri Lanka districts Colombo.svg|thumb|A map of Sri Lanka highlighting the Colombo District, where the capital is located. 
|right|250px]]Mapother's insights on the mental healthcare structure in British Ceylon proved to be the catalyst of significant renovations. In 1939, the first outpatient clinic was established in the [[w:National_Hospital_of_Sri_Lanka|National Hospital of Sri Lanka]] in Colombo. The first trained Ceylonese psychiatrists began practice in the 1940s, leading to the establishment of the first neuropsychiatric clinic in Colombo in 1943. Treatments for the mentally ill improved dramatically, as [[w:insulin_shock_therapy|insulin shock therapy]] and [[w:Electroconvulsive_therapy|cardiazol convulsive therapy]] were utilized&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Kathriarachchi|first=Samudra T.|last2=Seneviratne|first2=V. Lakmi|last3=Amarakoon|first3=Luckshika|date=2019-06|title=Development of Mental Health Care in Sri Lanka: Lessons Learned|url=https://journals.lww.com/tpsy/fulltext/2019/33020/development_of_mental_health_care_in_sri_lanka_.1.aspx|journal=Taiwanese Journal of Psychiatry|language=en-US|volume=33|issue=2|pages=55|doi=10.4103/TPSY.TPSY_15_19|issn=1028-3684}}&lt;/ref&gt;. Mapother's advocation for the decentralization of services were further honored through the 1947 establishment of a first child guidance clinic in Colombo General Hospital&lt;ref name=&quot;:0&quot; /&gt;. 

In 1948, British Ceylon was granted independence after the [[w:Sri_Lankan_independence_movement|Sri Lankan independence movement]]. Changes in the mental healthcare structure were not immediate following independence, but rapid expansions of mental healthcare services were continuing to actualize.  

The following decades saw positive institutional developments, such as the creation of a second hospital in [[w:Mulleriyawa|Mulleriyawa]] in 1957, and the creation of a psychiatric inpatient unit in Colombo General Hospital in 1967—effectively granting the city of Colombo the luxury of hosting the top psychiatric care in the country&lt;ref name=&quot;:5&quot;&gt;{{Cite book|url=http://link.springer.com/10.1007/978-1-4899-7999-5_4|title=Mental Health System Development in Sri Lanka|last=Minas|first=Harry|last2=Mendis|first2=Jayan|last3=Hall|first3=Teresa|date=2017|publisher=Springer US|isbn=978-1-4899-7997-1|editor-last=Minas|editor-first=Harry|location=Boston, MA|pages=59–77|language=en|doi=10.1007/978-1-4899-7999-5_4|editor-last2=Lewis|editor-first2=Milton}}&lt;/ref&gt;. The 1950s was also the start of psychopharmacological innovations, with the introduction of [[w:Lithium_(medication)|lithium]] and long-acting injectable antipsychotics ([[w:Depot_injection|depot]] [[w:Antipsychotic|neuroleptics]]) in the succeeding years&lt;ref name=&quot;:4&quot; /&gt;. Additionally, the number of public psychiatrist positions increased by 400% from 1953 to 1967&lt;ref name=&quot;:5&quot; /&gt;.

After 1960, mental health services were expanded from beyond the capital to other cities in the country&lt;ref name=&quot;:2&quot; /&gt;.

In 1980, the [[w:Postgraduate_Institute_of_Medicine|Postgraduate Institute of Medicine]] initiated a program where students would enroll in a 5-year medical course and attain an MD in psychiatry, curbing the need for Sri Lankan medical students to be sent abroad to complete their training. Many of the medical students sent abroad for training never returned to Sri Lanka to practice, resulting in a &quot;1:500,000 to 1000,000&quot; ratio of psychiatrists to patients on &quot;most occasions&quot;&lt;ref name=&quot;:0&quot; /&gt;.

=== Mental Disease Ordinance of 1956 ===
In 1956, the 1873 Ordinance was revised a second time. The Mental Disease Ordinance of 1956 featured another linguistic development, as &quot;lunacy&quot; was replaced with &quot;mental disease&quot;&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|last=Hapangama|first=Aruni|last2=Mendis|first2=Jayan|last3=Kuruppuarachchi|first3=K. a. L. A.|date=2023-02|title=Why are we still living in the past? Sri Lanka needs urgent and timely reforms of its archaic mental health laws|url=https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4|journal=BJPsych International|language=en|volume=20|issue=1|pages=4–6|doi=10.1192/bji.2022.26|issn=2056-4740|pmc=9909436|pmid=36812028}}&lt;/ref&gt;. The Ordinance paved way for community-based services to be delivered to patients closer to their residences, rather than strictly allocating services to just hospitals. This led to the creation of a [[w:WHO|WHO]]-backed community clinic near the [[w:University_of_Colombo|University of Colombo]] in the 1970s, where the focus was to eventually ease patients in the Angoda Mental Hospital back into the general population&lt;ref name=&quot;:5&quot; /&gt;.

=== Developments from the 1990s ===
The 1990s and onwards saw further positive developments in framing the mental healthcare system, including the establishment of the [https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=featured&amp;Itemid=101&amp;lang=en Directorate of Mental Health] in 1998. The Directorate of Mental Health is a part of the [[w:Ministry_of_Health_(Sri_Lanka)|Ministry of Health]] and is responsible for the monitoring and implementation of mental health programs across the country&lt;ref&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?lang=en|title=Home - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}&lt;/ref&gt;.  As of 2025, the current director of the Directorate of Mental Health is Dr. Chithramalee de Silva&lt;ref name=&quot;:2&quot; /&gt;.

On November 11, 2005, the Mental Health Policy was approved by the Government of Sri Lanka, advocating for establishments of more de-centralized, community-based mental health services across the country. The policy aimed to concisely define the rigorous standards needed to be met for each respected medical professional, including psychiatrists and clinical psychologists&lt;ref&gt;{{Cite journal|last=Rajapakshe|first=Onali Bimalka Wickramaseckara|last2=Mohan|first2=Mohapradeep|last3=Singh|first3=Swaran Preet|date=2023-05|title=Development of adolescent mental health services in Sri Lanka|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC10895478/|journal=BJPsych international|volume=20|issue=2|pages=41–43|doi=10.1192/bji.2022.32|issn=2056-4740|pmc=10895478|pmid=38414998}}&lt;/ref&gt;. The policy also included a new position, the &quot;Medical Officer of Mental Health&quot;, tasked with overseeing and assisting in creating community-based mental health services&lt;ref name=&quot;:0&quot; /&gt;. In the same year, the Sri Lankan government began implementing psychological services in state institutions, such as the military&lt;ref name=&quot;:8&quot; /&gt;.  

In 2007, the National Mental Health Advisory Council (NMHAC) was created to serve as an 'advisory' board for the Ministry of Health&lt;ref name=&quot;:7&quot;&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=article&amp;id=9&amp;Itemid=220&amp;lang=en|title=Introduction - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}&lt;/ref&gt;. 

In 2008, the Angoda Mental Hospital was restructured and renamed as the National Institute of Mental Health (NIMH)&lt;ref name=&quot;:7&quot; /&gt;. 

=== Modern-day Sri Lanka ===
[[File:Feeding Children in Sri Lanka.jpg|left|thumb|Despite the noteworthy improvements in mental healthcare services in recent decades, mental health remains a significant issue due to rising poverty. ]]
As of 2025, the Mental Health Act (mental health legislation) has been undergoing development since 2005 and is currently awaiting to be considered for the final stage of approval. This is expected to replace the 1956 Mental Health Ordinance&lt;ref name=&quot;:7&quot; /&gt;.

Currently, there are 7 tertiary care hospitals, 61 adult patient units, 3 child inpatient units, and 1 forensic unit with over 100 psychiatrists all throughout the 22 districts&lt;ref name=&quot;:4&quot; /&gt;. The [[w:Lady_Ridgeway_Hospital_for_Children|Lady Ridgeway Hospital]] in Colombo and the Sirimavo Bandaranayke Specialized Children Hospital in Kandy are specialized in treating children with [[w:Learning_disability|SLD]], [[w:ADHD|ADHD]], [[w:Autism_Spectrum_Disorder|ASD]], and provides family support for patients. As of 2017, 22 rehabilitation centers exist through the country, including 7 alcohol rehab centers&lt;ref name=&quot;:7&quot; /&gt;.   

Despite the impressive advancements in mental healthcare in the last couple of decades, Sri Lanka still suffers significant mental health issues due to increasing poverty levels in the country. The [[w:World_Bank|World Bank]] reported that [https://www.wsws.org/en/articles/2024/04/08/eesc-a08.html the poverty levels in Sri Lanka increased from 11% in 2019 to 26% in 2024], with 60% of Sri Lankan households facing &quot;decreased incomes&quot;&lt;ref&gt;Lakhtakia, Shruti, Atapattu Mudiyanselage, Udahiruni Shashadari Atapat, Walker, Richard Ancrum. ''Sri Lanka Development Update - Bridge to Recovery (English).'' Washington, D.C.: World Bank Group. &lt;nowiki&gt;http://documents.worldbank.org/curated/en/099634104012434919&lt;/nowiki&gt;&lt;/ref&gt;. This was exacerbated by Sri Lanka's excessive foreign debt, economic troubles stemming from [[w:Gotabaya_Rajapaksa|Gotabaya Rajapaksa]]'s presidential term, the COVID-19 pandemic, and the [[w:Russian_invasion_of_Ukraine|ongoing invasion of Ukraine by Russia (2022)]].  

According to [[w:NYU|New York University]] graduate student [https://gc-cuny.academia.edu/NadiaAugustyniak Nadia Augustyniak] in her 2025 overview of Sri Lanka's public mental healthcare system, poverty-induced financial precarity remains a major obstacle to receiving access to mental healthcare services. Even though trauma from adverse weather and conflict is deleterious to mental health, issues originating from every-day struggles, especially struggles related to poverty, could arguably play a more significant role&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}&lt;/ref&gt;.  

== Impact of Conflicts, Terrorism, Political Instability &amp; Natural Disasters ==

=== Sri Lankan Civil War ===
The '''Sri Lankan Civil War''' was a domestic conflict between the Sri Lankan government and the Liberation Tigers of Tamil Eelam (abbreviated as the ''LTTE),'' a militant group formed in the 1970s as a byproduct of rising tensions between the majority Sinhalese and minority Tamil population. The group is considered a terrorist organization&lt;ref&gt;{{Cite web|url=https://www.start.umd.edu/baad/database/liberation-tigers-tamil-eelam-ltte-1998.html|title=BAAD - Liberation Tigers of Tamil Eelam (LTTE) - 1998 {{!}} START.umd.edu|website=www.start.umd.edu|access-date=2025-06-09}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.cfr.org/backgrounder/liberation-tigers-tamil-eelam-aka-tamil-tigers-sri-lanka-separatists|title=Liberation Tigers of Tamil Eelam (aka Tamil Tigers) (Sri Lanka, separatists) {{!}} Council on Foreign Relations|last=Bhattacharji|first=Preeti|website=www.cfr.org|language=en|access-date=2025-06-09}}&lt;/ref&gt;. The LTTE conducted decades of massacres, assassinations of political figures, and suicide bombings to achieve ''[[w:Tamil_Eelam|Tamil Eelam]],'' leading to civilian displacement, infrastructure collapse, and the reduction of mental health services available in the northern region.[[File:DFID-funded, UNHCR emergency shelter tents, in the IDP camp at Menik Farm, Sri Lanka (3694081492).jpg|thumb|350x350px|An IDP camp in Menik Farm, Sri Lanka in 2009 ([https://www.bbc.com/news/world-asia-19703826 now closed]). Suicide rates in IDP camps were three times the general population.]]The civil war mainly affected the northeastern portion of the country, including the [[w:Vanni_(Sri_Lanka)|Vanni region]]. The conflict caused mass destruction to local mental healthcare facilities. Local residents described the conflict as ''varthayal varnicca mudiyathavai'', roughly translating into English as 'beyond description by words'&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|language=en|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}&lt;/ref&gt;. In 2003, only two psychiatrists were found in the region, operating on extremely limited resources. This furthered long-term trauma and mental health deterioration in the population&lt;ref name=&quot;:5&quot; /&gt;. 

In 2002, the humanitarian organization [https://www.msf.org/ Médecins Sans Frontières] (MSF) conducted an investigation on mental health needs in the [[w:Vavuniya|Vavuniya]] area, the site of intense conflict during the civil war (including the [[w:1985_Vavuniya_massacre|1985 Vavuniya massacre]]), and found that many of the residents suffered from high suicide rates, alcohol abuse, domestic violence, grief, and a &quot;sense of ‘learnt helplessness’&quot;&lt;ref name=&quot;:5&quot; /&gt;. A team from the University of Konstanz in Germany found that 92% of grade school children in the region were exposed to &quot;combat, shelling, and witnessing the death of loved ones&quot;&lt;ref name=&quot;:9&quot; /&gt;. 
[[File:Tractors. Jan 2009 displacement in the Vanni.jpg|left|thumb|350x350px|Displaced civilians evacuating from the Kilinochchi and Mullaitivu Districts due to military campaigns initiated by the Sri Lankan military (January 2009).]]
Additionally, accusations of war crimes have been made against [[w:War_crimes_during_the_final_stages_of_the_Sri_Lankan_civil_war|the Sri Lankan government]]&lt;ref&gt;See also [[w:Sexual violence in the Sri Lankan civil war]].&lt;/ref&gt;. A 2009 HRW report alleged that the Sri Lankan government considered the native Tamil population residing in war zones to be &quot;siding with the LTTE and [therefore, were] treated as combatants&quot;, and that the government conducted numerous shellings of &quot;areas crowded with civilians&quot;&lt;ref&gt;{{Cite journal|date=2009-02-19|title=War on the Displaced|url=https://www.hrw.org/report/2009/02/19/war-displaced/sri-lankan-army-and-ltte-abuses-against-civilians-vanni|journal=Human Rights Watch|language=en}}&lt;/ref&gt;. Furthermore, the LTTE conducted recruitment campaigns on the Vanni population where recruited men, women, and even children with minimal training, were recruited for war efforts.  

Over 200,000 Tamil civilians were moved into [[w:Internally_displaced_persons_in_Sri_Lanka|designated displacement camps during the war]], where conditions were poor&lt;ref&gt;{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000}}&lt;/ref&gt;. The suicide rate in these displacement camps was three times the community-level (2002), with a ratio of 103.5 suicides per 10,000 persons, compared to the general population's rate of 37.5 suicides per 10,000 persons. Almost all suicide attempts involved poisonous substances. Other forms of violence included domestic violence and child abuse.  

Local health officials in Vavuniya admitted that mental health concerns were a major problem, but were unable to address these concerns due to a lack of resources and support from the government. During the [[wikipedia:Sri_Lankan_civil_war#2002_peace_process_(2002%E2%80%932006)|brief 2002 ceasefire]], the MSF implemented a &quot;community-based programme&quot; which included &quot;increasing awareness, community strengthening, reinforcing coping-strategies for long-term war-affected communities, and counselling&quot;. The MSF also advocated for restrictions of poisonous substances due its means for suicide attempts, and stressed that &quot;much more [than resettlement]&quot; would need to be done to help alleviate the psychological pain the northern population had faced due to the war&lt;ref&gt;{{Cite journal|last=de Jong|first=Kaz|last2=Mulhern|first2=Maureen|last3=Ford|first3=Nathan|last4=Simpson|first4=Isabel|last5=Swan|first5=Alison|last6=van der Kam|first6=Saskia|date=2002-04|title=Psychological trauma of the civil war in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S0140673602084209|journal=The Lancet|language=en|volume=359|issue=9316|pages=1517–1518|doi=10.1016/S0140-6736(02)08420-9}}&lt;/ref&gt;. The ceasefire ended in 2006 and led to the [[w:Eelam_War_IV|final phase of the civil war]], eventually ending in 2009 with the [[w:https://en.wikipedia.org/wiki/Velupillai_Prabhakaran#Sri_Lankan_Army_Northern_offensive_and_death|death of the LTTE's leader]]. 

'''Post-war'''
[[File:Puttalam district.svg|left|thumb|Puttalam District, unlike its northern counterparts, was largely spared from the intense conflict, possibly explaining the lower rates of common mental disorders (CMDs).]]
The first district-wide cross-sectional multistage cluster sample survey was conducted in the [[w:Jaffna_District|Jaffna District]] shortly after the war ended in 2009. The study's sample included 1517 households and 2 internally displaced peoples camps. With a response rate of 92%, the study found that symptoms for PTSD were found in 7% of participants, symptoms of anxiety were found in 32.6% of participants, and symptoms of depression were found in 22.2% of participants. 2% of respondents were being placed in internally displaced peoples camps at the time of the study, 29.5% were freshly resettled from the internally displaced peoples camps, and the rest of the participants (68.5%) were never placed into camps. In comparison to residents who were never placed into camps, participants that were actively held in camps generally reported more symptoms of PTSD, anxiety, and depression. The researchers also found that women were especially vulnerable to deteriorating mental health conditions. This was explained by two factors: women having to assume the roles of both the father and the mother in the family setting after the, either voluntary or forced, departure of their husband to war, and sexist violence&lt;ref&gt;{{Cite journal|last=Husain|first=Farah|last2=Anderson|first2=Mark|last3=Lopes Cardozo|first3=Barbara|last4=Becknell|first4=Kristin|last5=Blanton|first5=Curtis|last6=Araki|first6=Diane|last7=Kottegoda Vithana|first7=Eeshara|date=2011-08-03|title=Prevalence of War-Related Mental Health Conditions and Association With Displacement Status in Postwar Jaffna District, Sri Lanka|url=https://doi.org/10.1001/jama.2011.1052|journal=JAMA|volume=306|issue=5|pages=522–531|doi=10.1001/jama.2011.1052|issn=0098-7484}}&lt;/ref&gt;. A 2013 study on adult patients in [https://www.ncbi.nlm.nih.gov/books/NBK232631/ primary care settings] (divisional hospitals, primary medical care units) found major depression to be significantly higher in females (5.1%) than males (3.6%), bolstering the findings from the 2009 study&lt;ref&gt;{{Cite journal|last=Senarath|first=Upul|last2=Wickramage|first2=Kolitha|last3=Peiris|first3=Sharika Lasanthi|date=2014-03-24|title=Prevalence of depression and its associated factors among patients attending primary care settings in the post-conflict Northern Province in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/1471-244X-14-85|journal=BMC Psychiatry|language=en|volume=14|issue=1|pages=85|doi=10.1186/1471-244X-14-85|issn=1471-244X|pmc=3987835|pmid=24661436}}&lt;/ref&gt;.

Muslims in Northern Sri Lanka also faced violence and discrimination during the conflict. Most notable incidents include [[w:Expulsion_of_Muslims_from_the_Northern_Province_of_Sri_Lanka|the October 1990 expulsion of Muslims from the North to the Puttalam District or Jaffna]] and the [[w:Kattankudy_mosque_massacre|1990 Kattankudy mosque massacre]]. The only study testing the displaced Muslim population post-civil war was completed in 2011, where a cross-sectional survey of 450 internally displaced people or people born into displacement (ages 18 - 65) revealed 18.8% of the sample suffering from common mental health disorders (CMD), including [[w:Somatoform_disorder|somatoform disorder]] (14%), &quot;other depressive syndromes&quot; (7.3%), major depression (5.1%), and anxiety disorder (2.8%). The percentages found in this study for somatoform disorder and major depression were &quot;considerably higher&quot; than the national percentages, though the researchers noted that the prevalence of CMD was lower in comparison to other countries marred with conflict, including Palestine (40.3%) and Ethiopia (27.8%). The researchers explained that the lower rate of CMD may be attributed to the [[w:Puttalam_District|serenity of the post-settlement destination]], as conflict was mainly centered in the North and East. In contrast to earlier findings, this study did not observe a higher prevalence of CMDs among women, although increased rates of somatoform disorders were noted (though the researchers did not reveal the data behind this)&lt;ref&gt;{{Cite journal|last=Siriwardhana|first=Chesmal|last2=Adikari|first2=Anushka|last3=Pannala|first3=Gayani|last4=Siribaddana|first4=Sisira|last5=Abas|first5=Melanie|last6=Sumathipala|first6=Athula|last7=Stewart|first7=Robert|date=2013-05-22|title=Prolonged Internal Displacement and Common Mental Disorders in Sri Lanka: The COMRAID Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0064742|journal=PLOS ONE|language=en|volume=8|issue=5|pages=e64742|doi=10.1371/journal.pone.0064742|issn=1932-6203|pmc=3661540|pmid=23717656}}&lt;/ref&gt;.

Research on the mental state of combatants has been limited, but a post-war 2009 study done between soldiers of the [[w:Sri_Lanka_Army_Special_Forces_Regiment|Special Forces]] and regular soldiers showed higher levels of exposure to traumatic events for units of the Special Forces, yet the former exhibited significantly less symptoms of CMDs compared to the latter. The authors of this study, [https://scholar.google.co.uk/citations?user=cVKEBdwAAAAJ&amp;hl=en&amp;oi=ao Raveen Hanwella] and [https://scholar.google.co.uk/citations?user=ZRj74qMAAAAJ&amp;hl=en&amp;oi=sra Varuni de Silva], offered the camaraderie of the military unit as an explanation for the discrepancy&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|date=2012-08|title=Mental health of Special Forces personnel deployed in battle|url=https://pubmed.ncbi.nlm.nih.gov/22038567|journal=Social Psychiatry and Psychiatric Epidemiology|volume=47|issue=8|pages=1343–1351|doi=10.1007/s00127-011-0442-0|issn=1433-9285|pmid=22038567}}&lt;/ref&gt;. A follow-up study was completed by the pair (with the addition of former Director-General of the Health Services of the Sri Lanka Navy [[w:Nicholas_Jayasekera|Nicholas Jayasekera]]), where the findings were similar, though the statistically significant bridge between the two cohorts in the previous study evaporated in the follow-up study. This may be due to the significant decline in mental health problems observed in the regular unit forces, potentially reflecting resilience in the aftermath of the conflict&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=Jayasekera|first2=Nicholas E. L. W.|last3=Silva|first3=Varuni A. de|date=2014-09-25|title=Mental Health Status of Sri Lanka Navy Personnel Three Years after End of Combat Operations: A Follow Up Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0108113|journal=PLOS ONE|language=en|volume=9|issue=9|pages=e108113|doi=10.1371/journal.pone.0108113|issn=1932-6203|pmc=4177866|pmid=25254557}}&lt;/ref&gt;. Amputees or soldiers with spinal injuries exhibited drastically different numbers, with approximately 40% of nearly 100 male-veterans in a post-war 2009 study displaying PTSD-like symptoms&lt;ref&gt;{{Cite journal|last=Abeyasinghe|first=N. L.|last2=de Zoysa|first2=P.|last3=Bandara|first3=K.M.K.C.|last4=Bartholameuz|first4=N. A.|last5=Bandara|first5=J. M.U.J.|date=2012-05-01|title=The prevalence of symptoms of Post-Traumatic Stress Disorder among soldiers with amputation of a limb or spinal injury: A report from a rehabilitation centre in Sri Lanka|url=https://doi.org/10.1080/13548506.2011.608805|journal=Psychology, Health &amp; Medicine|volume=17|issue=3|pages=376–381|doi=10.1080/13548506.2011.608805|issn=1354-8506|pmid=21942815}}&lt;/ref&gt;.

About a decade after the conflict ceased, a few notable studies have emerged to help guide understanding on the longer-term mental health effects on victims of the civil war. 

From July 2019 to October 2020, a study conducted on 585 local adolescents (ages 12-19) in the Vavuniya district revealed that despite 15.6% of the statistic having faced one or more war-related events, only 3.9% of the participants had moderate to severe depression. In addition to considerably low depression rates, only 5.7% of participants age 17+ were found to have moderate to severe hopelessness&lt;ref&gt;{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000|pmc=10472617|pmid=37653394}}&lt;/ref&gt;. The authors referenced a 2010 observation by psychiatrist [https://us.sagepub.com/en-us/nam/author/daya-somasundaram Daya Somasundaram], who noted that many Tamil IDPs presented &quot;remarkable resilience and post-traumatic growth&quot; after the civil war—an outcome he attributed to the close-knit, family-centered nature of Tamil communities&lt;ref&gt;{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}&lt;/ref&gt;.  

However, findings originating from a 2019 study, undertook by several faculty members from the University of Kelaniya, the University of Jaffna, the [[w:Gampaha_Wickramarachchi_University_of_Indigenous_Medicine|Gampaha Wickramarachchi University of Indigenous Medicine]], and the [https://onur.gov.lk/ Office for National Unity and Reconciliation (ONUR)] in Jaffna, found contrasting results. Out of 336 participants from districts which faced significant ramifications of the conflict (Jaffna, Kilinochchi, Mullaithivu, Vavuniya, and Mannar districts), 50.5% had extreme anxiety symptoms and 36.5% exhibited &quot;extremely severe&quot; symptoms of depression. 92.5% of families in the sample experienced suicidal ideation, with an observed negative correlation between trauma exposure and life satisfaction with families. Drug abuse (86.2%) and alcohol abuse (84.5%) were the two highest problematic behaviors recorded on a community-level, suggesting that the negative consequences of the civil war still persist, possibly on a substantial scale than previously recognized, in Tamil communities residing in the North&lt;ref&gt;{{Cite journal|last=Thamotharampillai|first=Umaharan|last2=Perera|first2=Ruwanthi|last3=Wickremasinghe|first3=Rajitha|last4=Williams|first4=Shehan|last5=Vijayasangar|first5=Thedsanamoorthy|last6=Sivatharsan|first6=Balasubramaniam|last7=Hilbert|first7=Vanceline|last8=Somasundaram|first8=Daya|date=2025-05-06|title=Collective Trauma- Psychosocial consequences of war in northern Sri Lanka 10 years on, a mixed methods study|url=https://www.sciencedirect.com/science/article/pii/S2666560325000696|journal=SSM - Mental Health|pages=100457|doi=10.1016/j.ssmmh.2025.100457|issn=2666-5603}}&lt;/ref&gt;. Further research should be conducted on Northern Tamil populations to assess the extent of mental health issues stemming from the conflict. 

In 2019, [https://www.researchgate.net/scientific-contributions/R-M-M-Monaragala-2087692299 Dr. R. M. M. Monaragala] conducted a study on 1,845 soldiers with combat experience, finding that 3.9% of the sample suffered from PTSD. Dr. Monaragala noted that &quot;probable depression, fatigue, aggression, and family history of mental disorder&quot; were correlative of PTSD presence. He suggested that &quot;screening and psychosocial intervention[s]&quot; could alleviate CMDs of former combatants&lt;ref&gt;{{Cite journal|last=Monaragala|first=R. M. M.|date=2024-04-19|title=Exploring the effects of the past civil war in terms of the prevalence and associating factors of PTSD|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v14i2.8465|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=14|issue=2|doi=10.4038/sljpsyc.v14i2.8465|issn=2012-6883}}&lt;/ref&gt;. 

=== 2004 Boxing Day Tsunami ===
The '''2004 Boxing Day Tsunami''' was a natural disaster where a tsunami spawned off a 9.2–9.3 magnitude earthquake off the coast of Aceh in Indonesia on December 26. The tsunami greatly affected the coastlines of the country, with the death toll reaching to around 35,000 deaths. In addition, 90,000 houses were destroyed and 516,000 people were forced to migrate due to severe infrastructural damage&lt;ref name=&quot;:5&quot; /&gt;. It stands as the [http://www.china.org.cn/english/features/tsunami_relief/119821.htm worst natural disaster to have ever hit Sri Lanka].  
[[File:Tsunami relief 2004 02.jpg|thumb|300x300px|Volunteers from [[w:Royal_College,_Colombo|Royal College in Colombo]] assisting in tsunami relief efforts (Sarvodaya Headquaters, Moratuwa).]]
A survey conducted on schoolchildren (ages 8-14) in Manadkadu (a Tamil-majority village in the northern coast), [[w:Kosgoda|Kosgoda]] (western coast), and [[w:Galle|Galle]] (southern coast), just a few weeks after the tsunami hit Sri Lanka, revealed that 33.8%, 13.9%, and 38.8% of children interviewed exhibited signs of PTSD (according to the DSM-IV's criteria), respectively (minus the time criteria, as the DSM-IV does not permit diagnosis of PTSD within 4 weeks of a traumatic incident). The loss of family members and exposure to previously traumatic incidents appeared to be highly correlate with PTSD development&lt;ref&gt;{{Cite journal|last=Neuner|first=Frank|last2=Schauer|first2=Elisabeth|last3=Catani|first3=Claudia|last4=Ruf|first4=Martina|last5=Elbert|first5=Thomas|date=2006|title=Post-tsunami stress: A study of posttraumatic stress disorder in children living in three severely affected regions in Sri Lanka|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/jts.20121|journal=Journal of Traumatic Stress|language=en|volume=19|issue=3|pages=339–347|doi=10.1002/jts.20121|issn=1573-6598}}&lt;/ref&gt;. 

Many victims in the Jaffna area suffered with &quot;[https://www.psychiatry.org/patients-families/prolonged-grief-disorder pathological grief], phobias, depression and PTSD&quot; post-tsunami. Schizophrenia in the Jaffna Tamil community, which had already suffered elevated prevalence of PTSD prior to the tsunami, had worsened—highlighting the need for specialized care in response to cumulative exposures to chronic and acute traumas. In a study published in ''International Psychiatry'' (2006), Jaffna-based researchers noted that, contrary to their initial inclinations, there was not a &quot;large[r] (than expected) rise in [the] number of people&quot; seeking mental health support 3 months after the tsunami. However, 10 months after the disaster, the researchers anticipated that &quot;more psychiatric disorders&quot; would emerge due to &quot;very little rebuilding [efforts]&quot; and an apparent &quot;unfairness in the aid system&quot;.&lt;ref&gt;{{Cite journal|last=Somasundaram|first=D. J.|last2=Yoganathan|first2=S.|last3=Ganesvaran|first3=T.|date=1993-09|title=Schizophrenia in northern Sri Lanka|url=https://pubmed.ncbi.nlm.nih.gov/7828234|journal=The Ceylon Medical Journal..|volume=38|issue=3|pages=131–135|issn=0009-0875|pmid=7828234}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Danvers|first=K.|last2=Sivayokan|first2=S.|last3=Somasundaram|first3=D. J.|last4=Sivashankar|first4=R.|date=2006-07|title=Ten months on: qualitative assessment of psychosocial issues in northern Sri Lanka following the tsunami|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6734678/|journal=International Psychiatry: Bulletin of the Board of International Affairs of the Royal College of Psychiatrists|volume=3|issue=3|pages=5–8|issn=1749-3676|pmc=6734678|pmid=31507850}}&lt;/ref&gt; 

At the February 2005 ''After the Tsunami: Mental Health Challenges to the Community for Today and Tomorrow'' conference in Thailand, [https://www.researchgate.net/profile/Chandanie-Hewage Dr. Chandanie Hewage] of the [[w:University_of_Ruhuna|University of Ruhuna]] commentated that measures taken to assist the affected were &quot;not coordinated&quot; due to poor &quot;communication systems and road [conditions].&quot; Regardless, efforts were continued by the government and health professionals to alleviate the struggles the victims were facing, including the psychological ramifications of the disaster. 

Several issues in the delivery of these services were highlighted by Dr. Hewage, including poor maintenance of health records, lack of awareness on drug consumption by the patients themselves, and shortages of health professionals. Dr. Hewage points out that personnel had &quot;little&quot; mental health training prior to the disaster, suggesting increased &quot;research&quot; and adequate &quot;provision[ing] and training of staff&quot; for the long-term&lt;ref&gt;{{Cite journal|last=Davidson|first=Jonathan R. T.|date=2006|title=Foreword. After the tsunami: mental health challenges to the community for today and tomorrow|url=https://pubmed.ncbi.nlm.nih.gov/16602809|journal=The Journal of Clinical Psychiatry|volume=67 Suppl 2|pages=3–8|issn=0160-6689|pmid=16602809}}&lt;/ref&gt;. With inadequate documentation, no systematic procedures in place, and insufficient personnel, tsunami victims with mental health concerns may not receive the services they need, further compacting neuropsychological ailments.  

In 2008 (about 3-4 years after the tsunami), researchers in the hard-hit village of [[w:Peraliya|Peraliya]] (Galle District) found that from a sample of approximately 90 adults, 25% suffered from moderate–severe PTSD, with women scoring &quot;above the cut-off for anxiety&quot; and reporting more &quot;somatic symptoms&quot;, though researchers inferred that the PTSD rate found in the study may be influenced by other factors, including war or economic hardship&lt;ref&gt;{{Cite journal|last=Hollifield|first=Michael|last2=Hewage|first2=Chandanie|last3=Gunawardena|first3=Charlotte N.|last4=Kodituwakku|first4=Piyadasa|last5=Bopagoda|first5=Kalum|last6=Weerarathnege|first6=Krishantha|last7=Group|first7=International Post-Tsunami Study|date=2008-01|title=Symptoms and coping in Sri Lanka 20–21 months after the 2004 tsunami|url=https://www.cambridge.org/core/journals/the-british-journal-of-psychiatry/article/symptoms-and-coping-in-sri-lanka-2021-months-after-the-2004-tsunami/CB33752239AF362A0BFD55B3668D60B0|journal=The British Journal of Psychiatry|language=en|volume=192|issue=1|pages=39–44|doi=10.1192/bjp.bp.107.038422|issn=0007-1250}}&lt;/ref&gt;.  

=== 2019 Easter Bombings ===
The '''2019 Easter Bombings''' were a series of coordinated attacks perpetrated by the Islamic extremist group, [[w:National_Thowheeth_Jama'ath|National Thowheeth Jama'ath]], on April 21, 2019. The attack targeted three churches and three hotels in the Colombo area, killing nearly 300 people and injuring over 500. The attacks were also attributed to the incompetency of the Sri Lankan government, who ignored [https://www.bbc.com/news/world-asia-48044636 multiple warnings preceding the attacks]. The attacks negatively affected the Sri Lankan Catholic community and further weakened relations between the major religious groups&lt;ref&gt;{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}&lt;/ref&gt;.  

In the aftermath of the attacks, professionals in the [[w:Gampaha_District|Gampaha District]] resorted to &quot;low-cost methodologies&quot; for children and adolescents affected by the attack, as a &quot;severe shortage&quot; of children and adolescent mental health experts were exposed&lt;ref&gt;{{Cite journal|last=Chandradasa|first=Miyuru|last2=Rathnayake|first2=Layani C|last3=Rowel|first3=Madushi|last4=Fernando|first4=Lalin|date=2020-06-01|title=Early phase child and adolescent psychiatry response after mass trauma: Lessons learned from the Easter Sunday attack in Sri Lanka|url=https://doi.org/10.1177/0020764020913314|journal=International Journal of Social Psychiatry|language=EN|volume=66|issue=4|pages=331–334|doi=10.1177/0020764020913314|issn=0020-7640}}&lt;/ref&gt;. In a qualitative study of 8 survivors of the attacks receiving grief counseling, [[w:University_of_Ruhuna|University of Ruhuna]] assistant professor [https://www.researchgate.net/profile/Virasha-Godakanda Virasha Godakanda] observed that 70% of the sample size expressed a lack of confidence in adequate mental health interventions from the government, reducing the quality of such services. Professor Godakanda strongly endorsed for &quot;culturally-sensitive&quot; programs, a diversity in therapeutic approaches (including nature-based therapy), and &quot;prolonged investigations&quot; to track developments in mental health resources and impacts of implemented interventions&lt;ref&gt;{{Cite journal|last=Godakanda|first=Virasha|date=2025-01-29|title=A GRIEF COUNSELING INTERVENTION AFTER THE MASS TRAUMA: LESSONS LEARNED FROM THE VICTIMS OF THE EASTER SUNDAY ATTACK IN SRI LANKA|url=https://kjmr.com.pk/kjmr/article/view/216|journal=Kashf Journal of Multidisciplinary Research|language=en|volume=2|issue=01|pages=13–32|doi=10.71146/kjmr216|issn=3007-200X}}&lt;/ref&gt;. 

A few weeks following the attacks, Muslims in Sri Lanka were subjected to [[w:2019_anti-Muslim_riots_in_Sri_Lanka|violent, coordinated riots]] masterminded by Sinhalese national forces&lt;ref&gt;{{Cite journal|last=Mujahidin|first=Muhammad Saekul|date=2023-07-03|title=Extremism and Islamophobia Against the Muslim Minority in Sri Lanka|url=https://www.ajis.org/|journal=American Journal of Islam and Society|language=en|volume=40|issue=1-2|pages=213–241|doi=10.35632/ajis.v40i1-2.3135|issn=2690-3741}}&lt;/ref&gt;. Riots were mainly centered in the [[w:Kurunegala_District|Kurunegala]], Gampaha, and [[w:Kandy_District|Kandy]] Districts. At least [https://www.aljazeera.com/news/2019/5/21/in-sri-lanka-muslims-say-sinhala-neighbours-turned-against-them one confirmed death was reported]. Calls for vague ''niqab'' and ''burqa'' bans were increasingly prominent, eventually leading to the 2021 burqa ban by the Sri Lankan government. Pakistani and Afghani refugees fleeing religious persecution in Negombo were forced to be &quot;made refugees again&quot; after local protests were orchestrated against their settlement. Anti-Muslim sentiment was &quot;unleashed online, in the law, and on the street&quot;&lt;ref&gt;{{Cite book|title=CARTOGRAPHIC JOURNEY OF RACE, GENDER AND POWER: global identity|date=2021|publisher=CAMBRIDGE SCHOLARS PUBLIS|isbn=978-1-5275-6965-2|location=S.l.}}&lt;/ref&gt;. Albeit its relevancy to the attacks, no in-depth mental health studies have took place on the minority Muslim population following the Easter bombings. Further research is imperative in exploring the sustained psychological effects of Islamophobia and its effect on the Muslim minority community in the aftermath of the 2019 Easter attacks. 

Literature on the impact of the 2019 Easter Bombings on mental health is limited and further research should be conducted.  

=== 2019-2024 Economic Crisis ===
The '''2019-2024 Economic Crisis''' refers to a 5 year period where the Sri Lankan economy experienced significant inflation and an abrupt hike in prices on basic, everyday items. It is the worse economic crisis the country has faced since the Sri Lankans were granted independence in 1948. Schools in Sri Lanka were forced to postpone examinations due to paper shortages. Gas shortages led to long lines at gas stations, some lasting for days, throughout the island. Shortages in electricity, cooking gas, and aviation feul were additional consequences of the economic crisis.

Healthcare workers faced a barrage of impediments in their line of work during the crisis, including a lopsided work-life balance due to unprecedented demand, increased stress and mental fatigue from a lack of resources and personnel, unhealthy coping mechanisms, job dissatisfaction, and a reduction in work quality. Such effects perpetuated a self-enforcing cycle of psychologically distressed mental healthcare workers providing subpar services, affecting patients and amplifying mental health issues experienced by both the workforce and their patients&lt;ref&gt;{{Cite journal|last=Dilogini|first=S.|last2=Grace|first2=H. H.|last3=Thasika|first3=T.|date=2024|title=Exploring The Mental Health and Well-Being of Public Healthcare Workers (HCWs) Amid Economic Crisis in Sri Lanka|url=http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/11092|language=en|publisher=Chartered Institute of Personnel Management}}&lt;/ref&gt;. 

Medical students from the Faculty of Medicine at the University of Colombo reported that the economic crisis forced abrupt changes in dietary consumption, increased hopelessness in the future, increased stress and anxiety, and a decrease in interest in pursuing a &quot;clinical post-graduate career&quot;&lt;ref&gt;{{Cite journal|last=Adikaranayake|first=Pesala Randika|last2=Perera|first2=Anusha Nimrod|last3=Nilaweera|first3=Akhila Imantha|last4=Fernando|first4=Desha Rajni|last5=Wijayaratne|first5=Dilushi Rowena|date=2025-07-01|title=Effects of Sri Lankan economic crisis on health, lifestyle and education of medical students in Faculty of Medicine, University of Colombo – an online survey|url=https://doi.org/10.1186/s12909-025-07506-y|journal=BMC Medical Education|language=en|volume=25|issue=1|pages=938|doi=10.1186/s12909-025-07506-y|issn=1472-6920|pmc=12211748}}&lt;/ref&gt;. 283 government-school teachers completed a web-based cross-sectional survey in April 2024, with majority of the participants reporting a severe reduction in monthly income &amp; 1/3 of participants exhibiting &quot;clinical levels of psychological distress&quot;&lt;ref&gt;{{Cite journal|last=Senevirathne|first=C. P.|last2=Senarathne|first2=D. L. P.|last3=Fernando|first3=M. S.|last4=Senevirathne|first4=S. P.|date=2025-05-28|title=Examining the economic burden and mental health distress among government school teachers in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/s40359-025-02921-8|journal=BMC Psychology|language=en|volume=13|issue=1|pages=572|doi=10.1186/s40359-025-02921-8|issn=2050-7283}}&lt;/ref&gt;. A study published in that same year reported that out of 261 nurses working in teaching hospitals, 91.6% were forced to allocate their finances to strictly &quot;general needs&quot;, while more than 50% looked into international opportunities for employment. Notably, the study reported an overall near &quot;twofold greater&quot; rate of depression, anxiety, and stress compared to previous studies on nurses in Sri Lanka&lt;ref&gt;{{Cite journal|last=Senevirathne|first=C.P|last2=Senarathne|first2=L.|last3=Fernando|first3=M.|date=2024-04-01|title=Exploring the Association Between Behavioural Modification in Response to the Prevailing Economic Crisis and Mental Health Outcomes of Nurses from Teaching Hospitals, Sri Lanka|url=https://doi.org/10.1177/23779608241272679|journal=SAGE Open Nursing|language=EN|volume=10|pages=23779608241272679|doi=10.1177/23779608241272679|issn=2377-9608|pmc=11311183}}&lt;/ref&gt;. 

The detrimental effects the crisis has had on the mental health sector reveal a concerning area of underappreciation and under compensation towards a critical sector for the well-being of the country. Adequate staffing, increased funding, and an improved work-life balance should be emphasized for the workers of health sector of the country.  

== Present-Day Challenges ==

=== Ethnic tension ===
Despite the ending of the Sri Lankan civil war and the introduction of pluralist policies (such as the [https://srilankaembassy.fr/sites/default/files/files/media/pdf/NationalPolicy-English.pdf 2017 National Policy on Reconciliation and Coexistence] under the Sirisena administration), tensions amongst members of the ethnic groups still persist. Evidence of these tensions was found in a 2022 study conducted in the Ratnapura district, where religious leaders expressed skepticism through semi-structured interviews on &quot;conflict transformation&quot;. A Tamil citizen of the Ratnapura community recounted that they were forced to &quot;hide in jungles&quot; and consume &quot;dirty water in drainage[s]&quot; due to scarcity of food and drinkable water as a result of the conflict. In certain personal accounts, ethnic conflicts appear to affect the social behavior and identity of the majority ethnic group. One Sinhala participant recounted his objection to the war-time retaliatory destruction of a shop run by a Tamil shopkeeper was met with interrogative questions about &quot;whether [he was] Sinhalese or not&quot;. Both accounts convey interethnic tensions stemming from decade-long conflicts&lt;ref&gt;Jayathilaka, Aruna &amp; Gamage, Sayuri. (2024). Role of Buddhist and Hindu Religious Leaders Role of Buddhist and Hindu Religious Leaders in the Post-War Conflict Transformation Process: A Study Based on Rathnapura District in Srilanka. ''Retrieved from'' https://gandhimargjournal.org/wp-content/uploads/2024/09/Volume-46-Issue-1-April-June-2024.pdf#page=66&lt;/ref&gt;. 

Beyond individual accounts and the official end of the civil war, the minority groups in the country continue to feel ostracized. The Sri Lankan Tamil population remains dissatisfied with the Sri Lankan government due to their alleged lack of accountability of perpetrators of war crimes and lack of information on the whereabouts of [[w:Enforced_disappearances_in_Sri_Lanka|thousands of enforced disappearances]] that took place from the 1980s. Additionally, rising anti-Muslim sentiment in recent years has contributed to increased ethnic tensions, a stark contrast to the previous centuries of peaceful co-existence between the groups. 
[[File:Bodu Bala Sena symbol.svg|thumb|The symbol for Bodu Bala Sena, a nationalistic Sinhala Buddhist group criticized for catalyzing ethnic tensions in Sri Lanka.]]
Laws passed by the Sri Lankan government, such as the [[w:Prevention_of_Terrorism_Act_(Sri_Lanka)|Prevention of Terrorism Act]] and [[wikipedia:Anti-conversion_law#Sri_Lanka|anti-conversion laws]], have forced the United States Commission on International Religious Freedom to label Sri Lanka as a nation that &quot;[engages] or [tolerates] severe violations of religious freedom&quot; in their 2024 report. The government has been criticized by human rights organizations for &quot;disproportionately targeting religious minorities&quot;&lt;ref&gt;{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}&lt;/ref&gt;. Additionally, the implementation of the three dominant languages, English, Sinhala, and Tamil, across formal education and government services have been lackadaisical, narrowing opportunities of foundational social interactions between the groups. Persistent discrimination and prejudice towards minority groups can lead to an array of complex and self-deprecating mental health issues.   

Efforts to mitigate ethnic tensions include strategies like [[w:Community-based_participatory_research|community-based participatory research]] (CBPR), task-sharing, and securing online mental health services in order to expand mental health services. However, the implementation of evidence-based plans has been met with difficulty due to inaccessibility, high costs, and shortages of adequately-trained personnel.  

Movements aiming for improved intra group and inter group coexistences, such as the Jaffna People’s Forum for Coexistence, should be emphasized on a systematic and multi-level basis, including but not limited to education, public sectors, and within communities. Pluralistic values are encouraged to be emphasized across both private and public schools to foster cultural sensitivity and tolerance. Measures should be taken against groups criticized for promoting sectarian hostility, such as the [[w:Bodu_Bala_Sena|Bodu Bala Sena]]. 

=== Poverty ===
It has been proven that poverty significantly increases the chances of developing mental illnesses. This is further amplified by possible discrimination&lt;ref&gt;{{Cite journal|last=Knifton|first=Lee|last2=Inglis|first2=Greig|date=2020-10|title=Poverty and mental health: policy, practice and research implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC7525587/|journal=BJPsych bulletin|volume=44|issue=5|pages=193–196|doi=10.1192/bjb.2020.78|issn=2056-4694|pmc=7525587|pmid=32744210}}&lt;/ref&gt;. Poverty also affects the ability for individuals with mental health concerns to receive the treatment they need. Due to the repercussions of the economic crisis, clients in Sri Lanka could not attend further counseling sessions&lt;ref name=&quot;:8&quot; /&gt;. Poverty from 2021 to 2022 [https://databankfiles.worldbank.org/public/ddpext_download/poverty/987B9C90-CB9F-4D93-AE8C-750588BF00QA/current/Global_POVEQ_LKA.pdf reportedly doubled], with future forecasts predicting the poverty line to &quot;remain above 25 percent&quot;. Suicide has been empirically linked to economic hardships in previous studies&lt;ref&gt;{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}&lt;/ref&gt;. A 2013 study done on suicidal patients in [[w:Batticaloa_Teaching_Hospital|Batticaloa Teaching Hospital]] revealed 76% of patients who attempted suicide were from rural areas while 15% were from urban areas&lt;ref&gt;{{Cite book|url=http://ir.lib.seu.ac.lk/handle/123456789/1457|title=The influence of common risk factors for the patient with attempted suicide hospitalized at the teaching hospital, Batticaloa|last=Kisokanth|first=G.|last2=Najeem|first2=M. M.|last3=Karunakaran|first3=K. E.|date=2014-08-02|publisher=South Eastern University of Sri Lanka, University Park, Oluvil #32360, Sri Lanka|isbn=978-955-627-053-2|language=en-US}}&lt;/ref&gt;. The Sri Lankan government should consider the economical impacts that poverty has on mental health and implement ways to aid poverty-stricken individuals with mental health concerns.   

=== Stigmas ===
Stigma consists of the &quot;combined effect of prejudice, ignorance and discrimination.&quot;&lt;ref name=&quot;:10&quot;&gt;{{Cite web|url=http://www.researchgate.net/publication/233990797_The_Stigma_of_Mental_Illness_in_Sri_Lanka_The_Perspectives_of_Community_Mental_Health_Workers|title=(PDF) The Stigma of Mental Illness in Sri Lanka: The Perspectives of Community Mental Health Workers|website=ResearchGate|language=en|access-date=2025-07-25}}&lt;/ref&gt;.

A 2012 interview consisting of nine participants (two doctors, three nurses, one occupational therapist, one development worker, and two volunteers) revealed a number of concerning societal viewpoints on individuals with mental health concerns. The interviews revealed that negative judgements were not only levied against the individual with the mental illness, but also the family. Families hid mentally ill family members from the public to avoid &quot;shame&quot; and possible hinderances in marriage proposals. Views that mentally ill individuals were &quot;violent&quot; served as the motivating factor behind socially isolating those with mental illness from their communities. Interviewees mentioned that individuals dealing with mental health challenges would be attacked with stones and called &quot;derogatory names.&quot; A lack of community awareness regarding mental health and negative portrayals of mentally ill individuals in media exacerbates stigmatization, though the researchers commented that the media was &quot;improving&quot; in their depiction of mental illness. Beliefs that illnesses are caused by &quot;spirits&quot; can be problematic for individuals dealing with mental health issues and suggests poor mental health awareness. Mental health workers themselves believed that they were being stigmatized, as mental health was reportedly not taken as seriously as physical health. Despite the intriguing perspectives provided, the small sample size and usage of snow sampling raise questionable concerns regarding the generalizability of the results&lt;ref name=&quot;:10&quot; /&gt;. 

Improving media portrayal of subjects concerning mental health and involving community members in interventions dealing with mental health issues are ways that could destigmatize mental health amongst communities in Sri Lanka. Tying collaborations between allopathic services and traditional healers instead of having these two services work individually could enhance engagement between traditional medicine and Western medicine.

=== Suicide Trends &amp; Risk Factors ===
Suicide is defined as &quot;the act of killing oneself deliberately, initiated and performed by the person concerned in the full knowledge or expectation of its fatal outcome&quot;&lt;ref name=&quot;:11&quot;&gt;{{Cite book|title=The neuroscience of suicidal behavior|last=Heeringen|first=Kees van|date=2018|publisher=Cambridge University Press|isbn=978-1-316-60290-4|series=Cambridge fundamentals of neuroscience in psychology|location=Cambridge, United Kingdom New York, NY, USA Port Melbourne, VIC, Australia New Delhi, India Singapore}}&lt;/ref&gt;. Although Sri Lanka has seen a significant reduction in suicide rates from the mid 1990s, largely stemming from its ban on extremely toxic pesticide products, suicide and self harm remains a significant issue. The suicide rate per 100,000 people increased from 14.0 in 2019 to [https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide 15.0 in 2022] (according to WHO). On average, 27 males per 100,000 males and 5 females per 100,000 females committed suicide in 2022&lt;ref&gt;{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}&lt;/ref&gt;. Hanging appears to be the most used method for suicide for both males and females, with studies revealing a steady increase in recent years&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Bandara|first=Piumee|last2=Wickrama|first2=Prabath|last3=Sivayokan|first3=Sambasivamoorthy|last4=Knipe|first4=Duleeka|last5=Rajapakse|first5=Thilini|date=2024-04-17|title=Reflections on the trends of suicide in Sri Lanka, 1997–2022: The need for continued vigilance|url=https://journals.plos.org/globalpublichealth/article?id=10.1371/journal.pgph.0003054|journal=PLOS Global Public Health|language=en|volume=4|issue=4|pages=e0003054|doi=10.1371/journal.pgph.0003054|issn=2767-3375|pmc=11023397|pmid=38630779}}&lt;/ref&gt;. 

From 2023 to 2024, a group of researchers from the [[w:Eastern_University,_Sri_Lanka|Eastern University in Sri Lanka]] assessed 828 patients admitted to the Teaching Hospital in [[w:Batticaloa,_Sri_Lanka|Batticaloa, Sri Lanka]] for attempted suicide. They concluded that suicide prevention programs should be attuned to younger people (ages 15 to 35 in the study), emphasize the importance of education and reducing unemployment, and increase social support in the Tamil community. Despite accounting for other factors that could lead to suicidal ideation (ie, poverty), the results from this study suffer in external validity as 90% of the patients were Tamil and over 50% were between 16 and 25 years. In addition, correlations between suicide and unemployment rates have been questioned, with [[w:Austerity|austerity]] being a more reliable indicator of suicide rates than unemployment rates&lt;ref name=&quot;:11&quot; /&gt;. Further comprehensive studies on risk factors relating to suicide should be studied to examine correlations between unemployment rates and austerity measures.  

The WHO suggests implementing evidence-based suicide prevention programs, such as [https://www.who.int/initiatives/live-life-initiative-for-suicide-prevention LIVE LIFE], to reduce the national suicide rate&lt;ref&gt;{{Cite web|url=https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide|title=World Suicide Prevention day 2024 “Changing the Narrative on Suicide”|website=www.who.int|language=en|access-date=2025-07-29}}&lt;/ref&gt;. Media depictions of suicidal methods, such as hanging, can lead to sensationalism and the media should be cautious of such displays in movies and TV shows&lt;ref name=&quot;:12&quot; /&gt;. Awareness of depression and other mental health issues can serve as a safeguard against suicidal ideation in Sri Lankan men and women.  

== Role of Religion ==
According to the last demographic report (2012), 70.2% of Sri Lankans are Buddhist, 12.6% are Hindus, 9.7% are Muslims, and 7.4% are Christians. The Theravada Buddhist community makes up the majority in several provinces throughout the country&lt;ref&gt;{{Cite web|url=https://www.state.gov/reports/2022-report-on-international-religious-freedom/sri-lanka/|title=Sri Lanka|website=United States Department of State|language=en-US|access-date=2025-08-07}}&lt;/ref&gt;. Religion, especially Theravada Buddhism, has had a significant influence on not only the historical treatment of mental health in the country, but also everyday life&lt;ref name=&quot;:15&quot; /&gt;. The [[w:Mahāvaṃsa|''Mahāvaṃsa'']] affirms hospitals treating patients suffering from mental health issues as early as the 4th century BC. Additionally, the 1700s Nayaka king [[w:Kirti_Sri_Rajasinha|Kirthi Sri Rajasinghe]] detailed the implementation of Buddhist philosophy in psychiatry&lt;ref name=&quot;:4&quot; /&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Alwis|first=L. A. P. De|date=2017-12-05|title=Development of civil commitment statutes (laws of involuntary detention and treatment) in Sri Lanka: a historical review|url=https://mljsl.sljol.info/articles/10.4038/mljsl.v5i1.7351|journal=Medico-Legal Journal of Sri Lanka|language=en|volume=5|issue=1|doi=10.4038/mljsl.v5i1.7351|issn=2012-8231}}&lt;/ref&gt;.

Modern-day empirical studies have attested to the usefulness of religion in mitigating stress and elevating mental health&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-94-007-4276-5_22|title=Religion and Mental Health|last=Schieman|first=Scott|last2=Bierman|first2=Alex|last3=Ellison|first3=Christopher G.|date=2013|publisher=Springer Netherlands|isbn=978-94-007-4276-5|editor-last=Aneshensel|editor-first=Carol S.|location=Dordrecht|pages=457–478|language=en|doi=10.1007/978-94-007-4276-5_22|editor-last2=Phelan|editor-first2=Jo C.|editor-last3=Bierman|editor-first3=Alex}}&lt;/ref&gt;. Religion has been found to be positively correlated with improved mental health, and more religious patients were concluded to have &quot;better mental health and adapt[ed] more quickly to health problems&quot; versus patients who weren't religious&lt;ref&gt;{{Cite journal|last=Koenig|first=Harold G.|date=2012|title=Religion, spirituality, and health: the research and clinical implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3671693/|journal=ISRN psychiatry|volume=2012|pages=278730|doi=10.5402/2012/278730|issn=2090-7966|pmc=3671693|pmid=23762764}}&lt;/ref&gt;. [https://www.researchgate.net/scientific-contributions/T-N-Wickramarathna-2247724082 Dr. Wickramarathna] of the University Psychiatry Unit (UPU) at the National Hospital of Sri Lanka (NHSL) argues that psychiatrists must strive for a balance in their approach to patients and &quot;make positive use of religion in [their] practice[s]&quot;&lt;ref&gt;{{Cite journal|last=Wickramarathna|first=T. N.|date=2022-12-31|title=Psychiatrists should stand far from the shrine: why and why not we should separate religion from psychiatry|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v13i2.8397|journal=Sri Lanka Journal of Psychiatry|language=en|volume=13|issue=2|doi=10.4038/sljpsyc.v13i2.8397|issn=2012-6883}}&lt;/ref&gt;.

=== Buddhism ===
27 Sinhalese Buddhists from four Buddhist temples were selected for a series of 70-minute interviews and focus group discussions with the aim of learning the Sinhala Buddhist understanding and experience of spiritual well-being and psychological well-being. The interviewees held spiritual wellness to be the &quot;center&quot; of overall wellness, the &quot;precondition for a successful life&quot;&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/articles/10.4038/sljss.v44i1.7990|journal=Sri Lanka Journal of Social Sciences|language=en-US|volume=44|issue=1|doi=10.4038/sljss.v44i1.7990|issn=0258-9710}}&lt;/ref&gt;. Sinhala Buddhists believe that wellness cannot be achieved without spiritual tranquility. The report states that participants emphasized that spirituality &quot;cannot be directly intervened&quot; and can only be seen through &quot;[interactions] with society&quot;&lt;ref name=&quot;:14&quot; /&gt;. Despite the ''athmaya'' (soul) being &quot;unreachable&quot;, it can be &quot;intervened&quot;, or treated, through the actions of the mind and body with society&lt;ref name=&quot;:14&quot; /&gt;. One being &quot;psychologically ill&quot; can affect one's spiritual being, as the participants reported in their interviews, and can be affected through &quot;lifestyle stressors, environmental and socio-cultural causes, non-human related causes and bad-karma in the past lives&quot;&lt;ref name=&quot;:14&quot; /&gt;.  

The researchers concluded that despite Sinhala Buddhists not being able to articulately decipher the discrepancies between psychological well-being and spiritual well-being, they are able to conceptualize and maintain a culturally embedded understanding between the two, serving as reputable evidence of the integration of mental health in Sinhala Buddhist practices. However, it is important to note that these results come from a very small sample size and cannot be generalized to all Sri Lankan Buddhists.   

In addition, a 2009 study found that a belief in karma was correlated with poor health. However, an earlier study found a positive correlation between the reliance on the [[w:Karma_in_Buddhism|Buddhist concept of karma]] and trauma, inferencing Buddhist karma being a prevalent response to trauma&lt;ref&gt;{{Cite journal|last=Levy|first=Becca R.|last2=Slade|first2=Martin D.|last3=Ranasinghe|first3=Padmini|date=2009-03|title=Causal thinking after a tsunami wave: karma beliefs, pessimistic explanatory style and health among Sri Lankan survivors|url=https://pubmed.ncbi.nlm.nih.gov/19229624|journal=Journal of Religion and Health|volume=48|issue=1|pages=38–45|doi=10.1007/s10943-008-9162-5|issn=1573-6571|pmid=19229624}}&lt;/ref&gt;. Overall, the effectiveness of karma as a coping mechanism appears to be conflicted.   

Studies indicate that other practices of Buddhism seem to be utilized by individuals affected by the war. 40% of Sri Lankan Buddhists affected by the 2004 tsunami found the Buddhist ritual ''Bodhipuja'' to be helpful in dealing with traumatic experiences&lt;ref&gt;{{Cite web|url=https://jmvh.org/article/mental-health-and-the-role-of-cultural-and-religious-support-in-the-assistance-of-disabled-veterans-in-sri-lanka/|title=Mental Health and the Role of Cultural and Religious Support in the Assistance of Disabled Veterans in Sri Lanka|website=JMVH|language=en-US|access-date=2025-08-12}}&lt;/ref&gt;.    

=== Catholicism ===
Catholic counseling refers to &quot;a nuanced and holistic mental health care paradigm that intricately weaves together psychological science with the moral, spiritual, and pastoral traditions of the Catholic Church&quot;&lt;ref name=&quot;:13&quot;&gt;Perera, U. [https://www.researchgate.net/profile/Udeshini-Perera/publication/394095042_Catholic_Counselling_in_Sri_Lanka_Integrating_Faith_Psychology_and_Cultural_Healing/links/6889303af8031739e6098c79/Catholic-Counselling-in-Sri-Lanka-Integrating-Faith-Psychology-and-Cultural-Healing.pdf Catholic Counselling in Sri Lanka: Integrating Faith, Psychology, and Cultural Healing]. July 2025.&lt;/ref&gt; and aims to assimilate Catholic theology and evidence-based psychological treatment while including Sri Lankan cultural elements. This is achieved through emphasis on community cohesion and a locally-based understanding of &quot;personhood&quot;&lt;ref name=&quot;:13&quot; /&gt;. 

The origins of Catholic counseling trace back to the introduction of Roman Catholicism to the island in the 1600s, with the focus of the early Sri Lankan Catholic community being on &quot;[[w:Evangelism|evangelization]], education, and sacramental formation&quot;. Demand for counseling services in general increased due to the impacts of the Sri Lankan Civil War, where Catholic organizations (Caritas Sri Lanka, Seth Sarana, Subodhi Integral Centre (Piliyandala), etc.) established several Catholic-based trauma-informed programmes for victims of the Civil War. Programmes use group therapy, forgiveness rituals, and narrative repairs to alleviate war trauma. 

Examples of integration of Catholic virtues and counseling can be seen in [[w:Cognitive_Behavioral_Therapy|Cognitive Behavioral Therapy]] (CBT), where &quot;hope&quot; and &quot;humility&quot; are used as the frameworks for creating spiritual resilience&lt;ref name=&quot;:13&quot; /&gt;. The general Christian call of &quot;agape love and acceptance&quot; is echoed by the concept of [[w:Unconditional_positive_regard|unconditional positive regard]]. ''[[w:Lectio_Divina|Lectio Divina]]'' (Catholic prayer and meditation) and ''Marian devotions'' are integrated into therapeutic practices to achieve emotional regulation and mindfulness.   

Senior Lecturer [https://www.researchgate.net/profile/Udeshini-Perera Udeshini Perera] of the University of Colombo articulates a critical role of Catholic counseling. She claims that secular counseling fails to address the &quot;spiritual roots of distress and moral confusion&quot;. Catholic counseling fills in this gap by integrating &quot;psychological insights with a transcendent orientation, supporting lasting transformation and integrity&quot;&lt;ref name=&quot;:13&quot; /&gt;.  

As of 2025, no formal accreditation or standardized training exists for [[w:Pastoral_counseling|pastoral counselors]] in Sri Lanka, hampering the legitimacy of Catholic counseling. Udeshini Perera remarks that mental health stigma, lack of standardized training, research regarding Catholic counseling effectiveness, and acceptance of the combination of religion and science in a professional setting present challenges for Catholic pastoral counseling in the country. Additionally, Catholic psychiatry in Sri Lanka appears to be under-researched, and evidence of its empirical effects on followers appears sparse. Further research is needed in assessing the empirical effects of Catholic counseling in Sri Lanka.  

=== Islam ===
The literature on the empirical effects of Islamic-based psychotherapy in Sri Lanka is limited. Research has revealed a 2012 case study where a 21-year-old Muslim woman was experiencing episodic possession states. The patient ceased attending psychiatric services and opted for religious rituals. The patient reported, in a follow-up visit, that the possession states had been absent for 3 months since her switch to religious rituals. The woman and her family attributed the apparent improvement of her condition to religious rituals&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|last3=Yoosuf|first3=Alam|last4=Karunaratne|first4=Sanjeewani|last5=de Silva|first5=Pushpa|date=2012|title=Religious Beliefs, Possession States, and Spirits: Three Case Studies from Sri Lanka|url=http://www.hindawi.com/journals/crips/2012/232740/|journal=Case Reports in Psychiatry|language=en|volume=2012|pages=1–3|doi=10.1155/2012/232740|issn=2090-682X|pmc=3437272|pmid=22970398}}&lt;/ref&gt;.

Future recommendations would be to employ resources to research the foundations of Islamic psychiatry in the country, and to observe the rituals employed and their effects on patients. Studies have found that Islamic prayer can be an effective means of &quot;support and coping&quot;&lt;ref name=&quot;:15&quot; /&gt;. Seven world-wide case studies using Islamic-based psychotherapy on patients, consisting of religious rituals such as scriptural reading from the [[w:Quran|Quran]], teaching of fundamental Islamic concepts (such as ''[[w:Tawakkul|tawakkul]]''), and active implementation of contemplation (''[[w:Tadabbur|tadabbur]]''), have reported positive effects in decreasing cognitive and emotional symptoms associated with &quot;religious, obsessive-compulsive disorder, depression, agoraphobia, generalized anxiety disorder, grief, and substance use disorder.”&lt;ref&gt;{{Cite journal|last=Kurhade|first=Chhaya Shantaram|last2=Jagannathan|first2=Aarti|last3=Varambally|first3=Shivarama|last4=Shivanna|first4=Sushrutha|date=2022-01|title=Religion-based interventions for mental health disorders: A systematic review|url=https://journals.lww.com/10.4103/ijoyppp.ijoyppp_14_21|journal=Journal of Applied Consciousness Studies|language=en|volume=10|issue=1|pages=20–33|doi=10.4103/ijoyppp.ijoyppp_14_21|issn=2949-6993}}&lt;/ref&gt; Additionally, a community-based study of elderly patients in Bangalore, India receiving Islamic-based psychotherapy observed decreased exhibitions of sleep disorders, eating disorders, and emotional distress&lt;ref&gt;{{Cite journal|last=Hafeez|first=Nimin|last2=Sanjay|first2=Thittamaranahalli Varadappa|last3=Puthussery|first3=Yannick Poulose|last4=Madhusudan|first4=Muralidhar|last5=Kariyappa|first5=Poornima Muddaiah|last6=Kulkarni|first6=Sridevi|last7=Raj|first7=Lavanya|date=2023-12-31|title=Spiritual practices among elderly, prevalence, pattern and associated factors: a community-based study from rural Bengaluru, India|url=https://jccpsl.sljol.info/articles/10.4038/jccpsl.v29i4.8610|journal=Journal of the College of Community Physicians of Sri Lanka|language=en|volume=29|issue=4|doi=10.4038/jccpsl.v29i4.8610|issn=1391-3174}}&lt;/ref&gt;. 

=== Hinduism ===
Despite Hindus being 12.6% of the population of Sri Lanka, the research on Hinduism-based therapy in the country is limited. Ayurvedic medicine, a form of medicine originating from ancient India, predominated the Sri Lankan medical landscape for over 2,000 years and even had a symbiotic relationship with Sinhalese medicine, which also played a significant and influential role in the country's medical framework&lt;ref name=&quot;:0&quot; /&gt;&lt;ref&gt;{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/article/10.4038/sljss.v44i1.7990/|journal=Sri Lanka Journal of Social Sciences|volume=44|issue=1|pages=33|doi=10.4038/sljss.v44i1.7990|issn=2478-1169}}&lt;/ref&gt;. Despite its historical dominance, Ayurvedic medicine has been challenged against modern evidence-based medical standards&lt;ref&gt;{{Cite book|url=https://philarchive.org/rec/DOMAAT|title=Ayurveda: Ancient Tradition or Pseudoscientific Practice? A Philosophical Inquiry|last=Dominic|first=Shubham K.}}&lt;/ref&gt;.

=== Comparative synthesis ===
Taking an overarching review of the role of religion in Sri Lanka, methods to improve mental well-being are practiced by adherents of Buddhism, Hinduism, Islam, and Christianity. These methods are practiced through karma, tawakkul, hope, and humility. Additionally, these practices are implemented in traditionally-oriented mental health care, which has been reported to be preferred over psychiatric care at times. These rituals practiced across these religions indicate a common theme of psychologically integrated aspects of well-being. Interpretation of trauma is a central use in religion, with religious principles, such as karma and ''tawakkul'', serving as psychologically analogous mechanisms during times of distress.  

In terms of methodological comparisons to the studies described, qualitative interviews have documented Buddhist practices and principles, like Bodhipuja and the belief in karma, in response to traumatic events, while case studies found religious practices by other religious groups, such as a Muslim patient reading Islamic scripture and observing prayer to reduce emotional distress. Peer-reviewed sources have documented Catholic practices and principles, such as ''Lectio Divina'' and unconditional positive regard, in improving mindfulness and emotional regulation. The paper acknowledges limitations in the evaluation of certain findings, such as in Islam and Hinduism. These shortcomings, however, are a reflection of the existing literature and its deficiencies. Empirical findings indicate mental health practices are complex and are multifaceted in their effects. 

Evidently, religion serves a parallel role to psychiatric services in improving mental health. Despite its perceived benefits, the findings surrounding religions' role in mental health suffer from conflicting, and sometimes contradictory, results. Additionally, a disproportionate amount of empirical findings seem to be Buddhist-predominant, while other religions are underrepresented in the research. Regarding research barriers, the methodological approaches implemented to study the practices of religious followers vary, though much of the research was brought from qualitative or case-based studies, impeding generalizability. Another noteworthy issue is that many studies do not utilize standardized, psychiatric measures.

== Future Outlook ==
Despite significant changes to the mental health environment in Sri Lanka, the current legal framework shaping mental health in the country has not been updated since 1956. A Cambridge University Press article detailed many limitations of the Mental Disease Ordinance of 1956, including discrepancies between the legal provisions of involuntary admissions and modern practices, potential exposure to trauma through extra-legal detentions of the mentally ill, and an absence of legal guidelines addressing the restraint of violent patients&lt;ref name=&quot;:6&quot; /&gt;. Participants from Sri Lanka reported in a comparative legislative questionnaire that they felt the mental health laws were &quot;outdated&quot; and descriptions of clinical roles remained ambiguous&lt;ref name=&quot;:16&quot; /&gt;. A draft mental health legislation from 2007 included provisions for human rights, but due to &quot;bureaucratic processes&quot; and a &quot;lack of consensus&quot;, the draft has not been officially approved.

These limitations pose challenges to the standardization of mental healthcare admissions and may impact the rights of detained patients. Detained patients may have their human rights violated due to a lack of an up-to-date legal framework, thereby impeding the identification of such violations. Additionally, with the lack of clarity on clinical roles, clinical responsibilities may not be routinely recognized and observed, leading to role confusion and potential legal ramifications&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|last=Dey|first=Sangeeta|last2=Mellsop|first2=Graham|last3=Diesfeld|first3=Kate|last4=Dharmawardene|first4=Vajira|last5=Mendis|first5=Susitha|last6=Chaudhuri|first6=Sreemanti|last7=Deb|first7=Aniruddha|last8=Huq|first8=Nafisa|last9=Ahmed|first9=Helal Uddin|date=2019-10-24|title=Comparing legislation for involuntary admission and treatment of mental illness in four South Asian countries|url=https://ijmhs.biomedcentral.com/articles/10.1186/s13033-019-0322-7|journal=International Journal of Mental Health Systems|volume=13|issue=1|pages=67|doi=10.1186/s13033-019-0322-7|issn=1752-4458|pmc=6813093|pmid=31666805}}&lt;/ref&gt;. Lastly, current efforts should ideally move beyond just addressing poverty-centered matters, but also expand efforts to domestic violence victims and children with disabilities, as shelters and specialized services are limited&lt;ref name=&quot;:82&quot;&gt;{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}&lt;/ref&gt;.

Stagnation in policy development leaves Sri Lanka without a practical, up-to-date, and comprehensive mental health framework, which could put both clinicians and patients at risk. Future reforms should include clarification on the treatment and detention process of involuntary admissions of patients and a clear delineation of clinical roles and their responsibilities. Without the necessary reforms to advance Sri Lankan mental health legislation, clinicians and vulnerable patients may suffer from a lack of comprehensive oversight. 

==Additional information==

===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.

===Competing interests===
No competing interests.

===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.

==References==
{{reflist|35em}}

[[Category:Mental health]]
[[Category:Sri Lanka]]</text>
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| journal = WikiJournal of Medicine      &lt;!-- WikiJournal of Medicine, Science, or Humanities --&gt;
| last1 = Azeez
| orcid1 = 0009-0007-9202-4614
| first1 = Aaqib
| last2 = 
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| first4 = &lt;!-- up to 9 authors can be added in this above format --&gt;
| et_al = &lt;!-- if there are &gt;9 authors, hyperlink to the list here --&gt;
| affiliation1 = Old Dominion University
| correspondence1 = aaqib.azeez@yahoo.com
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| abstract = Mental health issues continue to be a significant problem in Sri Lanka, with 2022 suicide rates in the country reporting 15 suicides per 100,000 people, above the global average of 10.5 suicides per 100,000 people. The barriers to mental healthcare on the island are multi-faceted and are best understood with historical context. This narrative review covers the historical developments of mental healthcare,  mental health impacts of historical events within the last 100 years, current challenges affecting mental health outcomes, the role of the island's major religions in mental health and mental healthcare, and recommendations for improving future mental healthcare. The author uses peer-reviewed journal articles, relevant books, historical documents, and governmental/non-governmental reports to support clinical and historical claims, though non-peer-reviewed sources were used to contextualize historical and non-clinical claims. The narrative review concludes that outdated legislation, impacts from recent conflicts or disasters, stigma surrounding mental health, and economic vulnerability contribute to mental health issues and the inefficiency of mental healthcare services. The author recommends updating legal frameworks, expanding services, and raising awareness to mitigate social stigma.
}}

== Introduction ==
Mental health continues to be a critically relevant topic as the island nation has experienced decades of [[w:Black_July|violent ethnic conflict]], terrorist attacks, alleged war crimes, and economic disruptions. Sri Lanka continues to recover from a [[w:Sri_Lankan_economic_crisis_(2019–2024)|severe economic crisis (2019 - 2024)]], a [[w:Sri_Lankan_civil_war|nearly 30-year civil war ending in 2009]], a [[w:2019_Sri_Lanka_Easter_bombings|2019 terrorist attack]], and the [[w:2004_Boxing_Day_tsunami|2004 Boxing Day tsunami]]. The exact effect these major events have had on mental health in the country is &quot;unknown&quot;, but the statistics remain concerning despite a declining trend in the overall suicide rate.

Suicide rates in the country during the mid-1990s were the second-highest in the world, with ingesting toxic products being the main suicide method. Despite the decline in suicide numbers since then—possibly attributed to Sri Lanka's ban on toxic products—evidence from a 2023 study reports an upward trend in suicide through hanging from 2016 to 2021—independent of the [[w:COVID-19_pandemic_in_Sri_Lanka|COVID-19 pandemic]]. Several risk factors for suicide, such as poverty and economic instability, are still prevalent and even increasing in the country&lt;ref&gt;{{Cite journal|last=Rajapakse|first=Thilini|last2=Silva|first2=Tharuka|last3=Hettiarachchi|first3=Nirosha Madhuwanthi|last4=Gunnell|first4=David|last5=Metcalfe|first5=Chris|last6=Spittal|first6=Matthew J.|last7=Knipe|first7=Duleeka|date=2023-01-19|title=The Impact of the COVID-19 Pandemic and Lockdowns on Self-Poisoning and Suicide in Sri Lanka: An Interrupted Time Series Analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC9914278/|journal=International Journal of Environmental Research and Public Health|volume=20|issue=3|pages=1833|doi=10.3390/ijerph20031833|issn=1660-4601|pmc=9914278|pmid=36767200}}&lt;/ref&gt;.

== Methods ==
A narrative review was conducted on mental health in Sri Lanka. Sources used included peer-reviewed journal articles, relevant books, historical documents, and governmental/non-governmental reports. These sources were found on Google Scholar, PubMed/PMC, Sri Lankan journals, and official Sri Lankan governmental websites showing relevant statistics/reports. Keywords used to conduct searches included, but were not limited to: &quot;Sri Lanka mental health&quot;, &quot;Sri Lanka civil war trauma&quot;, &quot;Sri Lanka suicide&quot;, &quot;Sri Lanka mental health ordinances&quot;, &quot;Sri Lanka religion and mental health&quot;, &quot;Sri Lanka public mental healthcare&quot;, and &quot;Sri Lanka poverty/economic crisis mental health impact.&quot; Studies that were included were relevant to the topic (Sri Lanka, South Asian mental health law, suicide, public mental health, conflict/disaster trauma, or cultural/religious practice), had full text available, and were in the English language. Non-peer-reviewed sources were primarily used to explain historical claims or contextualize non-clinical claims. 

==Historical Development of Mental Health Services==

Records attest to the care of the mentally ill through established hospitals in the island since the 4th century.&lt;ref name=&quot;:17&quot; /&gt; Prior to the incarceration of the mentally ill by the European colonizing forces, the mentally ill were regarded as ''Pissowetitch'', or people who had &quot;the spirit of the Gods within him&quot; and &quot;whatsoever he pronounceth, is looked upon as spoken by God himself, and the people will speak to him, as if it were the very person of God&quot;&lt;ref&gt;{{Cite web|url=https://www.gutenberg.org/files/14346/14346-h/14346-h.htm|title=An Historical Relation Of the Island Ceylon, in the East-Indies: Together, With an Account of the Detaining in Captivity the Author and divers other Englishmen now Living there, and of the Author’s Miraculous Escape.|last=Knox|first=Robert|website=www.gutenberg.org|language=en-us|access-date=2026-06-29}}&lt;/ref&gt;. With this religious understanding, Lucien de Alwis reasoned that the mentally ill in Sri Lanka were &quot;placed... at a higher social status than the mentally ill in the Western world&quot;, with this understanding correlating with the unsurprising absence of evidence of any &quot;large scale segregation[s] of [the] mentally ill from society&quot;&lt;ref name=&quot;:17&quot; /&gt;. 

In the 1800s, established care for mental health began shifting primarily from indigenous practices, mainly derived from [[w:Ayurveda|Ayurveda medicine]], [[w:Siddha_medicine|Siddha medicine]], and [[w:Unani_medicine|Unani medicine]], to a Western model by the British&lt;ref name=&quot;:17&quot; /&gt;&lt;ref name=&quot;:0&quot;&gt;Gambheera, H. (2011). [https://www.saarcpsychiatry.com/viewText?chapter=c6 The evolution of psychiatric services in Sri Lanka]. South Asian Journal of Psychiatry, 2(1), 25–27.&lt;/ref&gt;&lt;ref name=&quot;:15&quot;&gt;{{Cite book|url=https://doi.org/10.1007/978-981-96-8078-8_7|title=Social Psychiatry in Sri Lanka|last=Baminiwatta|first=Anuradha|last2=Williams|first2=Shehan|date=2025|publisher=Springer Nature|isbn=978-981-96-8078-8|editor-last=Arafat|editor-first=S. M. Yasir|location=Singapore|pages=141–158|language=en|doi=10.1007/978-981-96-8078-8_7|editor-last2=Singh|editor-first2=Amit|editor-last3=Kar|editor-first3=Sujita Kumar}}&lt;/ref&gt;. 

=== Adoption of a Western-based mental healthcare model and ordinances ===
In 1839, [[w:James_Alexander_Stewart-Mackenzie|James Alexander Stewart-Mackenzie]], the 7th Governor of British Ceylon, released the Lunacy Ordinance, authorizing municipal authorities to create lunatic asylums for the mentally ill&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=article&amp;id=6&amp;Itemid=125&amp;lang=en|title=History - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-10}}&lt;/ref&gt;. The ordinance was concerned with the legal frameworks of detaining individuals considered dangerous to others or individuals falsely presenting themselves as mentally ill, and not on medical treatments to alleviate the conditions of detained individuals. UK psychiatrist [[w:Edward_Mapother|Edward Mapother]] critiqued the ordinance during his 1937 inspection of British Ceylon's mental health institutions in a series of reports titled ''A Disgrace to a Civilised Community'', remarking that the ordinance &quot;[did] not seem to have contemplated treatment as a contingency to be considered&quot;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|title=Permeable walls: historical perspectives on hospital and asylum visiting|date=2009|publisher=Rodopi|isbn=978-90-420-2599-8|editor-last=Mooney|editor-first=Graham|series=Clio medica|location=Amsterdam New York, NY|editor-last2=Reinarz|editor-first2=Jonathan}}&lt;/ref&gt;.

The 1839 Ordinance was repealed and replaced by the 1840 Ordinance, which removed two requirements from the previous Ordinance: the requirement for official medical diagnoses of the mentally ill and the mandate to maintain adequate staff-to-patient ratios within lunatic asylums&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Alwis|first=L. A. P. de|last2=Seneviratne|first2=V. L.|last3=Mendis|first3=T. S. S.|last4=Abhayanayaka|first4=C.|date=2024-12-31|title=The development of laws related to the disposal of forensic patients in Sri Lanka: A historical review|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v15i2.8569|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=15|issue=2|doi=10.4038/sljpsyc.v15i2.8569|issn=2012-6883}}&lt;/ref&gt;.

In 1873, a third Ordinance was released. It included linguistic changes, where the term, &quot;insane&quot;, was replaced with &quot;of unsound mind&quot;. The Ordinance also gave more power to medical professionals in determining insanity diagnoses, and more power to detainees in appealing their commitment to the mental asylum. Despite the increased granted authority, the legal frameworks behind the detainment of the criminally insane were left identical to previous ordinances&lt;ref name=&quot;:3&quot; /&gt;.

=== Development of mental asylums ===
At the time the 1839 ordinance was released, mentally ill patients were placed either in prisons throughout the country or leprosy hospitals, such as the [[w:Hendala_Leprosy_Hospital|Hendala Leprosy Hospital]] in the Gampaha district&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;. After the creation of the first mental asylum in Borella in 1846, patients from the Hendala Leprosy Hospital were transferred to Borella. Overcrowding soon became an issue, which led to patients being sent to prisons. [[File:Edward Mapother.jpg|thumb|A portrait taken of Edward Mapother during his time working at [[w:Maudsley_Hospital|Maudsley Hospital]] in London.
]]

As medical institutions were being made to house the mentally ill, another mental asylum was created in the [[w:Cinnamon_Gardens|Cinnamon Gardens]] area of Colombo in 1884, though this mental asylum faced overcrowding issues in just one year&lt;ref name=&quot;:0&quot; /&gt;. Treatment in these asylums was limited to occupational and protection therapy, failing to provide treatment for the root causes.

In 1926, the Angoda Mental Hospital was established, marginally alleviating the severe overcrowding issues that were plaguing the preceding mental asylums. Despite the addition of 1,700 beds to the facility, treatment was still vastly limited and the patients were left in significantly poor conditions. 

=== Edward Mapother and his 1937 inspection of British Ceylon ===
Edward Mapother was born in Dublin, Ireland, on July 12, 1881 and moved to London when he was 7 years old&lt;ref&gt;{{Cite book|title=Madness to mental illness: a history of the Royal College of Psychiatrists|last=Bewley|first=Thomas|date=2008|publisher=RCPsych Publications ; Distributed in North America by Balogh International|isbn=978-1-904671-35-0|location=London : [S.l.]}}&lt;/ref&gt;. Mapother attained his M.D. in 1908. While Mapother was the Medical Superintendent of Maudsley Hospital in London, England, he was invited to inspect British Ceylon's mental health institutions by Dr S. T. Gunasekara, the first Medical Director of British Ceylon&lt;ref name=&quot;:1&quot; /&gt;.

In Mapother's visit, he commented that the Angoda Mental Hospital had the atmosphere of &quot;a prison that is neglected and dilapidated&quot;&lt;ref name=&quot;:1&quot; /&gt;. Overcrowding was still a major issue, with the institute hosting 3,000 patients—more than double the intended capacity. Patients were sleeping on mats and were clearly out of reach of adequate treatment. Mapother also noted that only 4% of public health expenditure in the country was being set for hospitals, drawing a stark comparison to London's 25%&lt;ref name=&quot;:1&quot; /&gt;. Mapother offered a vivid and grim account of the hospital in his reports:
&lt;blockquote&gt;
The floor, roof and walls of each cell consist alike of drab cement without any attempt at colouring or decoration. High up in one wall is a small window with stout iron bars. In the floor is a large hole into which the patient may pass his motion and urine. These cells are incompletely divided from one another by a partition which does not reach the roof so that the noise and stink from any one cell may reach at least all the others of the same row. Into these empty cells I was informed that the most noisy and troublesome patients in the hospital; were turned at night completely naked. The doors of the cell contain no observation window, and considering the violent character of many of these patients there is every ground for believing that the doors are rarely opened in the night by the solitary attendant on duty. It needs little imagination to picture the suffering of any patient in an early stage of bodily illness passing a night under such conditions, a situation which must frequently arise. I am told that the noise proceeding from this building is like that on a bad night in a menagerie&lt;ref name=&quot;:0&quot; /&gt;.&lt;/blockquote&gt;Mapother proposed a series of reinforcements to the legal, institutional, and medical frameworks of mental health care in British Ceylon. This included the decentralization of the psychiatric services, a reworking of the Lunacy Ordinance to incorporate treatment into the legal framework, and the establishment of a separate service of medical professionals dedicated to psychiatry. Mapother's recommendations led to several of the best local medical professionals to be sent to London for extensive training in psychiatry, while nurses from England were sent to British Ceylon to supervise hospital operations and train local staff&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;.

On August 25, 1938, the Executive Committee of Health approved the strategies proposed by Mapother, though the Government was unable to fully implement all of Mapother's interventions due to the 'heavy cost'. In fact, the Government decided to forego one of his proposals at the beheast of the &quot;Visiting Committee&quot;, a committee that was tasked to &quot;meet at the hospital, carry out inspections, and make recommendations&quot; to the Executive Committee of Health&lt;ref name=&quot;:1&quot; /&gt;. The Government believed that deficiencies in their mental healthcare system could prove to be &quot;costly&quot; for their reputation, which enraged Maptoher. Mapother intended to contact the Secretary of State regarding the &quot;distortion&quot; of his plans, but was interrupted by events preceding [[w:World_War_II|World War II]]&lt;ref name=&quot;:1&quot; /&gt;. Mapother passed away on March 20, 1940, without materializing his follow-up plans.

=== Post-Mapother developments and further innovations ===
[[File:Sri Lanka districts Colombo.svg|thumb|A map of Sri Lanka highlighting the Colombo District, where the capital is located. 
|right|250px]]Mapother's insights on the mental healthcare structure in British Ceylon proved to be the catalyst of significant renovations. In 1939, the first outpatient clinic was established in the [[w:National_Hospital_of_Sri_Lanka|National Hospital of Sri Lanka]] in Colombo. The first trained Ceylonese psychiatrists began practice in the 1940s, leading to the establishment of the first neuropsychiatric clinic in Colombo in 1943. Treatments for the mentally ill improved dramatically, as [[w:insulin_shock_therapy|insulin shock therapy]] and [[w:Electroconvulsive_therapy|cardiazol convulsive therapy]] were utilized&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Kathriarachchi|first=Samudra T.|last2=Seneviratne|first2=V. Lakmi|last3=Amarakoon|first3=Luckshika|date=2019-06|title=Development of Mental Health Care in Sri Lanka: Lessons Learned|url=https://journals.lww.com/tpsy/fulltext/2019/33020/development_of_mental_health_care_in_sri_lanka_.1.aspx|journal=Taiwanese Journal of Psychiatry|language=en-US|volume=33|issue=2|pages=55|doi=10.4103/TPSY.TPSY_15_19|issn=1028-3684}}&lt;/ref&gt;. Mapother's advocation for the decentralization of services were further honored through the 1947 establishment of a first child guidance clinic in Colombo General Hospital&lt;ref name=&quot;:0&quot; /&gt;. 

In 1948, British Ceylon was granted independence after the [[w:Sri_Lankan_independence_movement|Sri Lankan independence movement]]. Changes in the mental healthcare structure were not immediate following independence, but rapid expansions of mental healthcare services were continuing to actualize.  

The following decades saw positive institutional developments, such as the creation of a second hospital in [[w:Mulleriyawa|Mulleriyawa]] in 1957, and the creation of a psychiatric inpatient unit in Colombo General Hospital in 1967—effectively granting the city of Colombo the luxury of hosting the top psychiatric care in the country&lt;ref name=&quot;:5&quot;&gt;{{Cite book|url=http://link.springer.com/10.1007/978-1-4899-7999-5_4|title=Mental Health System Development in Sri Lanka|last=Minas|first=Harry|last2=Mendis|first2=Jayan|last3=Hall|first3=Teresa|date=2017|publisher=Springer US|isbn=978-1-4899-7997-1|editor-last=Minas|editor-first=Harry|location=Boston, MA|pages=59–77|language=en|doi=10.1007/978-1-4899-7999-5_4|editor-last2=Lewis|editor-first2=Milton}}&lt;/ref&gt;. The 1950s was also the start of psychopharmacological innovations, with the introduction of [[w:Lithium_(medication)|lithium]] and long-acting injectable antipsychotics ([[w:Depot_injection|depot]] [[w:Antipsychotic|neuroleptics]]) in the succeeding years&lt;ref name=&quot;:4&quot; /&gt;. Additionally, the number of public psychiatrist positions increased by 400% from 1953 to 1967&lt;ref name=&quot;:5&quot; /&gt;.

After 1960, mental health services were expanded from beyond the capital to other cities in the country&lt;ref name=&quot;:2&quot; /&gt;.

In 1980, the [[w:Postgraduate_Institute_of_Medicine|Postgraduate Institute of Medicine]] initiated a program where students would enroll in a 5-year medical course and attain an MD in psychiatry, curbing the need for Sri Lankan medical students to be sent abroad to complete their training. Many of the medical students sent abroad for training never returned to Sri Lanka to practice, resulting in a &quot;1:500,000 to 1000,000&quot; ratio of psychiatrists to patients on &quot;most occasions&quot;&lt;ref name=&quot;:0&quot; /&gt;.

=== Mental Disease Ordinance of 1956 ===
In 1956, the 1873 Ordinance was revised a second time. The Mental Disease Ordinance of 1956 featured another linguistic development, as &quot;lunacy&quot; was replaced with &quot;mental disease&quot;&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|last=Hapangama|first=Aruni|last2=Mendis|first2=Jayan|last3=Kuruppuarachchi|first3=K. a. L. A.|date=2023-02|title=Why are we still living in the past? Sri Lanka needs urgent and timely reforms of its archaic mental health laws|url=https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4|journal=BJPsych International|language=en|volume=20|issue=1|pages=4–6|doi=10.1192/bji.2022.26|issn=2056-4740|pmc=9909436|pmid=36812028}}&lt;/ref&gt;. The Ordinance paved way for community-based services to be delivered to patients closer to their residences, rather than strictly allocating services to just hospitals. This led to the creation of a [[w:WHO|WHO]]-backed community clinic near the [[w:University_of_Colombo|University of Colombo]] in the 1970s, where the focus was to eventually ease patients in the Angoda Mental Hospital back into the general population&lt;ref name=&quot;:5&quot; /&gt;.

=== Developments from the 1990s ===
The 1990s and onwards saw further positive developments in framing the mental healthcare system, including the establishment of the [https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=featured&amp;Itemid=101&amp;lang=en Directorate of Mental Health] in 1998. The Directorate of Mental Health is a part of the [[w:Ministry_of_Health_(Sri_Lanka)|Ministry of Health]] and is responsible for the monitoring and implementation of mental health programs across the country&lt;ref&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?lang=en|title=Home - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}&lt;/ref&gt;.  As of 2025, the current director of the Directorate of Mental Health is Dr. Chithramalee de Silva&lt;ref name=&quot;:2&quot; /&gt;.

On November 11, 2005, the Mental Health Policy was approved by the Government of Sri Lanka, advocating for establishments of more de-centralized, community-based mental health services across the country. The policy aimed to concisely define the rigorous standards needed to be met for each respected medical professional, including psychiatrists and clinical psychologists&lt;ref&gt;{{Cite journal|last=Rajapakshe|first=Onali Bimalka Wickramaseckara|last2=Mohan|first2=Mohapradeep|last3=Singh|first3=Swaran Preet|date=2023-05|title=Development of adolescent mental health services in Sri Lanka|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC10895478/|journal=BJPsych international|volume=20|issue=2|pages=41–43|doi=10.1192/bji.2022.32|issn=2056-4740|pmc=10895478|pmid=38414998}}&lt;/ref&gt;. The policy also included a new position, the &quot;Medical Officer of Mental Health&quot;, tasked with overseeing and assisting in creating community-based mental health services&lt;ref name=&quot;:0&quot; /&gt;. In the same year, the Sri Lankan government began implementing psychological services in state institutions, such as the military&lt;ref name=&quot;:8&quot; /&gt;.  

In 2007, the National Mental Health Advisory Council (NMHAC) was created to serve as an 'advisory' board for the Ministry of Health&lt;ref name=&quot;:7&quot;&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=article&amp;id=9&amp;Itemid=220&amp;lang=en|title=Introduction - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}&lt;/ref&gt;. 

In 2008, the Angoda Mental Hospital was restructured and renamed as the National Institute of Mental Health (NIMH)&lt;ref name=&quot;:7&quot; /&gt;. 

=== Modern-day Sri Lanka ===
[[File:Feeding Children in Sri Lanka.jpg|left|thumb|Despite the noteworthy improvements in mental healthcare services in recent decades, mental health remains a significant issue due to rising poverty. ]]
As of 2025, the Mental Health Act (mental health legislation) has been undergoing development since 2005 and is currently awaiting to be considered for the final stage of approval. This is expected to replace the 1956 Mental Health Ordinance&lt;ref name=&quot;:7&quot; /&gt;.

Currently, there are 7 tertiary care hospitals, 61 adult patient units, 3 child inpatient units, and 1 forensic unit with over 100 psychiatrists all throughout the 22 districts&lt;ref name=&quot;:4&quot; /&gt;. The [[w:Lady_Ridgeway_Hospital_for_Children|Lady Ridgeway Hospital]] in Colombo and the Sirimavo Bandaranayke Specialized Children Hospital in Kandy are specialized in treating children with [[w:Learning_disability|SLD]], [[w:ADHD|ADHD]], [[w:Autism_Spectrum_Disorder|ASD]], and provides family support for patients. As of 2017, 22 rehabilitation centers exist through the country, including 7 alcohol rehab centers&lt;ref name=&quot;:7&quot; /&gt;.   

Despite the impressive advancements in mental healthcare in the last couple of decades, Sri Lanka still suffers significant mental health issues due to increasing poverty levels in the country. The [[w:World_Bank|World Bank]] reported that [https://www.wsws.org/en/articles/2024/04/08/eesc-a08.html the poverty levels in Sri Lanka increased from 11% in 2019 to 26% in 2024], with 60% of Sri Lankan households facing &quot;decreased incomes&quot;&lt;ref&gt;Lakhtakia, Shruti, Atapattu Mudiyanselage, Udahiruni Shashadari Atapat, Walker, Richard Ancrum. ''Sri Lanka Development Update - Bridge to Recovery (English).'' Washington, D.C.: World Bank Group. &lt;nowiki&gt;http://documents.worldbank.org/curated/en/099634104012434919&lt;/nowiki&gt;&lt;/ref&gt;. This was exacerbated by Sri Lanka's excessive foreign debt, economic troubles stemming from [[w:Gotabaya_Rajapaksa|Gotabaya Rajapaksa]]'s presidential term, the COVID-19 pandemic, and the [[w:Russian_invasion_of_Ukraine|ongoing invasion of Ukraine by Russia (2022)]].  

According to [[w:NYU|New York University]] graduate student [https://gc-cuny.academia.edu/NadiaAugustyniak Nadia Augustyniak] in her 2025 overview of Sri Lanka's public mental healthcare system, poverty-induced financial precarity remains a major obstacle to receiving access to mental healthcare services. Even though trauma from adverse weather and conflict is deleterious to mental health, issues originating from every-day struggles, especially struggles related to poverty, could arguably play a more significant role&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}&lt;/ref&gt;.  

== Impact of Conflicts, Terrorism, Political Instability &amp; Natural Disasters ==

=== Sri Lankan Civil War ===
The '''Sri Lankan Civil War''' was a domestic conflict between the Sri Lankan government and the Liberation Tigers of Tamil Eelam (abbreviated as the ''LTTE),'' a militant group formed in the 1970s as a byproduct of rising tensions between the majority Sinhalese and minority Tamil population. The group is considered a terrorist organization&lt;ref&gt;{{Cite web|url=https://www.start.umd.edu/baad/database/liberation-tigers-tamil-eelam-ltte-1998.html|title=BAAD - Liberation Tigers of Tamil Eelam (LTTE) - 1998 {{!}} START.umd.edu|website=www.start.umd.edu|access-date=2025-06-09}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.cfr.org/backgrounder/liberation-tigers-tamil-eelam-aka-tamil-tigers-sri-lanka-separatists|title=Liberation Tigers of Tamil Eelam (aka Tamil Tigers) (Sri Lanka, separatists) {{!}} Council on Foreign Relations|last=Bhattacharji|first=Preeti|website=www.cfr.org|language=en|access-date=2025-06-09}}&lt;/ref&gt;. The LTTE conducted decades of massacres, assassinations of political figures, and suicide bombings to achieve ''[[w:Tamil_Eelam|Tamil Eelam]],'' leading to civilian displacement, infrastructure collapse, and the reduction of mental health services available in the northern region.[[File:DFID-funded, UNHCR emergency shelter tents, in the IDP camp at Menik Farm, Sri Lanka (3694081492).jpg|thumb|350x350px|An IDP camp in Menik Farm, Sri Lanka in 2009 ([https://www.bbc.com/news/world-asia-19703826 now closed]). Suicide rates in IDP camps were three times the general population.]]The civil war mainly affected the northeastern portion of the country, including the [[w:Vanni_(Sri_Lanka)|Vanni region]]. The conflict caused mass destruction to local mental healthcare facilities. Local residents described the conflict as ''varthayal varnicca mudiyathavai'', roughly translating into English as 'beyond description by words'&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|language=en|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}&lt;/ref&gt;. In 2003, only two psychiatrists were found in the region, operating on extremely limited resources. This furthered long-term trauma and mental health deterioration in the population&lt;ref name=&quot;:5&quot; /&gt;. 

In 2002, the humanitarian organization [https://www.msf.org/ Médecins Sans Frontières] (MSF) conducted an investigation on mental health needs in the [[w:Vavuniya|Vavuniya]] area, the site of intense conflict during the civil war (including the [[w:1985_Vavuniya_massacre|1985 Vavuniya massacre]]), and found that many of the residents suffered from high suicide rates, alcohol abuse, domestic violence, grief, and a &quot;sense of ‘learnt helplessness’&quot;&lt;ref name=&quot;:5&quot; /&gt;. A team from the University of Konstanz in Germany found that 92% of grade school children in the region were exposed to &quot;combat, shelling, and witnessing the death of loved ones&quot;&lt;ref name=&quot;:9&quot; /&gt;. 
[[File:Tractors. Jan 2009 displacement in the Vanni.jpg|left|thumb|350x350px|Displaced civilians evacuating from the Kilinochchi and Mullaitivu Districts due to military campaigns initiated by the Sri Lankan military (January 2009).]]
Additionally, accusations of war crimes have been made against [[w:War_crimes_during_the_final_stages_of_the_Sri_Lankan_civil_war|the Sri Lankan government]]&lt;ref&gt;See also [[w:Sexual violence in the Sri Lankan civil war]].&lt;/ref&gt;. A 2009 HRW report alleged that the Sri Lankan government considered the native Tamil population residing in war zones to be &quot;siding with the LTTE and [therefore, were] treated as combatants&quot;, and that the government conducted numerous shellings of &quot;areas crowded with civilians&quot;&lt;ref&gt;{{Cite journal|date=2009-02-19|title=War on the Displaced|url=https://www.hrw.org/report/2009/02/19/war-displaced/sri-lankan-army-and-ltte-abuses-against-civilians-vanni|journal=Human Rights Watch|language=en}}&lt;/ref&gt;. Furthermore, the LTTE conducted recruitment campaigns on the Vanni population where recruited men, women, and even children with minimal training, were recruited for war efforts.  

Over 200,000 Tamil civilians were moved into [[w:Internally_displaced_persons_in_Sri_Lanka|designated displacement camps during the war]], where conditions were poor&lt;ref&gt;{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000}}&lt;/ref&gt;. The suicide rate in these displacement camps was three times the community-level (2002), with a ratio of 103.5 suicides per 10,000 persons, compared to the general population's rate of 37.5 suicides per 10,000 persons. Almost all suicide attempts involved poisonous substances. Other forms of violence included domestic violence and child abuse.  

Local health officials in Vavuniya admitted that mental health concerns were a major problem, but were unable to address these concerns due to a lack of resources and support from the government. During the [[wikipedia:Sri_Lankan_civil_war#2002_peace_process_(2002%E2%80%932006)|brief 2002 ceasefire]], the MSF implemented a &quot;community-based programme&quot; which included &quot;increasing awareness, community strengthening, reinforcing coping-strategies for long-term war-affected communities, and counselling&quot;. The MSF also advocated for restrictions of poisonous substances due its means for suicide attempts, and stressed that &quot;much more [than resettlement]&quot; would need to be done to help alleviate the psychological pain the northern population had faced due to the war&lt;ref&gt;{{Cite journal|last=de Jong|first=Kaz|last2=Mulhern|first2=Maureen|last3=Ford|first3=Nathan|last4=Simpson|first4=Isabel|last5=Swan|first5=Alison|last6=van der Kam|first6=Saskia|date=2002-04|title=Psychological trauma of the civil war in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S0140673602084209|journal=The Lancet|language=en|volume=359|issue=9316|pages=1517–1518|doi=10.1016/S0140-6736(02)08420-9}}&lt;/ref&gt;. The ceasefire ended in 2006 and led to the [[w:Eelam_War_IV|final phase of the civil war]], eventually ending in 2009 with the [[w:https://en.wikipedia.org/wiki/Velupillai_Prabhakaran#Sri_Lankan_Army_Northern_offensive_and_death|death of the LTTE's leader]]. 

'''Post-war'''
[[File:Puttalam district.svg|left|thumb|Puttalam District, unlike its northern counterparts, was largely spared from the intense conflict, possibly explaining the lower rates of common mental disorders (CMDs).]]
The first district-wide cross-sectional multistage cluster sample survey was conducted in the [[w:Jaffna_District|Jaffna District]] shortly after the war ended in 2009. The study's sample included 1517 households and 2 internally displaced peoples camps. With a response rate of 92%, the study found that symptoms for PTSD were found in 7% of participants, symptoms of anxiety were found in 32.6% of participants, and symptoms of depression were found in 22.2% of participants. 2% of respondents were being placed in internally displaced peoples camps at the time of the study, 29.5% were freshly resettled from the internally displaced peoples camps, and the rest of the participants (68.5%) were never placed into camps. In comparison to residents who were never placed into camps, participants that were actively held in camps generally reported more symptoms of PTSD, anxiety, and depression. The researchers also found that women were especially vulnerable to deteriorating mental health conditions. This was explained by two factors: women having to assume the roles of both the father and the mother in the family setting after the, either voluntary or forced, departure of their husband to war, and sexist violence&lt;ref&gt;{{Cite journal|last=Husain|first=Farah|last2=Anderson|first2=Mark|last3=Lopes Cardozo|first3=Barbara|last4=Becknell|first4=Kristin|last5=Blanton|first5=Curtis|last6=Araki|first6=Diane|last7=Kottegoda Vithana|first7=Eeshara|date=2011-08-03|title=Prevalence of War-Related Mental Health Conditions and Association With Displacement Status in Postwar Jaffna District, Sri Lanka|url=https://doi.org/10.1001/jama.2011.1052|journal=JAMA|volume=306|issue=5|pages=522–531|doi=10.1001/jama.2011.1052|issn=0098-7484}}&lt;/ref&gt;. A 2013 study on adult patients in [https://www.ncbi.nlm.nih.gov/books/NBK232631/ primary care settings] (divisional hospitals, primary medical care units) found major depression to be significantly higher in females (5.1%) than males (3.6%), bolstering the findings from the 2009 study&lt;ref&gt;{{Cite journal|last=Senarath|first=Upul|last2=Wickramage|first2=Kolitha|last3=Peiris|first3=Sharika Lasanthi|date=2014-03-24|title=Prevalence of depression and its associated factors among patients attending primary care settings in the post-conflict Northern Province in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/1471-244X-14-85|journal=BMC Psychiatry|language=en|volume=14|issue=1|pages=85|doi=10.1186/1471-244X-14-85|issn=1471-244X|pmc=3987835|pmid=24661436}}&lt;/ref&gt;.

Muslims in Northern Sri Lanka also faced violence and discrimination during the conflict. Most notable incidents include [[w:Expulsion_of_Muslims_from_the_Northern_Province_of_Sri_Lanka|the October 1990 expulsion of Muslims from the North to the Puttalam District or Jaffna]] and the [[w:Kattankudy_mosque_massacre|1990 Kattankudy mosque massacre]]. The only study testing the displaced Muslim population post-civil war was completed in 2011, where a cross-sectional survey of 450 internally displaced people or people born into displacement (ages 18 - 65) revealed 18.8% of the sample suffering from common mental health disorders (CMD), including [[w:Somatoform_disorder|somatoform disorder]] (14%), &quot;other depressive syndromes&quot; (7.3%), major depression (5.1%), and anxiety disorder (2.8%). The percentages found in this study for somatoform disorder and major depression were &quot;considerably higher&quot; than the national percentages, though the researchers noted that the prevalence of CMD was lower in comparison to other countries marred with conflict, including Palestine (40.3%) and Ethiopia (27.8%). The researchers explained that the lower rate of CMD may be attributed to the [[w:Puttalam_District|serenity of the post-settlement destination]], as conflict was mainly centered in the North and East. In contrast to earlier findings, this study did not observe a higher prevalence of CMDs among women, although increased rates of somatoform disorders were noted (though the researchers did not reveal the data behind this)&lt;ref&gt;{{Cite journal|last=Siriwardhana|first=Chesmal|last2=Adikari|first2=Anushka|last3=Pannala|first3=Gayani|last4=Siribaddana|first4=Sisira|last5=Abas|first5=Melanie|last6=Sumathipala|first6=Athula|last7=Stewart|first7=Robert|date=2013-05-22|title=Prolonged Internal Displacement and Common Mental Disorders in Sri Lanka: The COMRAID Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0064742|journal=PLOS ONE|language=en|volume=8|issue=5|pages=e64742|doi=10.1371/journal.pone.0064742|issn=1932-6203|pmc=3661540|pmid=23717656}}&lt;/ref&gt;.

Research on the mental state of combatants has been limited, but a post-war 2009 study done between soldiers of the [[w:Sri_Lanka_Army_Special_Forces_Regiment|Special Forces]] and regular soldiers showed higher levels of exposure to traumatic events for units of the Special Forces, yet the former exhibited significantly less symptoms of CMDs compared to the latter. The authors of this study, [https://scholar.google.co.uk/citations?user=cVKEBdwAAAAJ&amp;hl=en&amp;oi=ao Raveen Hanwella] and [https://scholar.google.co.uk/citations?user=ZRj74qMAAAAJ&amp;hl=en&amp;oi=sra Varuni de Silva], offered the camaraderie of the military unit as an explanation for the discrepancy&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|date=2012-08|title=Mental health of Special Forces personnel deployed in battle|url=https://pubmed.ncbi.nlm.nih.gov/22038567|journal=Social Psychiatry and Psychiatric Epidemiology|volume=47|issue=8|pages=1343–1351|doi=10.1007/s00127-011-0442-0|issn=1433-9285|pmid=22038567}}&lt;/ref&gt;. A follow-up study was completed by the pair (with the addition of former Director-General of the Health Services of the Sri Lanka Navy [[w:Nicholas_Jayasekera|Nicholas Jayasekera]]), where the findings were similar, though the statistically significant bridge between the two cohorts in the previous study evaporated in the follow-up study. This may be due to the significant decline in mental health problems observed in the regular unit forces, potentially reflecting resilience in the aftermath of the conflict&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=Jayasekera|first2=Nicholas E. L. W.|last3=Silva|first3=Varuni A. de|date=2014-09-25|title=Mental Health Status of Sri Lanka Navy Personnel Three Years after End of Combat Operations: A Follow Up Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0108113|journal=PLOS ONE|language=en|volume=9|issue=9|pages=e108113|doi=10.1371/journal.pone.0108113|issn=1932-6203|pmc=4177866|pmid=25254557}}&lt;/ref&gt;. Amputees or soldiers with spinal injuries exhibited drastically different numbers, with approximately 40% of nearly 100 male-veterans in a post-war 2009 study displaying PTSD-like symptoms&lt;ref&gt;{{Cite journal|last=Abeyasinghe|first=N. L.|last2=de Zoysa|first2=P.|last3=Bandara|first3=K.M.K.C.|last4=Bartholameuz|first4=N. A.|last5=Bandara|first5=J. M.U.J.|date=2012-05-01|title=The prevalence of symptoms of Post-Traumatic Stress Disorder among soldiers with amputation of a limb or spinal injury: A report from a rehabilitation centre in Sri Lanka|url=https://doi.org/10.1080/13548506.2011.608805|journal=Psychology, Health &amp; Medicine|volume=17|issue=3|pages=376–381|doi=10.1080/13548506.2011.608805|issn=1354-8506|pmid=21942815}}&lt;/ref&gt;.

About a decade after the conflict ceased, a few notable studies have emerged to help guide understanding on the longer-term mental health effects on victims of the civil war. 

From July 2019 to October 2020, a study conducted on 585 local adolescents (ages 12-19) in the Vavuniya district revealed that despite 15.6% of the statistic having faced one or more war-related events, only 3.9% of the participants had moderate to severe depression. In addition to considerably low depression rates, only 5.7% of participants age 17+ were found to have moderate to severe hopelessness&lt;ref&gt;{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000|pmc=10472617|pmid=37653394}}&lt;/ref&gt;. The authors referenced a 2010 observation by psychiatrist [https://us.sagepub.com/en-us/nam/author/daya-somasundaram Daya Somasundaram], who noted that many Tamil IDPs presented &quot;remarkable resilience and post-traumatic growth&quot; after the civil war—an outcome he attributed to the close-knit, family-centered nature of Tamil communities&lt;ref&gt;{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}&lt;/ref&gt;.  

However, findings originating from a 2019 study, undertook by several faculty members from the University of Kelaniya, the University of Jaffna, the [[w:Gampaha_Wickramarachchi_University_of_Indigenous_Medicine|Gampaha Wickramarachchi University of Indigenous Medicine]], and the [https://onur.gov.lk/ Office for National Unity and Reconciliation (ONUR)] in Jaffna, found contrasting results. Out of 336 participants from districts which faced significant ramifications of the conflict (Jaffna, Kilinochchi, Mullaithivu, Vavuniya, and Mannar districts), 50.5% had extreme anxiety symptoms and 36.5% exhibited &quot;extremely severe&quot; symptoms of depression. 92.5% of families in the sample experienced suicidal ideation, with an observed negative correlation between trauma exposure and life satisfaction with families. Drug abuse (86.2%) and alcohol abuse (84.5%) were the two highest problematic behaviors recorded on a community-level, suggesting that the negative consequences of the civil war still persist, possibly on a substantial scale than previously recognized, in Tamil communities residing in the North&lt;ref&gt;{{Cite journal|last=Thamotharampillai|first=Umaharan|last2=Perera|first2=Ruwanthi|last3=Wickremasinghe|first3=Rajitha|last4=Williams|first4=Shehan|last5=Vijayasangar|first5=Thedsanamoorthy|last6=Sivatharsan|first6=Balasubramaniam|last7=Hilbert|first7=Vanceline|last8=Somasundaram|first8=Daya|date=2025-05-06|title=Collective Trauma- Psychosocial consequences of war in northern Sri Lanka 10 years on, a mixed methods study|url=https://www.sciencedirect.com/science/article/pii/S2666560325000696|journal=SSM - Mental Health|pages=100457|doi=10.1016/j.ssmmh.2025.100457|issn=2666-5603}}&lt;/ref&gt;. Further research should be conducted on Northern Tamil populations to assess the extent of mental health issues stemming from the conflict. 

In 2019, [https://www.researchgate.net/scientific-contributions/R-M-M-Monaragala-2087692299 Dr. R. M. M. Monaragala] conducted a study on 1,845 soldiers with combat experience, finding that 3.9% of the sample suffered from PTSD. Dr. Monaragala noted that &quot;probable depression, fatigue, aggression, and family history of mental disorder&quot; were correlative of PTSD presence. He suggested that &quot;screening and psychosocial intervention[s]&quot; could alleviate CMDs of former combatants&lt;ref&gt;{{Cite journal|last=Monaragala|first=R. M. M.|date=2024-04-19|title=Exploring the effects of the past civil war in terms of the prevalence and associating factors of PTSD|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v14i2.8465|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=14|issue=2|doi=10.4038/sljpsyc.v14i2.8465|issn=2012-6883}}&lt;/ref&gt;. 

=== 2004 Boxing Day Tsunami ===
The '''2004 Boxing Day Tsunami''' was a natural disaster where a tsunami spawned off a 9.2–9.3 magnitude earthquake off the coast of Aceh in Indonesia on December 26. The tsunami greatly affected the coastlines of the country, with the death toll reaching to around 35,000 deaths. In addition, 90,000 houses were destroyed and 516,000 people were forced to migrate due to severe infrastructural damage&lt;ref name=&quot;:5&quot; /&gt;. It stands as the [http://www.china.org.cn/english/features/tsunami_relief/119821.htm worst natural disaster to have ever hit Sri Lanka].  
[[File:Tsunami relief 2004 02.jpg|thumb|300x300px|Volunteers from [[w:Royal_College,_Colombo|Royal College in Colombo]] assisting in tsunami relief efforts (Sarvodaya Headquaters, Moratuwa).]]
A survey conducted on schoolchildren (ages 8-14) in Manadkadu (a Tamil-majority village in the northern coast), [[w:Kosgoda|Kosgoda]] (western coast), and [[w:Galle|Galle]] (southern coast), just a few weeks after the tsunami hit Sri Lanka, revealed that 33.8%, 13.9%, and 38.8% of children interviewed exhibited signs of PTSD (according to the DSM-IV's criteria), respectively (minus the time criteria, as the DSM-IV does not permit diagnosis of PTSD within 4 weeks of a traumatic incident). The loss of family members and exposure to previously traumatic incidents appeared to be highly correlate with PTSD development&lt;ref&gt;{{Cite journal|last=Neuner|first=Frank|last2=Schauer|first2=Elisabeth|last3=Catani|first3=Claudia|last4=Ruf|first4=Martina|last5=Elbert|first5=Thomas|date=2006|title=Post-tsunami stress: A study of posttraumatic stress disorder in children living in three severely affected regions in Sri Lanka|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/jts.20121|journal=Journal of Traumatic Stress|language=en|volume=19|issue=3|pages=339–347|doi=10.1002/jts.20121|issn=1573-6598}}&lt;/ref&gt;. 

Many victims in the Jaffna area suffered with &quot;[https://www.psychiatry.org/patients-families/prolonged-grief-disorder pathological grief], phobias, depression and PTSD&quot; post-tsunami. Schizophrenia in the Jaffna Tamil community, which had already suffered elevated prevalence of PTSD prior to the tsunami, had worsened—highlighting the need for specialized care in response to cumulative exposures to chronic and acute traumas. In a study published in ''International Psychiatry'' (2006), Jaffna-based researchers noted that, contrary to their initial inclinations, there was not a &quot;large[r] (than expected) rise in [the] number of people&quot; seeking mental health support 3 months after the tsunami. However, 10 months after the disaster, the researchers anticipated that &quot;more psychiatric disorders&quot; would emerge due to &quot;very little rebuilding [efforts]&quot; and an apparent &quot;unfairness in the aid system&quot;.&lt;ref&gt;{{Cite journal|last=Somasundaram|first=D. J.|last2=Yoganathan|first2=S.|last3=Ganesvaran|first3=T.|date=1993-09|title=Schizophrenia in northern Sri Lanka|url=https://pubmed.ncbi.nlm.nih.gov/7828234|journal=The Ceylon Medical Journal..|volume=38|issue=3|pages=131–135|issn=0009-0875|pmid=7828234}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Danvers|first=K.|last2=Sivayokan|first2=S.|last3=Somasundaram|first3=D. J.|last4=Sivashankar|first4=R.|date=2006-07|title=Ten months on: qualitative assessment of psychosocial issues in northern Sri Lanka following the tsunami|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6734678/|journal=International Psychiatry: Bulletin of the Board of International Affairs of the Royal College of Psychiatrists|volume=3|issue=3|pages=5–8|issn=1749-3676|pmc=6734678|pmid=31507850}}&lt;/ref&gt; 

At the February 2005 ''After the Tsunami: Mental Health Challenges to the Community for Today and Tomorrow'' conference in Thailand, [https://www.researchgate.net/profile/Chandanie-Hewage Dr. Chandanie Hewage] of the [[w:University_of_Ruhuna|University of Ruhuna]] commentated that measures taken to assist the affected were &quot;not coordinated&quot; due to poor &quot;communication systems and road [conditions].&quot; Regardless, efforts were continued by the government and health professionals to alleviate the struggles the victims were facing, including the psychological ramifications of the disaster. 

Several issues in the delivery of these services were highlighted by Dr. Hewage, including poor maintenance of health records, lack of awareness on drug consumption by the patients themselves, and shortages of health professionals. Dr. Hewage points out that personnel had &quot;little&quot; mental health training prior to the disaster, suggesting increased &quot;research&quot; and adequate &quot;provision[ing] and training of staff&quot; for the long-term&lt;ref&gt;{{Cite journal|last=Davidson|first=Jonathan R. T.|date=2006|title=Foreword. After the tsunami: mental health challenges to the community for today and tomorrow|url=https://pubmed.ncbi.nlm.nih.gov/16602809|journal=The Journal of Clinical Psychiatry|volume=67 Suppl 2|pages=3–8|issn=0160-6689|pmid=16602809}}&lt;/ref&gt;. With inadequate documentation, no systematic procedures in place, and insufficient personnel, tsunami victims with mental health concerns may not receive the services they need, further compacting neuropsychological ailments.  

In 2008 (about 3-4 years after the tsunami), researchers in the hard-hit village of [[w:Peraliya|Peraliya]] (Galle District) found that from a sample of approximately 90 adults, 25% suffered from moderate–severe PTSD, with women scoring &quot;above the cut-off for anxiety&quot; and reporting more &quot;somatic symptoms&quot;, though researchers inferred that the PTSD rate found in the study may be influenced by other factors, including war or economic hardship&lt;ref&gt;{{Cite journal|last=Hollifield|first=Michael|last2=Hewage|first2=Chandanie|last3=Gunawardena|first3=Charlotte N.|last4=Kodituwakku|first4=Piyadasa|last5=Bopagoda|first5=Kalum|last6=Weerarathnege|first6=Krishantha|last7=Group|first7=International Post-Tsunami Study|date=2008-01|title=Symptoms and coping in Sri Lanka 20–21 months after the 2004 tsunami|url=https://www.cambridge.org/core/journals/the-british-journal-of-psychiatry/article/symptoms-and-coping-in-sri-lanka-2021-months-after-the-2004-tsunami/CB33752239AF362A0BFD55B3668D60B0|journal=The British Journal of Psychiatry|language=en|volume=192|issue=1|pages=39–44|doi=10.1192/bjp.bp.107.038422|issn=0007-1250}}&lt;/ref&gt;.  

=== 2019 Easter Bombings ===
The '''2019 Easter Bombings''' were a series of coordinated attacks perpetrated by the Islamic extremist group, [[w:National_Thowheeth_Jama'ath|National Thowheeth Jama'ath]], on April 21, 2019. The attack targeted three churches and three hotels in the Colombo area, killing nearly 300 people and injuring over 500. The attacks were also attributed to the incompetency of the Sri Lankan government, who ignored [https://www.bbc.com/news/world-asia-48044636 multiple warnings preceding the attacks]. The attacks negatively affected the Sri Lankan Catholic community and further weakened relations between the major religious groups&lt;ref&gt;{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}&lt;/ref&gt;.  

In the aftermath of the attacks, professionals in the [[w:Gampaha_District|Gampaha District]] resorted to &quot;low-cost methodologies&quot; for children and adolescents affected by the attack, as a &quot;severe shortage&quot; of children and adolescent mental health experts were exposed&lt;ref&gt;{{Cite journal|last=Chandradasa|first=Miyuru|last2=Rathnayake|first2=Layani C|last3=Rowel|first3=Madushi|last4=Fernando|first4=Lalin|date=2020-06-01|title=Early phase child and adolescent psychiatry response after mass trauma: Lessons learned from the Easter Sunday attack in Sri Lanka|url=https://doi.org/10.1177/0020764020913314|journal=International Journal of Social Psychiatry|language=EN|volume=66|issue=4|pages=331–334|doi=10.1177/0020764020913314|issn=0020-7640}}&lt;/ref&gt;. In a qualitative study of 8 survivors of the attacks receiving grief counseling, [[w:University_of_Ruhuna|University of Ruhuna]] assistant professor [https://www.researchgate.net/profile/Virasha-Godakanda Virasha Godakanda] observed that 70% of the sample size expressed a lack of confidence in adequate mental health interventions from the government, reducing the quality of such services. Professor Godakanda strongly endorsed for &quot;culturally-sensitive&quot; programs, a diversity in therapeutic approaches (including nature-based therapy), and &quot;prolonged investigations&quot; to track developments in mental health resources and impacts of implemented interventions&lt;ref&gt;{{Cite journal|last=Godakanda|first=Virasha|date=2025-01-29|title=A GRIEF COUNSELING INTERVENTION AFTER THE MASS TRAUMA: LESSONS LEARNED FROM THE VICTIMS OF THE EASTER SUNDAY ATTACK IN SRI LANKA|url=https://kjmr.com.pk/kjmr/article/view/216|journal=Kashf Journal of Multidisciplinary Research|language=en|volume=2|issue=01|pages=13–32|doi=10.71146/kjmr216|issn=3007-200X}}&lt;/ref&gt;. 

A few weeks following the attacks, Muslims in Sri Lanka were subjected to [[w:2019_anti-Muslim_riots_in_Sri_Lanka|violent, coordinated riots]] masterminded by Sinhalese national forces&lt;ref&gt;{{Cite journal|last=Mujahidin|first=Muhammad Saekul|date=2023-07-03|title=Extremism and Islamophobia Against the Muslim Minority in Sri Lanka|url=https://www.ajis.org/|journal=American Journal of Islam and Society|language=en|volume=40|issue=1-2|pages=213–241|doi=10.35632/ajis.v40i1-2.3135|issn=2690-3741}}&lt;/ref&gt;. Riots were mainly centered in the [[w:Kurunegala_District|Kurunegala]], Gampaha, and [[w:Kandy_District|Kandy]] Districts. At least [https://www.aljazeera.com/news/2019/5/21/in-sri-lanka-muslims-say-sinhala-neighbours-turned-against-them one confirmed death was reported]. Calls for vague ''niqab'' and ''burqa'' bans were increasingly prominent, eventually leading to the 2021 burqa ban by the Sri Lankan government. Pakistani and Afghani refugees fleeing religious persecution in Negombo were forced to be &quot;made refugees again&quot; after local protests were orchestrated against their settlement. Anti-Muslim sentiment was &quot;unleashed online, in the law, and on the street&quot;&lt;ref&gt;{{Cite book|title=CARTOGRAPHIC JOURNEY OF RACE, GENDER AND POWER: global identity|date=2021|publisher=CAMBRIDGE SCHOLARS PUBLIS|isbn=978-1-5275-6965-2|location=S.l.}}&lt;/ref&gt;. Albeit its relevancy to the attacks, no in-depth mental health studies have took place on the minority Muslim population following the Easter bombings. Further research is imperative in exploring the sustained psychological effects of Islamophobia and its effect on the Muslim minority community in the aftermath of the 2019 Easter attacks. 

Literature on the impact of the 2019 Easter Bombings on mental health is limited and further research should be conducted.  

=== 2019-2024 Economic Crisis ===
The '''2019-2024 Economic Crisis''' refers to a 5 year period where the Sri Lankan economy experienced significant inflation and an abrupt hike in prices on basic, everyday items. It is the worse economic crisis the country has faced since the Sri Lankans were granted independence in 1948. Schools in Sri Lanka were forced to postpone examinations due to paper shortages. Gas shortages led to long lines at gas stations, some lasting for days, throughout the island. Shortages in electricity, cooking gas, and aviation feul were additional consequences of the economic crisis.

Healthcare workers faced a barrage of impediments in their line of work during the crisis, including a lopsided work-life balance due to unprecedented demand, increased stress and mental fatigue from a lack of resources and personnel, unhealthy coping mechanisms, job dissatisfaction, and a reduction in work quality. Such effects perpetuated a self-enforcing cycle of psychologically distressed mental healthcare workers providing subpar services, affecting patients and amplifying mental health issues experienced by both the workforce and their patients&lt;ref&gt;{{Cite journal|last=Dilogini|first=S.|last2=Grace|first2=H. H.|last3=Thasika|first3=T.|date=2024|title=Exploring The Mental Health and Well-Being of Public Healthcare Workers (HCWs) Amid Economic Crisis in Sri Lanka|url=http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/11092|language=en|publisher=Chartered Institute of Personnel Management}}&lt;/ref&gt;. 

Medical students from the Faculty of Medicine at the University of Colombo reported that the economic crisis forced abrupt changes in dietary consumption, increased hopelessness in the future, increased stress and anxiety, and a decrease in interest in pursuing a &quot;clinical post-graduate career&quot;&lt;ref&gt;{{Cite journal|last=Adikaranayake|first=Pesala Randika|last2=Perera|first2=Anusha Nimrod|last3=Nilaweera|first3=Akhila Imantha|last4=Fernando|first4=Desha Rajni|last5=Wijayaratne|first5=Dilushi Rowena|date=2025-07-01|title=Effects of Sri Lankan economic crisis on health, lifestyle and education of medical students in Faculty of Medicine, University of Colombo – an online survey|url=https://doi.org/10.1186/s12909-025-07506-y|journal=BMC Medical Education|language=en|volume=25|issue=1|pages=938|doi=10.1186/s12909-025-07506-y|issn=1472-6920|pmc=12211748}}&lt;/ref&gt;. 283 government-school teachers completed a web-based cross-sectional survey in April 2024, with majority of the participants reporting a severe reduction in monthly income &amp; 1/3 of participants exhibiting &quot;clinical levels of psychological distress&quot;&lt;ref&gt;{{Cite journal|last=Senevirathne|first=C. P.|last2=Senarathne|first2=D. L. P.|last3=Fernando|first3=M. S.|last4=Senevirathne|first4=S. P.|date=2025-05-28|title=Examining the economic burden and mental health distress among government school teachers in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/s40359-025-02921-8|journal=BMC Psychology|language=en|volume=13|issue=1|pages=572|doi=10.1186/s40359-025-02921-8|issn=2050-7283}}&lt;/ref&gt;. A study published in that same year reported that out of 261 nurses working in teaching hospitals, 91.6% were forced to allocate their finances to strictly &quot;general needs&quot;, while more than 50% looked into international opportunities for employment. Notably, the study reported an overall near &quot;twofold greater&quot; rate of depression, anxiety, and stress compared to previous studies on nurses in Sri Lanka&lt;ref&gt;{{Cite journal|last=Senevirathne|first=C.P|last2=Senarathne|first2=L.|last3=Fernando|first3=M.|date=2024-04-01|title=Exploring the Association Between Behavioural Modification in Response to the Prevailing Economic Crisis and Mental Health Outcomes of Nurses from Teaching Hospitals, Sri Lanka|url=https://doi.org/10.1177/23779608241272679|journal=SAGE Open Nursing|language=EN|volume=10|pages=23779608241272679|doi=10.1177/23779608241272679|issn=2377-9608|pmc=11311183}}&lt;/ref&gt;. 

The detrimental effects the crisis has had on the mental health sector reveal a concerning area of underappreciation and under compensation towards a critical sector for the well-being of the country. Adequate staffing, increased funding, and an improved work-life balance should be emphasized for the workers of health sector of the country.  

== Present-Day Challenges ==

=== Ethnic tension ===
Despite the ending of the Sri Lankan civil war and the introduction of pluralist policies (such as the [https://srilankaembassy.fr/sites/default/files/files/media/pdf/NationalPolicy-English.pdf 2017 National Policy on Reconciliation and Coexistence] under the Sirisena administration), tensions amongst members of the ethnic groups still persist. Evidence of these tensions was found in a 2022 study conducted in the Ratnapura district, where religious leaders expressed skepticism through semi-structured interviews on &quot;conflict transformation&quot;. A Tamil citizen of the Ratnapura community recounted that they were forced to &quot;hide in jungles&quot; and consume &quot;dirty water in drainage[s]&quot; due to scarcity of food and drinkable water as a result of the conflict. In certain personal accounts, ethnic conflicts appear to affect the social behavior and identity of the majority ethnic group. One Sinhala participant recounted his objection to the war-time retaliatory destruction of a shop run by a Tamil shopkeeper was met with interrogative questions about &quot;whether [he was] Sinhalese or not&quot;. Both accounts convey interethnic tensions stemming from decade-long conflicts&lt;ref&gt;Jayathilaka, Aruna &amp; Gamage, Sayuri. (2024). Role of Buddhist and Hindu Religious Leaders Role of Buddhist and Hindu Religious Leaders in the Post-War Conflict Transformation Process: A Study Based on Rathnapura District in Srilanka. ''Retrieved from'' https://gandhimargjournal.org/wp-content/uploads/2024/09/Volume-46-Issue-1-April-June-2024.pdf#page=66&lt;/ref&gt;. 

Beyond individual accounts and the official end of the civil war, the minority groups in the country continue to feel ostracized. The Sri Lankan Tamil population remains dissatisfied with the Sri Lankan government due to their alleged lack of accountability of perpetrators of war crimes and lack of information on the whereabouts of [[w:Enforced_disappearances_in_Sri_Lanka|thousands of enforced disappearances]] that took place from the 1980s. Additionally, rising anti-Muslim sentiment in recent years has contributed to increased ethnic tensions, a stark contrast to the previous centuries of peaceful co-existence between the groups. 
[[File:Bodu Bala Sena symbol.svg|thumb|The symbol for Bodu Bala Sena, a nationalistic Sinhala Buddhist group criticized for catalyzing ethnic tensions in Sri Lanka.]]
Laws passed by the Sri Lankan government, such as the [[w:Prevention_of_Terrorism_Act_(Sri_Lanka)|Prevention of Terrorism Act]] and [[wikipedia:Anti-conversion_law#Sri_Lanka|anti-conversion laws]], have forced the United States Commission on International Religious Freedom to label Sri Lanka as a nation that &quot;[engages] or [tolerates] severe violations of religious freedom&quot; in their 2024 report. The government has been criticized by human rights organizations for &quot;disproportionately targeting religious minorities&quot;&lt;ref&gt;{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}&lt;/ref&gt;. Additionally, the implementation of the three dominant languages, English, Sinhala, and Tamil, across formal education and government services have been lackadaisical, narrowing opportunities of foundational social interactions between the groups. Persistent discrimination and prejudice towards minority groups can lead to an array of complex and self-deprecating mental health issues.   

Efforts to mitigate ethnic tensions include strategies like [[w:Community-based_participatory_research|community-based participatory research]] (CBPR), task-sharing, and securing online mental health services in order to expand mental health services. However, the implementation of evidence-based plans has been met with difficulty due to inaccessibility, high costs, and shortages of adequately-trained personnel.  

Movements aiming for improved intra group and inter group coexistences, such as the Jaffna People’s Forum for Coexistence, should be emphasized on a systematic and multi-level basis, including but not limited to education, public sectors, and within communities. Pluralistic values are encouraged to be emphasized across both private and public schools to foster cultural sensitivity and tolerance. Measures should be taken against groups criticized for promoting sectarian hostility, such as the [[w:Bodu_Bala_Sena|Bodu Bala Sena]]. 

=== Poverty ===
It has been proven that poverty significantly increases the chances of developing mental illnesses. This is further amplified by possible discrimination&lt;ref&gt;{{Cite journal|last=Knifton|first=Lee|last2=Inglis|first2=Greig|date=2020-10|title=Poverty and mental health: policy, practice and research implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC7525587/|journal=BJPsych bulletin|volume=44|issue=5|pages=193–196|doi=10.1192/bjb.2020.78|issn=2056-4694|pmc=7525587|pmid=32744210}}&lt;/ref&gt;. Poverty also affects the ability for individuals with mental health concerns to receive the treatment they need. Due to the repercussions of the economic crisis, clients in Sri Lanka could not attend further counseling sessions&lt;ref name=&quot;:8&quot; /&gt;. Poverty from 2021 to 2022 [https://databankfiles.worldbank.org/public/ddpext_download/poverty/987B9C90-CB9F-4D93-AE8C-750588BF00QA/current/Global_POVEQ_LKA.pdf reportedly doubled], with future forecasts predicting the poverty line to &quot;remain above 25 percent&quot;. Suicide has been empirically linked to economic hardships in previous studies&lt;ref&gt;{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}&lt;/ref&gt;. A 2013 study done on suicidal patients in [[w:Batticaloa_Teaching_Hospital|Batticaloa Teaching Hospital]] revealed 76% of patients who attempted suicide were from rural areas while 15% were from urban areas&lt;ref&gt;{{Cite book|url=http://ir.lib.seu.ac.lk/handle/123456789/1457|title=The influence of common risk factors for the patient with attempted suicide hospitalized at the teaching hospital, Batticaloa|last=Kisokanth|first=G.|last2=Najeem|first2=M. M.|last3=Karunakaran|first3=K. E.|date=2014-08-02|publisher=South Eastern University of Sri Lanka, University Park, Oluvil #32360, Sri Lanka|isbn=978-955-627-053-2|language=en-US}}&lt;/ref&gt;. The Sri Lankan government should consider the economical impacts that poverty has on mental health and implement ways to aid poverty-stricken individuals with mental health concerns.   

=== Stigmas ===
Stigma consists of the &quot;combined effect of prejudice, ignorance and discrimination.&quot;&lt;ref name=&quot;:10&quot;&gt;{{Cite web|url=http://www.researchgate.net/publication/233990797_The_Stigma_of_Mental_Illness_in_Sri_Lanka_The_Perspectives_of_Community_Mental_Health_Workers|title=(PDF) The Stigma of Mental Illness in Sri Lanka: The Perspectives of Community Mental Health Workers|website=ResearchGate|language=en|access-date=2025-07-25}}&lt;/ref&gt;.

A 2012 interview consisting of nine participants (two doctors, three nurses, one occupational therapist, one development worker, and two volunteers) revealed a number of concerning societal viewpoints on individuals with mental health concerns. The interviews revealed that negative judgements were not only levied against the individual with the mental illness, but also the family. Families hid mentally ill family members from the public to avoid &quot;shame&quot; and possible hinderances in marriage proposals. Views that mentally ill individuals were &quot;violent&quot; served as the motivating factor behind socially isolating those with mental illness from their communities. Interviewees mentioned that individuals dealing with mental health challenges would be attacked with stones and called &quot;derogatory names.&quot; A lack of community awareness regarding mental health and negative portrayals of mentally ill individuals in media exacerbates stigmatization, though the researchers commented that the media was &quot;improving&quot; in their depiction of mental illness. Beliefs that illnesses are caused by &quot;spirits&quot; can be problematic for individuals dealing with mental health issues and suggests poor mental health awareness. Mental health workers themselves believed that they were being stigmatized, as mental health was reportedly not taken as seriously as physical health. Despite the intriguing perspectives provided, the small sample size and usage of snow sampling raise questionable concerns regarding the generalizability of the results&lt;ref name=&quot;:10&quot; /&gt;. 

Improving media portrayal of subjects concerning mental health and involving community members in interventions dealing with mental health issues are ways that could destigmatize mental health amongst communities in Sri Lanka. Tying collaborations between allopathic services and traditional healers instead of having these two services work individually could enhance engagement between traditional medicine and Western medicine.

=== Suicide Trends &amp; Risk Factors ===
Suicide is defined as &quot;the act of killing oneself deliberately, initiated and performed by the person concerned in the full knowledge or expectation of its fatal outcome&quot;&lt;ref name=&quot;:11&quot;&gt;{{Cite book|title=The neuroscience of suicidal behavior|last=Heeringen|first=Kees van|date=2018|publisher=Cambridge University Press|isbn=978-1-316-60290-4|series=Cambridge fundamentals of neuroscience in psychology|location=Cambridge, United Kingdom New York, NY, USA Port Melbourne, VIC, Australia New Delhi, India Singapore}}&lt;/ref&gt;. Although Sri Lanka has seen a significant reduction in suicide rates from the mid 1990s, largely stemming from its ban on extremely toxic pesticide products, suicide and self harm remains a significant issue. The suicide rate per 100,000 people increased from 14.0 in 2019 to [https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide 15.0 in 2022] (according to WHO). On average, 27 males per 100,000 males and 5 females per 100,000 females committed suicide in 2022&lt;ref&gt;{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}&lt;/ref&gt;. Hanging appears to be the most used method for suicide for both males and females, with studies revealing a steady increase in recent years&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Bandara|first=Piumee|last2=Wickrama|first2=Prabath|last3=Sivayokan|first3=Sambasivamoorthy|last4=Knipe|first4=Duleeka|last5=Rajapakse|first5=Thilini|date=2024-04-17|title=Reflections on the trends of suicide in Sri Lanka, 1997–2022: The need for continued vigilance|url=https://journals.plos.org/globalpublichealth/article?id=10.1371/journal.pgph.0003054|journal=PLOS Global Public Health|language=en|volume=4|issue=4|pages=e0003054|doi=10.1371/journal.pgph.0003054|issn=2767-3375|pmc=11023397|pmid=38630779}}&lt;/ref&gt;. 

From 2023 to 2024, a group of researchers from the [[w:Eastern_University,_Sri_Lanka|Eastern University in Sri Lanka]] assessed 828 patients admitted to the Teaching Hospital in [[w:Batticaloa,_Sri_Lanka|Batticaloa, Sri Lanka]] for attempted suicide. They concluded that suicide prevention programs should be attuned to younger people (ages 15 to 35 in the study), emphasize the importance of education and reducing unemployment, and increase social support in the Tamil community. Despite accounting for other factors that could lead to suicidal ideation (ie, poverty), the results from this study suffer in external validity as 90% of the patients were Tamil and over 50% were between 16 and 25 years. In addition, correlations between suicide and unemployment rates have been questioned, with [[w:Austerity|austerity]] being a more reliable indicator of suicide rates than unemployment rates&lt;ref name=&quot;:11&quot; /&gt;. Further comprehensive studies on risk factors relating to suicide should be studied to examine correlations between unemployment rates and austerity measures.  

The WHO suggests implementing evidence-based suicide prevention programs, such as [https://www.who.int/initiatives/live-life-initiative-for-suicide-prevention LIVE LIFE], to reduce the national suicide rate&lt;ref&gt;{{Cite web|url=https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide|title=World Suicide Prevention day 2024 “Changing the Narrative on Suicide”|website=www.who.int|language=en|access-date=2025-07-29}}&lt;/ref&gt;. Media depictions of suicidal methods, such as hanging, can lead to sensationalism and the media should be cautious of such displays in movies and TV shows&lt;ref name=&quot;:12&quot; /&gt;. Awareness of depression and other mental health issues can serve as a safeguard against suicidal ideation in Sri Lankan men and women.  

== Role of Religion ==
According to the last demographic report (2012), 70.2% of Sri Lankans are Buddhist, 12.6% are Hindus, 9.7% are Muslims, and 7.4% are Christians. The Theravada Buddhist community makes up the majority in several provinces throughout the country&lt;ref&gt;{{Cite web|url=https://www.state.gov/reports/2022-report-on-international-religious-freedom/sri-lanka/|title=Sri Lanka|website=United States Department of State|language=en-US|access-date=2025-08-07}}&lt;/ref&gt;. Religion, especially Theravada Buddhism, has had a significant influence on not only the historical treatment of mental health in the country, but also everyday life&lt;ref name=&quot;:15&quot; /&gt;. The [[w:Mahāvaṃsa|''Mahāvaṃsa'']] details hospitals treating patients suffering from mental health issues as early as the 4th century BC. Additionally, the 1700s Nayaka king [[w:Kirti_Sri_Rajasinha|Kirthi Sri Rajasinghe]] detailed the implementation of Buddhist philosophy in psychiatry&lt;ref name=&quot;:4&quot; /&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Alwis|first=L. A. P. De|date=2017-12-05|title=Development of civil commitment statutes (laws of involuntary detention and treatment) in Sri Lanka: a historical review|url=https://mljsl.sljol.info/articles/10.4038/mljsl.v5i1.7351|journal=Medico-Legal Journal of Sri Lanka|language=en|volume=5|issue=1|doi=10.4038/mljsl.v5i1.7351|issn=2012-8231}}&lt;/ref&gt;.

Modern-day empirical studies have attested to the usefulness of religion in mitigating stress and elevating mental health&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-94-007-4276-5_22|title=Religion and Mental Health|last=Schieman|first=Scott|last2=Bierman|first2=Alex|last3=Ellison|first3=Christopher G.|date=2013|publisher=Springer Netherlands|isbn=978-94-007-4276-5|editor-last=Aneshensel|editor-first=Carol S.|location=Dordrecht|pages=457–478|language=en|doi=10.1007/978-94-007-4276-5_22|editor-last2=Phelan|editor-first2=Jo C.|editor-last3=Bierman|editor-first3=Alex}}&lt;/ref&gt;. Religion has been found to be positively correlated with improved mental health, and more religious patients were concluded to have &quot;better mental health and adapt[ed] more quickly to health problems&quot; versus patients who weren't religious&lt;ref&gt;{{Cite journal|last=Koenig|first=Harold G.|date=2012|title=Religion, spirituality, and health: the research and clinical implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3671693/|journal=ISRN psychiatry|volume=2012|pages=278730|doi=10.5402/2012/278730|issn=2090-7966|pmc=3671693|pmid=23762764}}&lt;/ref&gt;. [https://www.researchgate.net/scientific-contributions/T-N-Wickramarathna-2247724082 Dr. Wickramarathna] of the University Psychiatry Unit (UPU) at the National Hospital of Sri Lanka (NHSL) argues that psychiatrists must strive for a balance in their approach to patients and &quot;make positive use of religion in [their] practice[s]&quot;&lt;ref&gt;{{Cite journal|last=Wickramarathna|first=T. N.|date=2022-12-31|title=Psychiatrists should stand far from the shrine: why and why not we should separate religion from psychiatry|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v13i2.8397|journal=Sri Lanka Journal of Psychiatry|language=en|volume=13|issue=2|doi=10.4038/sljpsyc.v13i2.8397|issn=2012-6883}}&lt;/ref&gt;.

=== Buddhism ===
27 Sinhalese Buddhists from four Buddhist temples were selected for a series of 70-minute interviews and focus group discussions with the aim of learning the Sinhala Buddhist understanding and experience of spiritual well-being and psychological well-being. The interviewees held spiritual wellness to be the &quot;center&quot; of overall wellness, the &quot;precondition for a successful life&quot;&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/articles/10.4038/sljss.v44i1.7990|journal=Sri Lanka Journal of Social Sciences|language=en-US|volume=44|issue=1|doi=10.4038/sljss.v44i1.7990|issn=0258-9710}}&lt;/ref&gt;. Sinhala Buddhists believe that wellness cannot be achieved without spiritual tranquility. The report states that participants emphasized that spirituality &quot;cannot be directly intervened&quot; and can only be seen through &quot;[interactions] with society&quot;&lt;ref name=&quot;:14&quot; /&gt;. Despite the ''athmaya'' (soul) being &quot;unreachable&quot;, it can be &quot;intervened&quot;, or treated, through the actions of the mind and body with society&lt;ref name=&quot;:14&quot; /&gt;. One being &quot;psychologically ill&quot; can affect one's spiritual being, as the participants reported in their interviews, and can be impaired through &quot;lifestyle stressors, environmental and socio-cultural causes, non-human related causes and bad-karma in the past lives&quot;&lt;ref name=&quot;:14&quot; /&gt;.  

The researchers concluded that despite Sinhala Buddhists not being able to articulately decipher the discrepancies between psychological well-being and spiritual well-being, they are able to conceptualize and maintain a culturally embedded understanding between the two, serving as reputable evidence of the integration of mental health in Sinhala Buddhist practices. However, it is important to note that these results come from a very small sample size and cannot be generalized to all Sri Lankan Buddhists.   

In addition, a 2009 study found that a belief in karma was correlated with poor health. However, an earlier study found a positive correlation between the reliance on the [[w:Karma_in_Buddhism|Buddhist concept of karma]] and trauma, inferencing Buddhist karma being a prevalent response to trauma&lt;ref&gt;{{Cite journal|last=Levy|first=Becca R.|last2=Slade|first2=Martin D.|last3=Ranasinghe|first3=Padmini|date=2009-03|title=Causal thinking after a tsunami wave: karma beliefs, pessimistic explanatory style and health among Sri Lankan survivors|url=https://pubmed.ncbi.nlm.nih.gov/19229624|journal=Journal of Religion and Health|volume=48|issue=1|pages=38–45|doi=10.1007/s10943-008-9162-5|issn=1573-6571|pmid=19229624}}&lt;/ref&gt;. Overall, the effectiveness of karma as a coping mechanism appears to be conflicted.   

Studies indicate that other practices of Buddhism seem to be utilized by individuals affected by the war. 40% of Sri Lankan Buddhists affected by the 2004 tsunami found the Buddhist ritual ''Bodhipuja'' to be helpful in dealing with traumatic experiences&lt;ref&gt;{{Cite web|url=https://jmvh.org/article/mental-health-and-the-role-of-cultural-and-religious-support-in-the-assistance-of-disabled-veterans-in-sri-lanka/|title=Mental Health and the Role of Cultural and Religious Support in the Assistance of Disabled Veterans in Sri Lanka|website=JMVH|language=en-US|access-date=2025-08-12}}&lt;/ref&gt;.    

=== Catholicism ===
Catholic counseling refers to &quot;a nuanced and holistic mental health care paradigm that intricately weaves together psychological science with the moral, spiritual, and pastoral traditions of the Catholic Church&quot;&lt;ref name=&quot;:13&quot;&gt;Perera, U. [https://www.researchgate.net/profile/Udeshini-Perera/publication/394095042_Catholic_Counselling_in_Sri_Lanka_Integrating_Faith_Psychology_and_Cultural_Healing/links/6889303af8031739e6098c79/Catholic-Counselling-in-Sri-Lanka-Integrating-Faith-Psychology-and-Cultural-Healing.pdf Catholic Counselling in Sri Lanka: Integrating Faith, Psychology, and Cultural Healing]. July 2025.&lt;/ref&gt; and aims to assimilate Catholic theology and evidence-based psychological treatment while including Sri Lankan cultural elements. This is achieved through emphasis on community cohesion and a locally-based understanding of &quot;personhood&quot;&lt;ref name=&quot;:13&quot; /&gt;. 

The origins of Catholic counseling trace back to the introduction of Roman Catholicism to the island in the 1600s, with the focus of the early Sri Lankan Catholic community being on &quot;[[w:Evangelism|evangelization]], education, and sacramental formation&quot;. Demand for counseling services in general increased due to the impacts of the Sri Lankan Civil War, where Catholic organizations (Caritas Sri Lanka, Seth Sarana, Subodhi Integral Centre (Piliyandala), etc.) established several Catholic-based trauma-informed programmes for victims of the Civil War. Programmes use group therapy, forgiveness rituals, and narrative repairs to alleviate war trauma. 

Examples of integration of Catholic virtues and counseling can be seen in [[w:Cognitive_Behavioral_Therapy|Cognitive Behavioral Therapy]] (CBT), where &quot;hope&quot; and &quot;humility&quot; are used as the frameworks for creating spiritual resilience&lt;ref name=&quot;:13&quot; /&gt;. The general Christian call for &quot;agape love and acceptance&quot; is echoed by the concept of [[w:Unconditional_positive_regard|unconditional positive regard]]. ''[[w:Lectio_Divina|Lectio Divina]]'' (Catholic prayer and meditation) and ''Marian devotions'' are integrated into therapeutic practices to achieve emotional regulation and mindfulness.   

Senior Lecturer [https://www.researchgate.net/profile/Udeshini-Perera Udeshini Perera] of the University of Colombo articulates a critical role of Catholic counseling. She claims that secular counseling fails to address the &quot;spiritual roots of distress and moral confusion&quot;. Catholic counseling fills in this gap by integrating &quot;psychological insights with a transcendent orientation, supporting lasting transformation and integrity&quot;&lt;ref name=&quot;:13&quot; /&gt;.  

As of 2025, no formal accreditation or standardized training exists for [[w:Pastoral_counseling|pastoral counselors]] in Sri Lanka, hampering the legitimacy of Catholic counseling. Udeshini Perera remarks that mental health stigma, lack of standardized training, research regarding Catholic counseling effectiveness, and acceptance of the combination of religion and science in a professional setting present challenges for Catholic pastoral counseling in the country. Additionally, Catholic psychiatry in Sri Lanka appears to be under-researched, and evidence of its empirical effects on followers appears sparse. Further research is needed in assessing the empirical effects of Catholic counseling in Sri Lanka.  

=== Islam ===
The literature on the empirical effects of Islamic-based psychotherapy in Sri Lanka is limited. Research is limited to a 2012 case study of a 21-year-old Muslim woman experiencing episodic possession states. The patient ceased attending psychiatric services and opted for religious rituals. The patient reported, in a follow-up visit, that the possession states had been absent for 3 months since her switch to religious rituals. The woman and her family attributed the apparent improvement of her condition to religious rituals&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|last3=Yoosuf|first3=Alam|last4=Karunaratne|first4=Sanjeewani|last5=de Silva|first5=Pushpa|date=2012|title=Religious Beliefs, Possession States, and Spirits: Three Case Studies from Sri Lanka|url=http://www.hindawi.com/journals/crips/2012/232740/|journal=Case Reports in Psychiatry|language=en|volume=2012|pages=1–3|doi=10.1155/2012/232740|issn=2090-682X|pmc=3437272|pmid=22970398}}&lt;/ref&gt;.

Future recommendations would be to conduct research on the foundations of Islamic psychiatry in the country, and to observe the rituals implemented and their effects on patients. Studies have found that Islamic prayer can be an effective means of &quot;support and coping&quot;&lt;ref name=&quot;:15&quot; /&gt;. Seven world-wide case studies using Islamic-based psychotherapy on patients, consisting of religious rituals such as scriptural reading from the [[w:Quran|Quran]], teaching of fundamental Islamic concepts (such as ''[[w:Tawakkul|tawakkul]]''), and active implementation of contemplation (''[[w:Tadabbur|tadabbur]]''), have reported positive effects in decreasing cognitive and emotional symptoms associated with &quot;religious, obsessive-compulsive disorder, depression, agoraphobia, generalized anxiety disorder, grief, and substance use disorder.”&lt;ref&gt;{{Cite journal|last=Kurhade|first=Chhaya Shantaram|last2=Jagannathan|first2=Aarti|last3=Varambally|first3=Shivarama|last4=Shivanna|first4=Sushrutha|date=2022-01|title=Religion-based interventions for mental health disorders: A systematic review|url=https://journals.lww.com/10.4103/ijoyppp.ijoyppp_14_21|journal=Journal of Applied Consciousness Studies|language=en|volume=10|issue=1|pages=20–33|doi=10.4103/ijoyppp.ijoyppp_14_21|issn=2949-6993}}&lt;/ref&gt; Additionally, a community-based study of elderly patients in Bangalore, India receiving Islamic-based psychotherapy observed decreased exhibitions of sleep disorders, eating disorders, and emotional distress&lt;ref&gt;{{Cite journal|last=Hafeez|first=Nimin|last2=Sanjay|first2=Thittamaranahalli Varadappa|last3=Puthussery|first3=Yannick Poulose|last4=Madhusudan|first4=Muralidhar|last5=Kariyappa|first5=Poornima Muddaiah|last6=Kulkarni|first6=Sridevi|last7=Raj|first7=Lavanya|date=2023-12-31|title=Spiritual practices among elderly, prevalence, pattern and associated factors: a community-based study from rural Bengaluru, India|url=https://jccpsl.sljol.info/articles/10.4038/jccpsl.v29i4.8610|journal=Journal of the College of Community Physicians of Sri Lanka|language=en|volume=29|issue=4|doi=10.4038/jccpsl.v29i4.8610|issn=1391-3174}}&lt;/ref&gt;. 

=== Hinduism ===
Despite Hindus being 12.6% of the population of Sri Lanka, the research on Hinduism-based therapy in the country is limited. Ayurvedic medicine, a form of medicine originating from ancient India, predominated the Sri Lankan medical landscape for over 2,000 years and even had a symbiotic relationship with Sinhalese medicine, which also played a significant and influential role in the country's medical framework&lt;ref name=&quot;:0&quot; /&gt;&lt;ref&gt;{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/article/10.4038/sljss.v44i1.7990/|journal=Sri Lanka Journal of Social Sciences|volume=44|issue=1|pages=33|doi=10.4038/sljss.v44i1.7990|issn=2478-1169}}&lt;/ref&gt;. Despite its historical dominance, Ayurvedic medicine has been challenged against modern evidence-based medical standards&lt;ref&gt;{{Cite book|url=https://philarchive.org/rec/DOMAAT|title=Ayurveda: Ancient Tradition or Pseudoscientific Practice? A Philosophical Inquiry|last=Dominic|first=Shubham K.}}&lt;/ref&gt;.

=== Comparative synthesis ===
Taking an overarching review of the role of religion in Sri Lanka, methods to improve mental well-being are practiced by adherents of Buddhism, Hinduism, Islam, and Christianity. These practices are implemented in traditionally-oriented mental health care, which has been reportedly preferred over psychiatric care at times. These rituals practiced across these religions indicate a common theme of psychologically integrated aspects of well-being. Interpretation of trauma is a central use in religion, with religious principles, such as karma and ''tawakkul'', serve as psychologically analogous mechanisms during times of distress.  

In terms of methodological comparisons to the studies described, qualitative interviews have documented Buddhist practices and principles, like Bodhipuja and the belief in karma, in response to traumatic events, while case studies found religious practices by other religious groups, such as a Muslim patient reading Islamic scripture and observing prayer, to reduce emotional distress. Peer-reviewed sources have documented Catholic practices and principles, such as ''Lectio Divina'' and unconditional positive regard, in improving mindfulness and emotional regulation. The paper acknowledges limitations in the evaluation of certain findings, such as in Islam and Hinduism. These shortcomings, however, are a reflection of the existing literature and its deficiencies. Empirical findings indicate mental health practices are complex and are multifaceted in their effects. 

Evidently, religion serves a parallel role to psychiatric services in improving mental health. Despite its perceived benefits, the findings surrounding religions' role in mental health suffer from conflicting, and sometimes contradictory, results. Additionally, a disproportionate amount of empirical findings seem to be Buddhist-predominant, while other religions are underrepresented in the research. Regarding research barriers, the methodological approaches implemented to study the practices of religious followers vary, though much of the research was brought from qualitative or case-based studies, impeding generalizability. Another noteworthy issue is that many studies do not utilize standardized, psychiatric measures.

== Future Outlook ==
Despite significant changes to the mental health environment in Sri Lanka, the current legal framework shaping mental health in the country has not been updated since 1956. A Cambridge University Press article detailed many limitations of the Mental Disease Ordinance of 1956, including discrepancies between the legal provisions of involuntary admissions and modern practices, potential exposure to trauma through extra-legal detentions of the mentally ill, and an absence of legal guidelines addressing the restraint of violent patients&lt;ref name=&quot;:6&quot; /&gt;. Participants from Sri Lanka reported in a comparative legislative questionnaire that they felt the mental health laws were &quot;outdated&quot; and descriptions of clinical roles remained ambiguous&lt;ref name=&quot;:16&quot; /&gt;. A draft mental health legislation from 2007 included provisions for human rights, but due to &quot;bureaucratic processes&quot; and a &quot;lack of consensus&quot;, the draft has not been officially approved.

These limitations pose challenges to the standardization of mental healthcare admissions and may impact the rights of detained patients. Detained patients may have their human rights violated due to a lack of updated legal framework, thereby impeding the identification of such violations. Additionally, with the lack of clarity on clinical roles, clinical responsibilities may not be routinely recognized and observed, leading to role confusion and potential legal ramifications&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|last=Dey|first=Sangeeta|last2=Mellsop|first2=Graham|last3=Diesfeld|first3=Kate|last4=Dharmawardene|first4=Vajira|last5=Mendis|first5=Susitha|last6=Chaudhuri|first6=Sreemanti|last7=Deb|first7=Aniruddha|last8=Huq|first8=Nafisa|last9=Ahmed|first9=Helal Uddin|date=2019-10-24|title=Comparing legislation for involuntary admission and treatment of mental illness in four South Asian countries|url=https://ijmhs.biomedcentral.com/articles/10.1186/s13033-019-0322-7|journal=International Journal of Mental Health Systems|volume=13|issue=1|pages=67|doi=10.1186/s13033-019-0322-7|issn=1752-4458|pmc=6813093|pmid=31666805}}&lt;/ref&gt;. Lastly, current efforts should ideally move beyond just addressing poverty-centered matters, and expand efforts to domestic violence victims and children with disabilities, as shelters and specialized services are limited&lt;ref name=&quot;:82&quot;&gt;{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}&lt;/ref&gt;.

Stagnation in policy development leaves Sri Lanka without a practical, up-to-date, and comprehensive mental health framework, which could put both clinicians and patients at risk. Future reforms should include clarification on the treatment and detention process of involuntary admissions of patients and a clear delineation of clinical roles and their responsibilities. Without the necessary reforms to advance Sri Lankan mental health legislation, clinicians and vulnerable patients may suffer from a lack of comprehensive oversight. 

==Additional information==

===Acknowledgements===
Any people, organisations, or funding sources that you would like to thank.

===Competing interests===
No competing interests.

===Ethics statement===
An ethics statement, if appropriate, on any animal or human research performed should be included here or in the methods section.

==References==
{{reflist|35em}}

[[Category:Mental health]]
[[Category:Sri Lanka]]</text>
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| journal = WikiJournal of Medicine      &lt;!-- WikiJournal of Medicine, Science, or Humanities --&gt;
| last1 = Azeez
| orcid1 = 0009-0007-9202-4614
| first1 = Aaqib
| last2 = 
| first2 = 
| last3 = 
| first3 = 
| last4 = 
| first4 = &lt;!-- up to 9 authors can be added in this above format --&gt;
| et_al = &lt;!-- if there are &gt;9 authors, hyperlink to the list here --&gt;
| affiliation1 = Old Dominion University
| correspondence1 = aaqib.azeez@yahoo.com
| affiliations = institutes / affiliations
| correspondence = email@address.com
| keywords = &lt;!-- up to 6 keywords --&gt;
| license = &lt;!-- default is CC-BY --&gt;
| abstract = Mental health issues continue to be a significant problem in Sri Lanka, with 2022 suicide rates in the country reporting 15 suicides per 100,000 people, above the global average of 10.5 suicides per 100,000 people. The barriers to mental healthcare on the island are multi-faceted and are best understood with historical context. This narrative review covers the historical developments of mental healthcare,  mental health impacts of historical events within the last 100 years, current challenges affecting mental health outcomes, the role of the island's major religions in mental health and mental healthcare, and recommendations for improving future mental healthcare. The author uses peer-reviewed journal articles, relevant books, historical documents, and governmental/non-governmental reports to support clinical and historical claims, though non-peer-reviewed sources were used to contextualize historical and non-clinical claims. The narrative review concludes that outdated legislation, impacts from recent conflicts or disasters, stigma surrounding mental health, and economic vulnerability contribute to mental health issues and the inefficiency of mental healthcare services. The author recommends updating legal frameworks, expanding services, and raising awareness to mitigate social stigma.
}}

== Introduction ==
Mental health continues to be a critically relevant topic as the island nation has experienced decades of [[w:Black_July|violent ethnic conflict]], terrorist attacks, alleged war crimes, and economic disruptions. Sri Lanka continues to recover from a [[w:Sri_Lankan_economic_crisis_(2019–2024)|severe economic crisis (2019 - 2024)]], a [[w:Sri_Lankan_civil_war|nearly 30-year civil war ending in 2009]], a [[w:2019_Sri_Lanka_Easter_bombings|2019 terrorist attack]], and the [[w:2004_Boxing_Day_tsunami|2004 Boxing Day tsunami]]. The exact effect these major events have had on mental health in the country is &quot;unknown&quot;, but the statistics remain concerning despite a declining trend in the overall suicide rate.

Suicide rates in the country during the mid-1990s were the second-highest in the world, with ingesting toxic products being the main suicide method. Despite the decline in suicide numbers since then—possibly attributed to Sri Lanka's ban on toxic products—evidence from a 2023 study reports an upward trend in suicide through hanging from 2016 to 2021—independent of the [[w:COVID-19_pandemic_in_Sri_Lanka|COVID-19 pandemic]]. Several risk factors for suicide, such as poverty and economic instability, are still prevalent and even increasing in the country&lt;ref&gt;{{Cite journal|last=Rajapakse|first=Thilini|last2=Silva|first2=Tharuka|last3=Hettiarachchi|first3=Nirosha Madhuwanthi|last4=Gunnell|first4=David|last5=Metcalfe|first5=Chris|last6=Spittal|first6=Matthew J.|last7=Knipe|first7=Duleeka|date=2023-01-19|title=The Impact of the COVID-19 Pandemic and Lockdowns on Self-Poisoning and Suicide in Sri Lanka: An Interrupted Time Series Analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC9914278/|journal=International Journal of Environmental Research and Public Health|volume=20|issue=3|pages=1833|doi=10.3390/ijerph20031833|issn=1660-4601|pmc=9914278|pmid=36767200}}&lt;/ref&gt;.

== Methods ==
A narrative review was conducted on mental health in Sri Lanka. Sources used included peer-reviewed journal articles, relevant books, historical documents, and governmental/non-governmental reports. These sources were found on Google Scholar, PubMed/PMC, Sri Lankan journals, and official Sri Lankan governmental websites showing relevant statistics/reports. Keywords used to conduct searches included, but were not limited to: &quot;Sri Lanka mental health&quot;, &quot;Sri Lanka civil war trauma&quot;, &quot;Sri Lanka suicide&quot;, &quot;Sri Lanka mental health ordinances&quot;, &quot;Sri Lanka religion and mental health&quot;, &quot;Sri Lanka public mental healthcare&quot;, and &quot;Sri Lanka poverty/economic crisis mental health impact.&quot; Studies that were included were relevant to the topic (Sri Lanka, South Asian mental health law, suicide, public mental health, conflict/disaster trauma, or cultural/religious practice), had full text available, and were in the English language. Non-peer-reviewed sources were primarily used to explain historical claims or contextualize non-clinical claims. 

==Historical Development of Mental Health Services==

Records attest to the care of the mentally ill through established hospitals in the island since the 4th century.&lt;ref name=&quot;:17&quot; /&gt; Prior to the incarceration of the mentally ill by the European colonizing forces, the mentally ill were regarded as ''Pissowetitch'', or people who had &quot;the spirit of the Gods within him&quot; and &quot;whatsoever he pronounceth, is looked upon as spoken by God himself, and the people will speak to him, as if it were the very person of God&quot;&lt;ref&gt;{{Cite web|url=https://www.gutenberg.org/files/14346/14346-h/14346-h.htm|title=An Historical Relation Of the Island Ceylon, in the East-Indies: Together, With an Account of the Detaining in Captivity the Author and divers other Englishmen now Living there, and of the Author’s Miraculous Escape.|last=Knox|first=Robert|website=www.gutenberg.org|language=en-us|access-date=2026-06-29}}&lt;/ref&gt;. With this religious understanding, Lucien de Alwis reasoned that the mentally ill in Sri Lanka were &quot;placed... at a higher social status than the mentally ill in the Western world&quot;, with this understanding correlating with the unsurprising absence of evidence of any &quot;large scale segregation[s] of [the] mentally ill from society&quot;&lt;ref name=&quot;:17&quot; /&gt;. 

In the 1800s, established care for mental health began shifting primarily from indigenous practices, mainly derived from [[w:Ayurveda|Ayurveda medicine]], [[w:Siddha_medicine|Siddha medicine]], and [[w:Unani_medicine|Unani medicine]], to a Western model by the British&lt;ref name=&quot;:17&quot; /&gt;&lt;ref name=&quot;:0&quot;&gt;Gambheera, H. (2011). [https://www.saarcpsychiatry.com/viewText?chapter=c6 The evolution of psychiatric services in Sri Lanka]. South Asian Journal of Psychiatry, 2(1), 25–27.&lt;/ref&gt;&lt;ref name=&quot;:15&quot;&gt;{{Cite book|url=https://doi.org/10.1007/978-981-96-8078-8_7|title=Social Psychiatry in Sri Lanka|last=Baminiwatta|first=Anuradha|last2=Williams|first2=Shehan|date=2025|publisher=Springer Nature|isbn=978-981-96-8078-8|editor-last=Arafat|editor-first=S. M. Yasir|location=Singapore|pages=141–158|language=en|doi=10.1007/978-981-96-8078-8_7|editor-last2=Singh|editor-first2=Amit|editor-last3=Kar|editor-first3=Sujita Kumar}}&lt;/ref&gt;. 

=== Adoption of a Western-based mental healthcare model and ordinances ===
In 1839, [[w:James_Alexander_Stewart-Mackenzie|James Alexander Stewart-Mackenzie]], the 7th Governor of British Ceylon, released the Lunacy Ordinance, authorizing municipal authorities to create lunatic asylums for the mentally ill&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=article&amp;id=6&amp;Itemid=125&amp;lang=en|title=History - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-10}}&lt;/ref&gt;. The ordinance was concerned with the legal frameworks of detaining individuals considered dangerous to others or individuals falsely presenting themselves as mentally ill, and not on medical treatments to alleviate the conditions of detained individuals. UK psychiatrist [[w:Edward_Mapother|Edward Mapother]] critiqued the ordinance during his 1937 inspection of British Ceylon's mental health institutions in a series of reports titled ''A Disgrace to a Civilised Community'', remarking that the ordinance &quot;[did] not seem to have contemplated treatment as a contingency to be considered&quot;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|title=Permeable walls: historical perspectives on hospital and asylum visiting|date=2009|publisher=Rodopi|isbn=978-90-420-2599-8|editor-last=Mooney|editor-first=Graham|series=Clio medica|location=Amsterdam New York, NY|editor-last2=Reinarz|editor-first2=Jonathan}}&lt;/ref&gt;.

The 1839 Ordinance was repealed and replaced by the 1840 Ordinance, which removed two requirements from the previous Ordinance: the requirement for official medical diagnoses of the mentally ill and the mandate to maintain adequate staff-to-patient ratios within lunatic asylums&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Alwis|first=L. A. P. de|last2=Seneviratne|first2=V. L.|last3=Mendis|first3=T. S. S.|last4=Abhayanayaka|first4=C.|date=2024-12-31|title=The development of laws related to the disposal of forensic patients in Sri Lanka: A historical review|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v15i2.8569|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=15|issue=2|doi=10.4038/sljpsyc.v15i2.8569|issn=2012-6883}}&lt;/ref&gt;.

In 1873, a third Ordinance was released. It included linguistic changes, where the term, &quot;insane&quot;, was replaced with &quot;of unsound mind&quot;. The Ordinance also gave more power to medical professionals in determining insanity diagnoses, and more power to detainees in appealing their commitment to the mental asylum. Despite the increased granted authority, the legal frameworks behind the detainment of the criminally insane were left identical to previous ordinances&lt;ref name=&quot;:3&quot; /&gt;.

=== Development of mental asylums ===
At the time the 1839 ordinance was released, mentally ill patients were placed either in prisons throughout the country or leprosy hospitals, such as the [[w:Hendala_Leprosy_Hospital|Hendala Leprosy Hospital]] in the Gampaha district&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;. After the creation of the first mental asylum in Borella in 1846, patients from the Hendala Leprosy Hospital were transferred to Borella. Overcrowding soon became an issue, which led to patients being sent to prisons. [[File:Edward Mapother.jpg|thumb|A portrait taken of Edward Mapother during his time working at [[w:Maudsley_Hospital|Maudsley Hospital]] in London.
]]

As medical institutions were being made to house the mentally ill, another mental asylum was created in the [[w:Cinnamon_Gardens|Cinnamon Gardens]] area of Colombo in 1884, though this mental asylum faced overcrowding issues in just one year&lt;ref name=&quot;:0&quot; /&gt;. Treatment in these asylums was limited to occupational and protection therapy, failing to provide treatment for the root causes.

In 1926, the Angoda Mental Hospital was established, marginally alleviating the severe overcrowding issues that were plaguing the preceding mental asylums. Despite the addition of 1,700 beds to the facility, treatment was still vastly limited and the patients were left in significantly poor conditions. 

=== Edward Mapother and his 1937 inspection of British Ceylon ===
Edward Mapother was born in Dublin, Ireland, on July 12, 1881 and moved to London when he was 7 years old&lt;ref&gt;{{Cite book|title=Madness to mental illness: a history of the Royal College of Psychiatrists|last=Bewley|first=Thomas|date=2008|publisher=RCPsych Publications ; Distributed in North America by Balogh International|isbn=978-1-904671-35-0|location=London : [S.l.]}}&lt;/ref&gt;. Mapother attained his M.D. in 1908. While Mapother was the Medical Superintendent of Maudsley Hospital in London, England, he was invited to inspect British Ceylon's mental health institutions by Dr S. T. Gunasekara, the first Medical Director of British Ceylon&lt;ref name=&quot;:1&quot; /&gt;.

In Mapother's visit, he commented that the Angoda Mental Hospital had the atmosphere of &quot;a prison that is neglected and dilapidated&quot;&lt;ref name=&quot;:1&quot; /&gt;. Overcrowding was still a major issue, with the institute hosting 3,000 patients—more than double the intended capacity. Patients were sleeping on mats and were clearly out of reach of adequate treatment. Mapother also noted that only 4% of public health expenditure in the country was being set for hospitals, drawing a stark comparison to London's 25%&lt;ref name=&quot;:1&quot; /&gt;. Mapother offered a vivid and grim account of the hospital in his reports:
&lt;blockquote&gt;
The floor, roof and walls of each cell consist alike of drab cement without any attempt at colouring or decoration. High up in one wall is a small window with stout iron bars. In the floor is a large hole into which the patient may pass his motion and urine. These cells are incompletely divided from one another by a partition which does not reach the roof so that the noise and stink from any one cell may reach at least all the others of the same row. Into these empty cells I was informed that the most noisy and troublesome patients in the hospital; were turned at night completely naked. The doors of the cell contain no observation window, and considering the violent character of many of these patients there is every ground for believing that the doors are rarely opened in the night by the solitary attendant on duty. It needs little imagination to picture the suffering of any patient in an early stage of bodily illness passing a night under such conditions, a situation which must frequently arise. I am told that the noise proceeding from this building is like that on a bad night in a menagerie&lt;ref name=&quot;:0&quot; /&gt;.&lt;/blockquote&gt;Mapother proposed a series of reinforcements to the legal, institutional, and medical frameworks of mental health care in British Ceylon. This included the decentralization of the psychiatric services, a reworking of the Lunacy Ordinance to incorporate treatment into the legal framework, and the establishment of a separate service of medical professionals dedicated to psychiatry. Mapother's recommendations led to several of the best local medical professionals to be sent to London for extensive training in psychiatry, while nurses from England were sent to British Ceylon to supervise hospital operations and train local staff&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;.

On August 25, 1938, the Executive Committee of Health approved the strategies proposed by Mapother, though the Government was unable to fully implement all of Mapother's interventions due to the 'heavy cost'. In fact, the Government decided to forego one of his proposals at the beheast of the &quot;Visiting Committee&quot;, a committee that was tasked to &quot;meet at the hospital, carry out inspections, and make recommendations&quot; to the Executive Committee of Health&lt;ref name=&quot;:1&quot; /&gt;. The Government believed that deficiencies in their mental healthcare system could prove to be &quot;costly&quot; for their reputation, which enraged Maptoher. Mapother intended to contact the Secretary of State regarding the &quot;distortion&quot; of his plans, but was interrupted by events preceding [[w:World_War_II|World War II]]&lt;ref name=&quot;:1&quot; /&gt;. Mapother passed away on March 20, 1940, without materializing his follow-up plans.

=== Post-Mapother developments and further innovations ===
[[File:Sri Lanka districts Colombo.svg|thumb|A map of Sri Lanka highlighting the Colombo District, where the capital is located. 
|right|250px]]Mapother's insights on the mental healthcare structure in British Ceylon proved to be the catalyst of significant renovations. In 1939, the first outpatient clinic was established in the [[w:National_Hospital_of_Sri_Lanka|National Hospital of Sri Lanka]] in Colombo. The first trained Ceylonese psychiatrists began practice in the 1940s, leading to the establishment of the first neuropsychiatric clinic in Colombo in 1943. Treatments for the mentally ill improved dramatically, as [[w:insulin_shock_therapy|insulin shock therapy]] and [[w:Electroconvulsive_therapy|cardiazol convulsive therapy]] were utilized&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Kathriarachchi|first=Samudra T.|last2=Seneviratne|first2=V. Lakmi|last3=Amarakoon|first3=Luckshika|date=2019-06|title=Development of Mental Health Care in Sri Lanka: Lessons Learned|url=https://journals.lww.com/tpsy/fulltext/2019/33020/development_of_mental_health_care_in_sri_lanka_.1.aspx|journal=Taiwanese Journal of Psychiatry|language=en-US|volume=33|issue=2|pages=55|doi=10.4103/TPSY.TPSY_15_19|issn=1028-3684}}&lt;/ref&gt;. Mapother's advocation for the decentralization of services were further honored through the 1947 establishment of a first child guidance clinic in Colombo General Hospital&lt;ref name=&quot;:0&quot; /&gt;. 

In 1948, British Ceylon was granted independence after the [[w:Sri_Lankan_independence_movement|Sri Lankan independence movement]]. Changes in the mental healthcare structure were not immediate following independence, but rapid expansions of mental healthcare services were continuing to actualize.  

The following decades saw positive institutional developments, such as the creation of a second hospital in [[w:Mulleriyawa|Mulleriyawa]] in 1957, and the creation of a psychiatric inpatient unit in Colombo General Hospital in 1967—effectively granting the city of Colombo the luxury of hosting the top psychiatric care in the country&lt;ref name=&quot;:5&quot;&gt;{{Cite book|url=http://link.springer.com/10.1007/978-1-4899-7999-5_4|title=Mental Health System Development in Sri Lanka|last=Minas|first=Harry|last2=Mendis|first2=Jayan|last3=Hall|first3=Teresa|date=2017|publisher=Springer US|isbn=978-1-4899-7997-1|editor-last=Minas|editor-first=Harry|location=Boston, MA|pages=59–77|language=en|doi=10.1007/978-1-4899-7999-5_4|editor-last2=Lewis|editor-first2=Milton}}&lt;/ref&gt;. The 1950s was also the start of psychopharmacological innovations, with the introduction of [[w:Lithium_(medication)|lithium]] and long-acting injectable antipsychotics ([[w:Depot_injection|depot]] [[w:Antipsychotic|neuroleptics]]) in the succeeding years&lt;ref name=&quot;:4&quot; /&gt;. Additionally, the number of public psychiatrist positions increased by 400% from 1953 to 1967&lt;ref name=&quot;:5&quot; /&gt;.

After 1960, mental health services were expanded from beyond the capital to other cities in the country&lt;ref name=&quot;:2&quot; /&gt;.

In 1980, the [[w:Postgraduate_Institute_of_Medicine|Postgraduate Institute of Medicine]] initiated a program where students would enroll in a 5-year medical course and attain an MD in psychiatry, curbing the need for Sri Lankan medical students to be sent abroad to complete their training. Many of the medical students sent abroad for training never returned to Sri Lanka to practice, resulting in a &quot;1:500,000 to 1000,000&quot; ratio of psychiatrists to patients on &quot;most occasions&quot;&lt;ref name=&quot;:0&quot; /&gt;.

=== Mental Disease Ordinance of 1956 ===
In 1956, the 1873 Ordinance was revised a second time. The Mental Disease Ordinance of 1956 featured another linguistic development, as &quot;lunacy&quot; was replaced with &quot;mental disease&quot;&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|last=Hapangama|first=Aruni|last2=Mendis|first2=Jayan|last3=Kuruppuarachchi|first3=K. a. L. A.|date=2023-02|title=Why are we still living in the past? Sri Lanka needs urgent and timely reforms of its archaic mental health laws|url=https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4|journal=BJPsych International|language=en|volume=20|issue=1|pages=4–6|doi=10.1192/bji.2022.26|issn=2056-4740|pmc=9909436|pmid=36812028}}&lt;/ref&gt;. The Ordinance paved way for community-based services to be delivered to patients closer to their residences, rather than strictly allocating services to just hospitals. This led to the creation of a [[w:WHO|WHO]]-backed community clinic near the [[w:University_of_Colombo|University of Colombo]] in the 1970s, where the focus was to eventually ease patients in the Angoda Mental Hospital back into the general population&lt;ref name=&quot;:5&quot; /&gt;.

=== Developments from the 1990s ===
The 1990s and onwards saw further positive developments in framing the mental healthcare system, including the establishment of the [https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=featured&amp;Itemid=101&amp;lang=en Directorate of Mental Health] in 1998. The Directorate of Mental Health is a part of the [[w:Ministry_of_Health_(Sri_Lanka)|Ministry of Health]] and is responsible for the monitoring and implementation of mental health programs across the country&lt;ref&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?lang=en|title=Home - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}&lt;/ref&gt;.  As of 2025, the current director of the Directorate of Mental Health is Dr. Chithramalee de Silva&lt;ref name=&quot;:2&quot; /&gt;.

On November 11, 2005, the Mental Health Policy was approved by the Government of Sri Lanka, advocating for establishments of more de-centralized, community-based mental health services across the country. The policy aimed to concisely define the rigorous standards needed to be met for each respected medical professional, including psychiatrists and clinical psychologists&lt;ref&gt;{{Cite journal|last=Rajapakshe|first=Onali Bimalka Wickramaseckara|last2=Mohan|first2=Mohapradeep|last3=Singh|first3=Swaran Preet|date=2023-05|title=Development of adolescent mental health services in Sri Lanka|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC10895478/|journal=BJPsych international|volume=20|issue=2|pages=41–43|doi=10.1192/bji.2022.32|issn=2056-4740|pmc=10895478|pmid=38414998}}&lt;/ref&gt;. The policy also included a new position, the &quot;Medical Officer of Mental Health&quot;, tasked with overseeing and assisting in creating community-based mental health services&lt;ref name=&quot;:0&quot; /&gt;. In the same year, the Sri Lankan government began implementing psychological services in state institutions, such as the military&lt;ref name=&quot;:8&quot; /&gt;.  

In 2007, the National Mental Health Advisory Council (NMHAC) was created to serve as an 'advisory' board for the Ministry of Health&lt;ref name=&quot;:7&quot;&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=article&amp;id=9&amp;Itemid=220&amp;lang=en|title=Introduction - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}&lt;/ref&gt;. 

In 2008, the Angoda Mental Hospital was restructured and renamed as the National Institute of Mental Health (NIMH)&lt;ref name=&quot;:7&quot; /&gt;. 

=== Modern-day Sri Lanka ===
[[File:Feeding Children in Sri Lanka.jpg|left|thumb|Despite the noteworthy improvements in mental healthcare services in recent decades, mental health remains a significant issue due to rising poverty. ]]
As of 2025, the Mental Health Act (mental health legislation) has been undergoing development since 2005 and is currently awaiting to be considered for the final stage of approval. This is expected to replace the 1956 Mental Health Ordinance&lt;ref name=&quot;:7&quot; /&gt;.

Currently, there are 7 tertiary care hospitals, 61 adult patient units, 3 child inpatient units, and 1 forensic unit with over 100 psychiatrists all throughout the 22 districts&lt;ref name=&quot;:4&quot; /&gt;. The [[w:Lady_Ridgeway_Hospital_for_Children|Lady Ridgeway Hospital]] in Colombo and the Sirimavo Bandaranayke Specialized Children Hospital in Kandy are specialized in treating children with [[w:Learning_disability|SLD]], [[w:ADHD|ADHD]], [[w:Autism_Spectrum_Disorder|ASD]], and provides family support for patients. As of 2017, 22 rehabilitation centers exist through the country, including 7 alcohol rehab centers&lt;ref name=&quot;:7&quot; /&gt;.   

Despite the impressive advancements in mental healthcare in the last couple of decades, Sri Lanka still suffers significant mental health issues due to increasing poverty levels in the country. The [[w:World_Bank|World Bank]] reported that [https://www.wsws.org/en/articles/2024/04/08/eesc-a08.html the poverty levels in Sri Lanka increased from 11% in 2019 to 26% in 2024], with 60% of Sri Lankan households facing &quot;decreased incomes&quot;&lt;ref&gt;Lakhtakia, Shruti, Atapattu Mudiyanselage, Udahiruni Shashadari Atapat, Walker, Richard Ancrum. ''Sri Lanka Development Update - Bridge to Recovery (English).'' Washington, D.C.: World Bank Group. &lt;nowiki&gt;http://documents.worldbank.org/curated/en/099634104012434919&lt;/nowiki&gt;&lt;/ref&gt;. This was exacerbated by Sri Lanka's excessive foreign debt, economic troubles stemming from [[w:Gotabaya_Rajapaksa|Gotabaya Rajapaksa]]'s presidential term, the COVID-19 pandemic, and the [[w:Russian_invasion_of_Ukraine|ongoing invasion of Ukraine by Russia (2022)]].  

According to [[w:NYU|New York University]] graduate student [https://gc-cuny.academia.edu/NadiaAugustyniak Nadia Augustyniak] in her 2025 overview of Sri Lanka's public mental healthcare system, poverty-induced financial precarity remains a major obstacle to receiving access to mental healthcare services. Even though trauma from adverse weather and conflict is deleterious to mental health, issues originating from every-day struggles, especially struggles related to poverty, could arguably play a more significant role&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}&lt;/ref&gt;.  

== Impact of Conflicts, Terrorism, Political Instability &amp; Natural Disasters ==

=== Sri Lankan Civil War ===
The '''Sri Lankan Civil War''' was a domestic conflict between the Sri Lankan government and the Liberation Tigers of Tamil Eelam (abbreviated as the ''LTTE),'' a militant group formed in the 1970s as a byproduct of rising tensions between the majority Sinhalese and minority Tamil population. The group is considered a terrorist organization&lt;ref&gt;{{Cite web|url=https://www.start.umd.edu/baad/database/liberation-tigers-tamil-eelam-ltte-1998.html|title=BAAD - Liberation Tigers of Tamil Eelam (LTTE) - 1998 {{!}} START.umd.edu|website=www.start.umd.edu|access-date=2025-06-09}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.cfr.org/backgrounder/liberation-tigers-tamil-eelam-aka-tamil-tigers-sri-lanka-separatists|title=Liberation Tigers of Tamil Eelam (aka Tamil Tigers) (Sri Lanka, separatists) {{!}} Council on Foreign Relations|last=Bhattacharji|first=Preeti|website=www.cfr.org|language=en|access-date=2025-06-09}}&lt;/ref&gt;. The LTTE conducted decades of massacres, assassinations of political figures, and suicide bombings to achieve ''[[w:Tamil_Eelam|Tamil Eelam]],'' leading to civilian displacement, infrastructure collapse, and the reduction of mental health services available in the northern region.[[File:DFID-funded, UNHCR emergency shelter tents, in the IDP camp at Menik Farm, Sri Lanka (3694081492).jpg|thumb|350x350px|An IDP camp in Menik Farm, Sri Lanka in 2009 ([https://www.bbc.com/news/world-asia-19703826 now closed]). Suicide rates in IDP camps were three times the general population.]]The civil war mainly affected the northeastern portion of the country, including the [[w:Vanni_(Sri_Lanka)|Vanni region]]. The conflict caused mass destruction to local mental healthcare facilities. Local residents described the conflict as ''varthayal varnicca mudiyathavai'', roughly translating into English as 'beyond description by words'&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|language=en|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}&lt;/ref&gt;. In 2003, only two psychiatrists were found in the region, operating on extremely limited resources. This furthered long-term trauma and mental health deterioration in the population&lt;ref name=&quot;:5&quot; /&gt;. 

In 2002, the humanitarian organization [https://www.msf.org/ Médecins Sans Frontières] (MSF) conducted an investigation on mental health needs in the [[w:Vavuniya|Vavuniya]] area, the site of intense conflict during the civil war (including the [[w:1985_Vavuniya_massacre|1985 Vavuniya massacre]]), and found that many of the residents suffered from high suicide rates, alcohol abuse, domestic violence, grief, and a &quot;sense of ‘learnt helplessness’&quot;&lt;ref name=&quot;:5&quot; /&gt;. A team from the University of Konstanz in Germany found that 92% of grade school children in the region were exposed to &quot;combat, shelling, and witnessing the death of loved ones&quot;&lt;ref name=&quot;:9&quot; /&gt;. 
[[File:Tractors. Jan 2009 displacement in the Vanni.jpg|left|thumb|350x350px|Displaced civilians evacuating from the Kilinochchi and Mullaitivu Districts due to military campaigns initiated by the Sri Lankan military (January 2009).]]
Additionally, accusations of war crimes have been made against [[w:War_crimes_during_the_final_stages_of_the_Sri_Lankan_civil_war|the Sri Lankan government]]&lt;ref&gt;See also [[w:Sexual violence in the Sri Lankan civil war]].&lt;/ref&gt;. A 2009 HRW report alleged that the Sri Lankan government considered the native Tamil population residing in war zones to be &quot;siding with the LTTE and [therefore, were] treated as combatants&quot;, and that the government conducted numerous shellings of &quot;areas crowded with civilians&quot;&lt;ref&gt;{{Cite journal|date=2009-02-19|title=War on the Displaced|url=https://www.hrw.org/report/2009/02/19/war-displaced/sri-lankan-army-and-ltte-abuses-against-civilians-vanni|journal=Human Rights Watch|language=en}}&lt;/ref&gt;. Furthermore, the LTTE conducted recruitment campaigns on the Vanni population where recruited men, women, and even children with minimal training, were recruited for war efforts.  

Over 200,000 Tamil civilians were moved into [[w:Internally_displaced_persons_in_Sri_Lanka|designated displacement camps during the war]], where conditions were poor&lt;ref&gt;{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000}}&lt;/ref&gt;. The suicide rate in these displacement camps was three times the community-level (2002), with a ratio of 103.5 suicides per 10,000 persons, compared to the general population's rate of 37.5 suicides per 10,000 persons. Almost all suicide attempts involved poisonous substances. Other forms of violence included domestic violence and child abuse.  

Local health officials in Vavuniya admitted that mental health concerns were a major problem, but were unable to address these concerns due to a lack of resources and support from the government. During the [[wikipedia:Sri_Lankan_civil_war#2002_peace_process_(2002%E2%80%932006)|brief 2002 ceasefire]], the MSF implemented a &quot;community-based programme&quot; which included &quot;increasing awareness, community strengthening, reinforcing coping-strategies for long-term war-affected communities, and counselling&quot;. The MSF also advocated for restrictions of poisonous substances due its means for suicide attempts, and stressed that &quot;much more [than resettlement]&quot; would need to be done to help alleviate the psychological pain the northern population had faced due to the war&lt;ref&gt;{{Cite journal|last=de Jong|first=Kaz|last2=Mulhern|first2=Maureen|last3=Ford|first3=Nathan|last4=Simpson|first4=Isabel|last5=Swan|first5=Alison|last6=van der Kam|first6=Saskia|date=2002-04|title=Psychological trauma of the civil war in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S0140673602084209|journal=The Lancet|language=en|volume=359|issue=9316|pages=1517–1518|doi=10.1016/S0140-6736(02)08420-9}}&lt;/ref&gt;. The ceasefire ended in 2006 and led to the [[w:Eelam_War_IV|final phase of the civil war]], eventually ending in 2009 with the [[w:https://en.wikipedia.org/wiki/Velupillai_Prabhakaran#Sri_Lankan_Army_Northern_offensive_and_death|death of the LTTE's leader]]. 

'''Post-war'''
[[File:Puttalam district.svg|left|thumb|Puttalam District, unlike its northern counterparts, was largely spared from the intense conflict, possibly explaining the lower rates of common mental disorders (CMDs).]]
The first district-wide cross-sectional multistage cluster sample survey was conducted in the [[w:Jaffna_District|Jaffna District]] shortly after the war ended in 2009. The study's sample included 1517 households and 2 internally displaced peoples camps. With a response rate of 92%, the study found that symptoms for PTSD were found in 7% of participants, symptoms of anxiety were found in 32.6% of participants, and symptoms of depression were found in 22.2% of participants. 2% of respondents were being placed in internally displaced peoples camps at the time of the study, 29.5% were freshly resettled from the internally displaced peoples camps, and the rest of the participants (68.5%) were never placed into camps. In comparison to residents who were never placed into camps, participants that were actively held in camps generally reported more symptoms of PTSD, anxiety, and depression. The researchers also found that women were especially vulnerable to deteriorating mental health conditions. This was explained by two factors: women having to assume the roles of both the father and the mother in the family setting after the, either voluntary or forced, departure of their husband to war, and sexist violence&lt;ref&gt;{{Cite journal|last=Husain|first=Farah|last2=Anderson|first2=Mark|last3=Lopes Cardozo|first3=Barbara|last4=Becknell|first4=Kristin|last5=Blanton|first5=Curtis|last6=Araki|first6=Diane|last7=Kottegoda Vithana|first7=Eeshara|date=2011-08-03|title=Prevalence of War-Related Mental Health Conditions and Association With Displacement Status in Postwar Jaffna District, Sri Lanka|url=https://doi.org/10.1001/jama.2011.1052|journal=JAMA|volume=306|issue=5|pages=522–531|doi=10.1001/jama.2011.1052|issn=0098-7484}}&lt;/ref&gt;. A 2013 study on adult patients in [https://www.ncbi.nlm.nih.gov/books/NBK232631/ primary care settings] (divisional hospitals, primary medical care units) found major depression to be significantly higher in females (5.1%) than males (3.6%), bolstering the findings from the 2009 study&lt;ref&gt;{{Cite journal|last=Senarath|first=Upul|last2=Wickramage|first2=Kolitha|last3=Peiris|first3=Sharika Lasanthi|date=2014-03-24|title=Prevalence of depression and its associated factors among patients attending primary care settings in the post-conflict Northern Province in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/1471-244X-14-85|journal=BMC Psychiatry|language=en|volume=14|issue=1|pages=85|doi=10.1186/1471-244X-14-85|issn=1471-244X|pmc=3987835|pmid=24661436}}&lt;/ref&gt;.

Muslims in Northern Sri Lanka also faced violence and discrimination during the conflict. Most notable incidents include [[w:Expulsion_of_Muslims_from_the_Northern_Province_of_Sri_Lanka|the October 1990 expulsion of Muslims from the North to the Puttalam District or Jaffna]] and the [[w:Kattankudy_mosque_massacre|1990 Kattankudy mosque massacre]]. The only study testing the displaced Muslim population post-civil war was completed in 2011, where a cross-sectional survey of 450 internally displaced people or people born into displacement (ages 18 - 65) revealed 18.8% of the sample suffering from common mental health disorders (CMD), including [[w:Somatoform_disorder|somatoform disorder]] (14%), &quot;other depressive syndromes&quot; (7.3%), major depression (5.1%), and anxiety disorder (2.8%). The percentages found in this study for somatoform disorder and major depression were &quot;considerably higher&quot; than the national percentages, though the researchers noted that the prevalence of CMD was lower in comparison to other countries marred with conflict, including Palestine (40.3%) and Ethiopia (27.8%). The researchers explained that the lower rate of CMD may be attributed to the [[w:Puttalam_District|serenity of the post-settlement destination]], as conflict was mainly centered in the North and East. In contrast to earlier findings, this study did not observe a higher prevalence of CMDs among women, although increased rates of somatoform disorders were noted (though the researchers did not reveal the data behind this)&lt;ref&gt;{{Cite journal|last=Siriwardhana|first=Chesmal|last2=Adikari|first2=Anushka|last3=Pannala|first3=Gayani|last4=Siribaddana|first4=Sisira|last5=Abas|first5=Melanie|last6=Sumathipala|first6=Athula|last7=Stewart|first7=Robert|date=2013-05-22|title=Prolonged Internal Displacement and Common Mental Disorders in Sri Lanka: The COMRAID Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0064742|journal=PLOS ONE|language=en|volume=8|issue=5|pages=e64742|doi=10.1371/journal.pone.0064742|issn=1932-6203|pmc=3661540|pmid=23717656}}&lt;/ref&gt;.

Research on the mental state of combatants has been limited, but a post-war 2009 study done between soldiers of the [[w:Sri_Lanka_Army_Special_Forces_Regiment|Special Forces]] and regular soldiers showed higher levels of exposure to traumatic events for units of the Special Forces, yet the former exhibited significantly less symptoms of CMDs compared to the latter. The authors of this study, [https://scholar.google.co.uk/citations?user=cVKEBdwAAAAJ&amp;hl=en&amp;oi=ao Raveen Hanwella] and [https://scholar.google.co.uk/citations?user=ZRj74qMAAAAJ&amp;hl=en&amp;oi=sra Varuni de Silva], offered the camaraderie of the military unit as an explanation for the discrepancy&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|date=2012-08|title=Mental health of Special Forces personnel deployed in battle|url=https://pubmed.ncbi.nlm.nih.gov/22038567|journal=Social Psychiatry and Psychiatric Epidemiology|volume=47|issue=8|pages=1343–1351|doi=10.1007/s00127-011-0442-0|issn=1433-9285|pmid=22038567}}&lt;/ref&gt;. A follow-up study was completed by the pair (with the addition of former Director-General of the Health Services of the Sri Lanka Navy [[w:Nicholas_Jayasekera|Nicholas Jayasekera]]), where the findings were similar, though the statistically significant bridge between the two cohorts in the previous study evaporated in the follow-up study. This may be due to the significant decline in mental health problems observed in the regular unit forces, potentially reflecting resilience in the aftermath of the conflict&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=Jayasekera|first2=Nicholas E. L. W.|last3=Silva|first3=Varuni A. de|date=2014-09-25|title=Mental Health Status of Sri Lanka Navy Personnel Three Years after End of Combat Operations: A Follow Up Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0108113|journal=PLOS ONE|language=en|volume=9|issue=9|pages=e108113|doi=10.1371/journal.pone.0108113|issn=1932-6203|pmc=4177866|pmid=25254557}}&lt;/ref&gt;. Amputees or soldiers with spinal injuries exhibited drastically different numbers, with approximately 40% of nearly 100 male-veterans in a post-war 2009 study displaying PTSD-like symptoms&lt;ref&gt;{{Cite journal|last=Abeyasinghe|first=N. L.|last2=de Zoysa|first2=P.|last3=Bandara|first3=K.M.K.C.|last4=Bartholameuz|first4=N. A.|last5=Bandara|first5=J. M.U.J.|date=2012-05-01|title=The prevalence of symptoms of Post-Traumatic Stress Disorder among soldiers with amputation of a limb or spinal injury: A report from a rehabilitation centre in Sri Lanka|url=https://doi.org/10.1080/13548506.2011.608805|journal=Psychology, Health &amp; Medicine|volume=17|issue=3|pages=376–381|doi=10.1080/13548506.2011.608805|issn=1354-8506|pmid=21942815}}&lt;/ref&gt;.

About a decade after the conflict ceased, a few notable studies have emerged to help guide understanding on the longer-term mental health effects on victims of the civil war. 

From July 2019 to October 2020, a study conducted on 585 local adolescents (ages 12-19) in the Vavuniya district revealed that despite 15.6% of the statistic having faced one or more war-related events, only 3.9% of the participants had moderate to severe depression. In addition to considerably low depression rates, only 5.7% of participants age 17+ were found to have moderate to severe hopelessness&lt;ref&gt;{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000|pmc=10472617|pmid=37653394}}&lt;/ref&gt;. The authors referenced a 2010 observation by psychiatrist [https://us.sagepub.com/en-us/nam/author/daya-somasundaram Daya Somasundaram], who noted that many Tamil IDPs presented &quot;remarkable resilience and post-traumatic growth&quot; after the civil war—an outcome he attributed to the close-knit, family-centered nature of Tamil communities&lt;ref&gt;{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}&lt;/ref&gt;.  

However, findings originating from a 2019 study, undertook by several faculty members from the University of Kelaniya, the University of Jaffna, the [[w:Gampaha_Wickramarachchi_University_of_Indigenous_Medicine|Gampaha Wickramarachchi University of Indigenous Medicine]], and the [https://onur.gov.lk/ Office for National Unity and Reconciliation (ONUR)] in Jaffna, found contrasting results. Out of 336 participants from districts which faced significant ramifications of the conflict (Jaffna, Kilinochchi, Mullaithivu, Vavuniya, and Mannar districts), 50.5% had extreme anxiety symptoms and 36.5% exhibited &quot;extremely severe&quot; symptoms of depression. 92.5% of families in the sample experienced suicidal ideation, with an observed negative correlation between trauma exposure and life satisfaction with families. Drug abuse (86.2%) and alcohol abuse (84.5%) were the two highest problematic behaviors recorded on a community-level, suggesting that the negative consequences of the civil war still persist, possibly on a substantial scale than previously recognized, in Tamil communities residing in the North&lt;ref&gt;{{Cite journal|last=Thamotharampillai|first=Umaharan|last2=Perera|first2=Ruwanthi|last3=Wickremasinghe|first3=Rajitha|last4=Williams|first4=Shehan|last5=Vijayasangar|first5=Thedsanamoorthy|last6=Sivatharsan|first6=Balasubramaniam|last7=Hilbert|first7=Vanceline|last8=Somasundaram|first8=Daya|date=2025-05-06|title=Collective Trauma- Psychosocial consequences of war in northern Sri Lanka 10 years on, a mixed methods study|url=https://www.sciencedirect.com/science/article/pii/S2666560325000696|journal=SSM - Mental Health|pages=100457|doi=10.1016/j.ssmmh.2025.100457|issn=2666-5603}}&lt;/ref&gt;. Further research should be conducted on Northern Tamil populations to assess the extent of mental health issues stemming from the conflict. 

In 2019, [https://www.researchgate.net/scientific-contributions/R-M-M-Monaragala-2087692299 Dr. R. M. M. Monaragala] conducted a study on 1,845 soldiers with combat experience, finding that 3.9% of the sample suffered from PTSD. Dr. Monaragala noted that &quot;probable depression, fatigue, aggression, and family history of mental disorder&quot; were correlative of PTSD presence. He suggested that &quot;screening and psychosocial intervention[s]&quot; could alleviate CMDs of former combatants&lt;ref&gt;{{Cite journal|last=Monaragala|first=R. M. M.|date=2024-04-19|title=Exploring the effects of the past civil war in terms of the prevalence and associating factors of PTSD|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v14i2.8465|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=14|issue=2|doi=10.4038/sljpsyc.v14i2.8465|issn=2012-6883}}&lt;/ref&gt;. 

=== 2004 Boxing Day Tsunami ===
The '''2004 Boxing Day Tsunami''' was a natural disaster where a tsunami spawned off a 9.2–9.3 magnitude earthquake off the coast of Aceh in Indonesia on December 26. The tsunami greatly affected the coastlines of the country, with the death toll reaching to around 35,000 deaths. In addition, 90,000 houses were destroyed and 516,000 people were forced to migrate due to severe infrastructural damage&lt;ref name=&quot;:5&quot; /&gt;. It stands as the [http://www.china.org.cn/english/features/tsunami_relief/119821.htm worst natural disaster to have ever hit Sri Lanka].  
[[File:Tsunami relief 2004 02.jpg|thumb|300x300px|Volunteers from [[w:Royal_College,_Colombo|Royal College in Colombo]] assisting in tsunami relief efforts (Sarvodaya Headquaters, Moratuwa).]]
A survey conducted on schoolchildren (ages 8-14) in Manadkadu (a Tamil-majority village in the northern coast), [[w:Kosgoda|Kosgoda]] (western coast), and [[w:Galle|Galle]] (southern coast), just a few weeks after the tsunami hit Sri Lanka, revealed that 33.8%, 13.9%, and 38.8% of children interviewed exhibited signs of PTSD (according to the DSM-IV's criteria), respectively (minus the time criteria, as the DSM-IV does not permit diagnosis of PTSD within 4 weeks of a traumatic incident). The loss of family members and exposure to previously traumatic incidents appeared to be highly correlate with PTSD development&lt;ref&gt;{{Cite journal|last=Neuner|first=Frank|last2=Schauer|first2=Elisabeth|last3=Catani|first3=Claudia|last4=Ruf|first4=Martina|last5=Elbert|first5=Thomas|date=2006|title=Post-tsunami stress: A study of posttraumatic stress disorder in children living in three severely affected regions in Sri Lanka|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/jts.20121|journal=Journal of Traumatic Stress|language=en|volume=19|issue=3|pages=339–347|doi=10.1002/jts.20121|issn=1573-6598}}&lt;/ref&gt;. 

Many victims in the Jaffna area suffered with &quot;[https://www.psychiatry.org/patients-families/prolonged-grief-disorder pathological grief], phobias, depression and PTSD&quot; post-tsunami. Schizophrenia in the Jaffna Tamil community, which had already suffered elevated prevalence of PTSD prior to the tsunami, had worsened—highlighting the need for specialized care in response to cumulative exposures to chronic and acute traumas. In a study published in ''International Psychiatry'' (2006), Jaffna-based researchers noted that, contrary to their initial inclinations, there was not a &quot;large[r] (than expected) rise in [the] number of people&quot; seeking mental health support 3 months after the tsunami. However, 10 months after the disaster, the researchers anticipated that &quot;more psychiatric disorders&quot; would emerge due to &quot;very little rebuilding [efforts]&quot; and an apparent &quot;unfairness in the aid system&quot;.&lt;ref&gt;{{Cite journal|last=Somasundaram|first=D. J.|last2=Yoganathan|first2=S.|last3=Ganesvaran|first3=T.|date=1993-09|title=Schizophrenia in northern Sri Lanka|url=https://pubmed.ncbi.nlm.nih.gov/7828234|journal=The Ceylon Medical Journal..|volume=38|issue=3|pages=131–135|issn=0009-0875|pmid=7828234}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Danvers|first=K.|last2=Sivayokan|first2=S.|last3=Somasundaram|first3=D. J.|last4=Sivashankar|first4=R.|date=2006-07|title=Ten months on: qualitative assessment of psychosocial issues in northern Sri Lanka following the tsunami|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6734678/|journal=International Psychiatry: Bulletin of the Board of International Affairs of the Royal College of Psychiatrists|volume=3|issue=3|pages=5–8|issn=1749-3676|pmc=6734678|pmid=31507850}}&lt;/ref&gt; 

At the February 2005 ''After the Tsunami: Mental Health Challenges to the Community for Today and Tomorrow'' conference in Thailand, [https://www.researchgate.net/profile/Chandanie-Hewage Dr. Chandanie Hewage] of the [[w:University_of_Ruhuna|University of Ruhuna]] commentated that measures taken to assist the affected were &quot;not coordinated&quot; due to poor &quot;communication systems and road [conditions].&quot; Regardless, efforts were continued by the government and health professionals to alleviate the struggles the victims were facing, including the psychological ramifications of the disaster. 

Several issues in the delivery of these services were highlighted by Dr. Hewage, including poor maintenance of health records, lack of awareness on drug consumption by the patients themselves, and shortages of health professionals. Dr. Hewage points out that personnel had &quot;little&quot; mental health training prior to the disaster, suggesting increased &quot;research&quot; and adequate &quot;provision[ing] and training of staff&quot; for the long-term&lt;ref&gt;{{Cite journal|last=Davidson|first=Jonathan R. T.|date=2006|title=Foreword. After the tsunami: mental health challenges to the community for today and tomorrow|url=https://pubmed.ncbi.nlm.nih.gov/16602809|journal=The Journal of Clinical Psychiatry|volume=67 Suppl 2|pages=3–8|issn=0160-6689|pmid=16602809}}&lt;/ref&gt;. With inadequate documentation, no systematic procedures in place, and insufficient personnel, tsunami victims with mental health concerns may not receive the services they need, further compacting neuropsychological ailments.  

In 2008 (about 3-4 years after the tsunami), researchers in the hard-hit village of [[w:Peraliya|Peraliya]] (Galle District) found that from a sample of approximately 90 adults, 25% suffered from moderate–severe PTSD, with women scoring &quot;above the cut-off for anxiety&quot; and reporting more &quot;somatic symptoms&quot;, though researchers inferred that the PTSD rate found in the study may be influenced by other factors, including war or economic hardship&lt;ref&gt;{{Cite journal|last=Hollifield|first=Michael|last2=Hewage|first2=Chandanie|last3=Gunawardena|first3=Charlotte N.|last4=Kodituwakku|first4=Piyadasa|last5=Bopagoda|first5=Kalum|last6=Weerarathnege|first6=Krishantha|last7=Group|first7=International Post-Tsunami Study|date=2008-01|title=Symptoms and coping in Sri Lanka 20–21 months after the 2004 tsunami|url=https://www.cambridge.org/core/journals/the-british-journal-of-psychiatry/article/symptoms-and-coping-in-sri-lanka-2021-months-after-the-2004-tsunami/CB33752239AF362A0BFD55B3668D60B0|journal=The British Journal of Psychiatry|language=en|volume=192|issue=1|pages=39–44|doi=10.1192/bjp.bp.107.038422|issn=0007-1250}}&lt;/ref&gt;.  

=== 2019 Easter Bombings ===
The '''2019 Easter Bombings''' were a series of coordinated attacks perpetrated by the Islamic extremist group, [[w:National_Thowheeth_Jama'ath|National Thowheeth Jama'ath]], on April 21, 2019. The attack targeted three churches and three hotels in the Colombo area, killing nearly 300 people and injuring over 500. The attacks were also attributed to the incompetency of the Sri Lankan government, who ignored [https://www.bbc.com/news/world-asia-48044636 multiple warnings preceding the attacks]. The attacks negatively affected the Sri Lankan Catholic community and further weakened relations between the major religious groups&lt;ref&gt;{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}&lt;/ref&gt;.  

In the aftermath of the attacks, professionals in the [[w:Gampaha_District|Gampaha District]] resorted to &quot;low-cost methodologies&quot; for children and adolescents affected by the attack, as a &quot;severe shortage&quot; of children and adolescent mental health experts were exposed&lt;ref&gt;{{Cite journal|last=Chandradasa|first=Miyuru|last2=Rathnayake|first2=Layani C|last3=Rowel|first3=Madushi|last4=Fernando|first4=Lalin|date=2020-06-01|title=Early phase child and adolescent psychiatry response after mass trauma: Lessons learned from the Easter Sunday attack in Sri Lanka|url=https://doi.org/10.1177/0020764020913314|journal=International Journal of Social Psychiatry|language=EN|volume=66|issue=4|pages=331–334|doi=10.1177/0020764020913314|issn=0020-7640}}&lt;/ref&gt;. In a qualitative study of 8 survivors of the attacks receiving grief counseling, [[w:University_of_Ruhuna|University of Ruhuna]] assistant professor [https://www.researchgate.net/profile/Virasha-Godakanda Virasha Godakanda] observed that 70% of the sample size expressed a lack of confidence in adequate mental health interventions from the government, reducing the quality of such services. Professor Godakanda strongly endorsed for &quot;culturally-sensitive&quot; programs, a diversity in therapeutic approaches (including nature-based therapy), and &quot;prolonged investigations&quot; to track developments in mental health resources and impacts of implemented interventions&lt;ref&gt;{{Cite journal|last=Godakanda|first=Virasha|date=2025-01-29|title=A GRIEF COUNSELING INTERVENTION AFTER THE MASS TRAUMA: LESSONS LEARNED FROM THE VICTIMS OF THE EASTER SUNDAY ATTACK IN SRI LANKA|url=https://kjmr.com.pk/kjmr/article/view/216|journal=Kashf Journal of Multidisciplinary Research|language=en|volume=2|issue=01|pages=13–32|doi=10.71146/kjmr216|issn=3007-200X}}&lt;/ref&gt;. 

A few weeks following the attacks, Muslims in Sri Lanka were subjected to [[w:2019_anti-Muslim_riots_in_Sri_Lanka|violent, coordinated riots]] masterminded by Sinhalese national forces&lt;ref&gt;{{Cite journal|last=Mujahidin|first=Muhammad Saekul|date=2023-07-03|title=Extremism and Islamophobia Against the Muslim Minority in Sri Lanka|url=https://www.ajis.org/|journal=American Journal of Islam and Society|language=en|volume=40|issue=1-2|pages=213–241|doi=10.35632/ajis.v40i1-2.3135|issn=2690-3741}}&lt;/ref&gt;. Riots were mainly centered in the [[w:Kurunegala_District|Kurunegala]], Gampaha, and [[w:Kandy_District|Kandy]] Districts. At least [https://www.aljazeera.com/news/2019/5/21/in-sri-lanka-muslims-say-sinhala-neighbours-turned-against-them one confirmed death was reported]. Calls for vague ''niqab'' and ''burqa'' bans were increasingly prominent, eventually leading to the 2021 burqa ban by the Sri Lankan government. Pakistani and Afghani refugees fleeing religious persecution in Negombo were forced to be &quot;made refugees again&quot; after local protests were orchestrated against their settlement. Anti-Muslim sentiment was &quot;unleashed online, in the law, and on the street&quot;&lt;ref&gt;{{Cite book|title=CARTOGRAPHIC JOURNEY OF RACE, GENDER AND POWER: global identity|date=2021|publisher=CAMBRIDGE SCHOLARS PUBLIS|isbn=978-1-5275-6965-2|location=S.l.}}&lt;/ref&gt;. Albeit its relevancy to the attacks, no in-depth mental health studies have took place on the minority Muslim population following the Easter bombings. Further research is imperative in exploring the sustained psychological effects of Islamophobia and its effect on the Muslim minority community in the aftermath of the 2019 Easter attacks. 

Literature on the impact of the 2019 Easter Bombings on mental health is limited and further research should be conducted.  

=== 2019-2024 Economic Crisis ===
The '''2019-2024 Economic Crisis''' refers to a 5 year period where the Sri Lankan economy experienced significant inflation and an abrupt hike in prices on basic, everyday items. It is the worse economic crisis the country has faced since the Sri Lankans were granted independence in 1948. Schools in Sri Lanka were forced to postpone examinations due to paper shortages. Gas shortages led to long lines at gas stations, some lasting for days, throughout the island. Shortages in electricity, cooking gas, and aviation feul were additional consequences of the economic crisis.

Healthcare workers faced a barrage of impediments in their line of work during the crisis, including a lopsided work-life balance due to unprecedented demand, increased stress and mental fatigue from a lack of resources and personnel, unhealthy coping mechanisms, job dissatisfaction, and a reduction in work quality. Such effects perpetuated a self-enforcing cycle of psychologically distressed mental healthcare workers providing subpar services, affecting patients and amplifying mental health issues experienced by both the workforce and their patients&lt;ref&gt;{{Cite journal|last=Dilogini|first=S.|last2=Grace|first2=H. H.|last3=Thasika|first3=T.|date=2024|title=Exploring The Mental Health and Well-Being of Public Healthcare Workers (HCWs) Amid Economic Crisis in Sri Lanka|url=http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/11092|language=en|publisher=Chartered Institute of Personnel Management}}&lt;/ref&gt;. 

Medical students from the Faculty of Medicine at the University of Colombo reported that the economic crisis forced abrupt changes in dietary consumption, increased hopelessness in the future, increased stress and anxiety, and a decrease in interest in pursuing a &quot;clinical post-graduate career&quot;&lt;ref&gt;{{Cite journal|last=Adikaranayake|first=Pesala Randika|last2=Perera|first2=Anusha Nimrod|last3=Nilaweera|first3=Akhila Imantha|last4=Fernando|first4=Desha Rajni|last5=Wijayaratne|first5=Dilushi Rowena|date=2025-07-01|title=Effects of Sri Lankan economic crisis on health, lifestyle and education of medical students in Faculty of Medicine, University of Colombo – an online survey|url=https://doi.org/10.1186/s12909-025-07506-y|journal=BMC Medical Education|language=en|volume=25|issue=1|pages=938|doi=10.1186/s12909-025-07506-y|issn=1472-6920|pmc=12211748}}&lt;/ref&gt;. 283 government-school teachers completed a web-based cross-sectional survey in April 2024, with majority of the participants reporting a severe reduction in monthly income &amp; 1/3 of participants exhibiting &quot;clinical levels of psychological distress&quot;&lt;ref&gt;{{Cite journal|last=Senevirathne|first=C. P.|last2=Senarathne|first2=D. L. P.|last3=Fernando|first3=M. S.|last4=Senevirathne|first4=S. P.|date=2025-05-28|title=Examining the economic burden and mental health distress among government school teachers in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/s40359-025-02921-8|journal=BMC Psychology|language=en|volume=13|issue=1|pages=572|doi=10.1186/s40359-025-02921-8|issn=2050-7283}}&lt;/ref&gt;. A study published in that same year reported that out of 261 nurses working in teaching hospitals, 91.6% were forced to allocate their finances to strictly &quot;general needs&quot;, while more than 50% looked into international opportunities for employment. Notably, the study reported an overall near &quot;twofold greater&quot; rate of depression, anxiety, and stress compared to previous studies on nurses in Sri Lanka&lt;ref&gt;{{Cite journal|last=Senevirathne|first=C.P|last2=Senarathne|first2=L.|last3=Fernando|first3=M.|date=2024-04-01|title=Exploring the Association Between Behavioural Modification in Response to the Prevailing Economic Crisis and Mental Health Outcomes of Nurses from Teaching Hospitals, Sri Lanka|url=https://doi.org/10.1177/23779608241272679|journal=SAGE Open Nursing|language=EN|volume=10|pages=23779608241272679|doi=10.1177/23779608241272679|issn=2377-9608|pmc=11311183}}&lt;/ref&gt;. 

The detrimental effects the crisis has had on the mental health sector reveal a concerning area of underappreciation and under compensation towards a critical sector for the well-being of the country. Adequate staffing, increased funding, and an improved work-life balance should be emphasized for the workers of health sector of the country.  

== Present-Day Challenges ==

=== Ethnic tension ===
Despite the ending of the Sri Lankan civil war and the introduction of pluralist policies (such as the [https://srilankaembassy.fr/sites/default/files/files/media/pdf/NationalPolicy-English.pdf 2017 National Policy on Reconciliation and Coexistence] under the Sirisena administration), tensions amongst members of the ethnic groups still persist. Evidence of these tensions was found in a 2022 study conducted in the Ratnapura district, where religious leaders expressed skepticism through semi-structured interviews on &quot;conflict transformation&quot;. A Tamil citizen of the Ratnapura community recounted that they were forced to &quot;hide in jungles&quot; and consume &quot;dirty water in drainage[s]&quot; due to scarcity of food and drinkable water as a result of the conflict. In certain personal accounts, ethnic conflicts appear to affect the social behavior and identity of the majority ethnic group. One Sinhala participant recounted his objection to the war-time retaliatory destruction of a shop run by a Tamil shopkeeper was met with interrogative questions about &quot;whether [he was] Sinhalese or not&quot;. Both accounts convey interethnic tensions stemming from decade-long conflicts&lt;ref&gt;Jayathilaka, Aruna &amp; Gamage, Sayuri. (2024). Role of Buddhist and Hindu Religious Leaders Role of Buddhist and Hindu Religious Leaders in the Post-War Conflict Transformation Process: A Study Based on Rathnapura District in Srilanka. ''Retrieved from'' https://gandhimargjournal.org/wp-content/uploads/2024/09/Volume-46-Issue-1-April-June-2024.pdf#page=66&lt;/ref&gt;. 

Beyond individual accounts and the official end of the civil war, the minority groups in the country continue to feel ostracized. The Sri Lankan Tamil population remains dissatisfied with the Sri Lankan government due to their alleged lack of accountability of perpetrators of war crimes and lack of information on the whereabouts of [[w:Enforced_disappearances_in_Sri_Lanka|thousands of enforced disappearances]] that took place from the 1980s. Additionally, rising anti-Muslim sentiment in recent years has contributed to increased ethnic tensions, a stark contrast to the previous centuries of peaceful co-existence between the groups. 
[[File:Bodu Bala Sena symbol.svg|thumb|The symbol for Bodu Bala Sena, a nationalistic Sinhala Buddhist group criticized for catalyzing ethnic tensions in Sri Lanka.]]
Laws passed by the Sri Lankan government, such as the [[w:Prevention_of_Terrorism_Act_(Sri_Lanka)|Prevention of Terrorism Act]] and [[wikipedia:Anti-conversion_law#Sri_Lanka|anti-conversion laws]], have forced the United States Commission on International Religious Freedom to label Sri Lanka as a nation that &quot;[engages] or [tolerates] severe violations of religious freedom&quot; in their 2024 report. The government has been criticized by human rights organizations for &quot;disproportionately targeting religious minorities&quot;&lt;ref&gt;{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}&lt;/ref&gt;. Additionally, the implementation of the three dominant languages, English, Sinhala, and Tamil, across formal education and government services have been lackadaisical, narrowing opportunities of foundational social interactions between the groups. Persistent discrimination and prejudice towards minority groups can lead to an array of complex and self-deprecating mental health issues.   

Efforts to mitigate ethnic tensions include strategies like [[w:Community-based_participatory_research|community-based participatory research]] (CBPR), task-sharing, and securing online mental health services in order to expand mental health services. However, the implementation of evidence-based plans has been met with difficulty due to inaccessibility, high costs, and shortages of adequately-trained personnel.  

Movements aiming for improved intra group and inter group coexistences, such as the Jaffna People’s Forum for Coexistence, should be emphasized on a systematic and multi-level basis, including but not limited to education, public sectors, and within communities. Pluralistic values are encouraged to be emphasized across both private and public schools to foster cultural sensitivity and tolerance. Measures should be taken against groups criticized for promoting sectarian hostility, such as the [[w:Bodu_Bala_Sena|Bodu Bala Sena]]. 

=== Poverty ===
It has been proven that poverty significantly increases the chances of developing mental illnesses. This is further amplified by possible discrimination&lt;ref&gt;{{Cite journal|last=Knifton|first=Lee|last2=Inglis|first2=Greig|date=2020-10|title=Poverty and mental health: policy, practice and research implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC7525587/|journal=BJPsych bulletin|volume=44|issue=5|pages=193–196|doi=10.1192/bjb.2020.78|issn=2056-4694|pmc=7525587|pmid=32744210}}&lt;/ref&gt;. Poverty also affects the ability for individuals with mental health concerns to receive the treatment they need. Due to the repercussions of the economic crisis, clients in Sri Lanka could not attend further counseling sessions&lt;ref name=&quot;:8&quot; /&gt;. Poverty from 2021 to 2022 [https://databankfiles.worldbank.org/public/ddpext_download/poverty/987B9C90-CB9F-4D93-AE8C-750588BF00QA/current/Global_POVEQ_LKA.pdf reportedly doubled], with future forecasts predicting the poverty line to &quot;remain above 25 percent&quot;. Suicide has been empirically linked to economic hardships in previous studies&lt;ref&gt;{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}&lt;/ref&gt;. A 2013 study done on suicidal patients in [[w:Batticaloa_Teaching_Hospital|Batticaloa Teaching Hospital]] revealed 76% of patients who attempted suicide were from rural areas while 15% were from urban areas&lt;ref&gt;{{Cite book|url=http://ir.lib.seu.ac.lk/handle/123456789/1457|title=The influence of common risk factors for the patient with attempted suicide hospitalized at the teaching hospital, Batticaloa|last=Kisokanth|first=G.|last2=Najeem|first2=M. M.|last3=Karunakaran|first3=K. E.|date=2014-08-02|publisher=South Eastern University of Sri Lanka, University Park, Oluvil #32360, Sri Lanka|isbn=978-955-627-053-2|language=en-US}}&lt;/ref&gt;. The Sri Lankan government should consider the economical impacts that poverty has on mental health and implement ways to aid poverty-stricken individuals with mental health concerns.   

=== Stigmas ===
Stigma consists of the &quot;combined effect of prejudice, ignorance and discrimination.&quot;&lt;ref name=&quot;:10&quot;&gt;{{Cite web|url=http://www.researchgate.net/publication/233990797_The_Stigma_of_Mental_Illness_in_Sri_Lanka_The_Perspectives_of_Community_Mental_Health_Workers|title=(PDF) The Stigma of Mental Illness in Sri Lanka: The Perspectives of Community Mental Health Workers|website=ResearchGate|language=en|access-date=2025-07-25}}&lt;/ref&gt;.

A 2012 interview consisting of nine participants (two doctors, three nurses, one occupational therapist, one development worker, and two volunteers) revealed a number of concerning societal viewpoints on individuals with mental health concerns. The interviews revealed that negative judgements were not only levied against the individual with the mental illness, but also the family. Families hid mentally ill family members from the public to avoid &quot;shame&quot; and possible hinderances in marriage proposals. Views that mentally ill individuals were &quot;violent&quot; served as the motivating factor behind socially isolating those with mental illness from their communities. Interviewees mentioned that individuals dealing with mental health challenges would be attacked with stones and called &quot;derogatory names.&quot; A lack of community awareness regarding mental health and negative portrayals of mentally ill individuals in media exacerbates stigmatization, though the researchers commented that the media was &quot;improving&quot; in their depiction of mental illness. Beliefs that illnesses are caused by &quot;spirits&quot; can be problematic for individuals dealing with mental health issues and suggests poor mental health awareness. Mental health workers themselves believed that they were being stigmatized, as mental health was reportedly not taken as seriously as physical health. Despite the intriguing perspectives provided, the small sample size and usage of snow sampling raise questionable concerns regarding the generalizability of the results&lt;ref name=&quot;:10&quot; /&gt;. 

Improving media portrayal of subjects concerning mental health and involving community members in interventions dealing with mental health issues are ways that could destigmatize mental health amongst communities in Sri Lanka. Tying collaborations between allopathic services and traditional healers instead of having these two services work individually could enhance engagement between traditional medicine and Western medicine.

=== Suicide Trends &amp; Risk Factors ===
Suicide is defined as &quot;the act of killing oneself deliberately, initiated and performed by the person concerned in the full knowledge or expectation of its fatal outcome&quot;&lt;ref name=&quot;:11&quot;&gt;{{Cite book|title=The neuroscience of suicidal behavior|last=Heeringen|first=Kees van|date=2018|publisher=Cambridge University Press|isbn=978-1-316-60290-4|series=Cambridge fundamentals of neuroscience in psychology|location=Cambridge, United Kingdom New York, NY, USA Port Melbourne, VIC, Australia New Delhi, India Singapore}}&lt;/ref&gt;. Although Sri Lanka has seen a significant reduction in suicide rates from the mid 1990s, largely stemming from its ban on extremely toxic pesticide products, suicide and self harm remains a significant issue. The suicide rate per 100,000 people increased from 14.0 in 2019 to [https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide 15.0 in 2022] (according to WHO). On average, 27 males per 100,000 males and 5 females per 100,000 females committed suicide in 2022&lt;ref&gt;{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}&lt;/ref&gt;. Hanging appears to be the most used method for suicide for both males and females, with studies revealing a steady increase in recent years&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Bandara|first=Piumee|last2=Wickrama|first2=Prabath|last3=Sivayokan|first3=Sambasivamoorthy|last4=Knipe|first4=Duleeka|last5=Rajapakse|first5=Thilini|date=2024-04-17|title=Reflections on the trends of suicide in Sri Lanka, 1997–2022: The need for continued vigilance|url=https://journals.plos.org/globalpublichealth/article?id=10.1371/journal.pgph.0003054|journal=PLOS Global Public Health|language=en|volume=4|issue=4|pages=e0003054|doi=10.1371/journal.pgph.0003054|issn=2767-3375|pmc=11023397|pmid=38630779}}&lt;/ref&gt;. 

From 2023 to 2024, a group of researchers from the [[w:Eastern_University,_Sri_Lanka|Eastern University in Sri Lanka]] assessed 828 patients admitted to the Teaching Hospital in [[w:Batticaloa,_Sri_Lanka|Batticaloa, Sri Lanka]] for attempted suicide. They concluded that suicide prevention programs should be attuned to younger people (ages 15 to 35 in the study), emphasize the importance of education and reducing unemployment, and increase social support in the Tamil community. Despite accounting for other factors that could lead to suicidal ideation (ie, poverty), the results from this study suffer in external validity as 90% of the patients were Tamil and over 50% were between 16 and 25 years. In addition, correlations between suicide and unemployment rates have been questioned, with [[w:Austerity|austerity]] being a more reliable indicator of suicide rates than unemployment rates&lt;ref name=&quot;:11&quot; /&gt;. Further comprehensive studies on risk factors relating to suicide should be studied to examine correlations between unemployment rates and austerity measures.  

The WHO suggests implementing evidence-based suicide prevention programs, such as [https://www.who.int/initiatives/live-life-initiative-for-suicide-prevention LIVE LIFE], to reduce the national suicide rate&lt;ref&gt;{{Cite web|url=https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide|title=World Suicide Prevention day 2024 “Changing the Narrative on Suicide”|website=www.who.int|language=en|access-date=2025-07-29}}&lt;/ref&gt;. Media depictions of suicidal methods, such as hanging, can lead to sensationalism and the media should be cautious of such displays in movies and TV shows&lt;ref name=&quot;:12&quot; /&gt;. Awareness of depression and other mental health issues can serve as a safeguard against suicidal ideation in Sri Lankan men and women.  

== Role of Religion ==
According to the last demographic report (2012), 70.2% of Sri Lankans are Buddhist, 12.6% are Hindus, 9.7% are Muslims, and 7.4% are Christians. The Theravada Buddhist community makes up the majority in several provinces throughout the country&lt;ref&gt;{{Cite web|url=https://www.state.gov/reports/2022-report-on-international-religious-freedom/sri-lanka/|title=Sri Lanka|website=United States Department of State|language=en-US|access-date=2025-08-07}}&lt;/ref&gt;. Religion, especially Theravada Buddhism, has had a significant influence on not only the historical treatment of mental health in the country, but also everyday life&lt;ref name=&quot;:15&quot; /&gt;. The [[w:Mahāvaṃsa|''Mahāvaṃsa'']] details hospitals treating patients suffering from mental health issues as early as the 4th century BC. Additionally, the 1700s Nayaka king [[w:Kirti_Sri_Rajasinha|Kirthi Sri Rajasinghe]] detailed the implementation of Buddhist philosophy in psychiatry&lt;ref name=&quot;:4&quot; /&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Alwis|first=L. A. P. De|date=2017-12-05|title=Development of civil commitment statutes (laws of involuntary detention and treatment) in Sri Lanka: a historical review|url=https://mljsl.sljol.info/articles/10.4038/mljsl.v5i1.7351|journal=Medico-Legal Journal of Sri Lanka|language=en|volume=5|issue=1|doi=10.4038/mljsl.v5i1.7351|issn=2012-8231}}&lt;/ref&gt;.

Modern-day empirical studies have attested to the usefulness of religion in mitigating stress and elevating mental health&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-94-007-4276-5_22|title=Religion and Mental Health|last=Schieman|first=Scott|last2=Bierman|first2=Alex|last3=Ellison|first3=Christopher G.|date=2013|publisher=Springer Netherlands|isbn=978-94-007-4276-5|editor-last=Aneshensel|editor-first=Carol S.|location=Dordrecht|pages=457–478|language=en|doi=10.1007/978-94-007-4276-5_22|editor-last2=Phelan|editor-first2=Jo C.|editor-last3=Bierman|editor-first3=Alex}}&lt;/ref&gt;. Religion has been found to be positively correlated with improved mental health, and more religious patients were concluded to have &quot;better mental health and adapt[ed] more quickly to health problems&quot; versus patients who weren't religious&lt;ref&gt;{{Cite journal|last=Koenig|first=Harold G.|date=2012|title=Religion, spirituality, and health: the research and clinical implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3671693/|journal=ISRN psychiatry|volume=2012|pages=278730|doi=10.5402/2012/278730|issn=2090-7966|pmc=3671693|pmid=23762764}}&lt;/ref&gt;. [https://www.researchgate.net/scientific-contributions/T-N-Wickramarathna-2247724082 Dr. Wickramarathna] of the University Psychiatry Unit (UPU) at the National Hospital of Sri Lanka (NHSL) argues that psychiatrists must strive for a balance in their approach to patients and &quot;make positive use of religion in [their] practice[s]&quot;&lt;ref&gt;{{Cite journal|last=Wickramarathna|first=T. N.|date=2022-12-31|title=Psychiatrists should stand far from the shrine: why and why not we should separate religion from psychiatry|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v13i2.8397|journal=Sri Lanka Journal of Psychiatry|language=en|volume=13|issue=2|doi=10.4038/sljpsyc.v13i2.8397|issn=2012-6883}}&lt;/ref&gt;.

=== Buddhism ===
27 Sinhalese Buddhists from four Buddhist temples were selected for a series of 70-minute interviews and focus group discussions with the aim of learning the Sinhala Buddhist understanding and experience of spiritual well-being and psychological well-being. The interviewees held spiritual wellness to be the &quot;center&quot; of overall wellness, the &quot;precondition for a successful life&quot;&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/articles/10.4038/sljss.v44i1.7990|journal=Sri Lanka Journal of Social Sciences|language=en-US|volume=44|issue=1|doi=10.4038/sljss.v44i1.7990|issn=0258-9710}}&lt;/ref&gt;. Sinhala Buddhists believe that wellness cannot be achieved without spiritual tranquility. The report states that participants emphasized that spirituality &quot;cannot be directly intervened&quot; and can only be seen through &quot;[interactions] with society&quot;&lt;ref name=&quot;:14&quot; /&gt;. Despite the ''athmaya'' (soul) being &quot;unreachable&quot;, it can be &quot;intervened&quot;, or treated, through the actions of the mind and body with society&lt;ref name=&quot;:14&quot; /&gt;. One being &quot;psychologically ill&quot; can affect one's spiritual being, as the participants reported in their interviews, and can be impaired through &quot;lifestyle stressors, environmental and socio-cultural causes, non-human related causes and bad-karma in the past lives&quot;&lt;ref name=&quot;:14&quot; /&gt;.  

The researchers concluded that despite Sinhala Buddhists not being able to articulately decipher the discrepancies between psychological well-being and spiritual well-being, they are able to conceptualize and maintain a culturally embedded understanding between the two, serving as reputable evidence of the integration of mental health in Sinhala Buddhist practices. However, it is important to note that these results come from a very small sample size and cannot be generalized to all Sri Lankan Buddhists.   

In addition, a 2009 study found that a belief in karma was correlated with poor health. However, an earlier study found a positive correlation between the reliance on the [[w:Karma_in_Buddhism|Buddhist concept of karma]] and trauma, inferencing Buddhist karma being a prevalent response to trauma&lt;ref&gt;{{Cite journal|last=Levy|first=Becca R.|last2=Slade|first2=Martin D.|last3=Ranasinghe|first3=Padmini|date=2009-03|title=Causal thinking after a tsunami wave: karma beliefs, pessimistic explanatory style and health among Sri Lankan survivors|url=https://pubmed.ncbi.nlm.nih.gov/19229624|journal=Journal of Religion and Health|volume=48|issue=1|pages=38–45|doi=10.1007/s10943-008-9162-5|issn=1573-6571|pmid=19229624}}&lt;/ref&gt;. Overall, the effectiveness of karma as a coping mechanism appears to be conflicted.   

Studies indicate that other practices of Buddhism seem to be utilized by individuals affected by the war. 40% of Sri Lankan Buddhists affected by the 2004 tsunami found the Buddhist ritual ''Bodhipuja'' to be helpful in dealing with traumatic experiences&lt;ref&gt;{{Cite web|url=https://jmvh.org/article/mental-health-and-the-role-of-cultural-and-religious-support-in-the-assistance-of-disabled-veterans-in-sri-lanka/|title=Mental Health and the Role of Cultural and Religious Support in the Assistance of Disabled Veterans in Sri Lanka|website=JMVH|language=en-US|access-date=2025-08-12}}&lt;/ref&gt;.    

=== Catholicism ===
Catholic counseling refers to &quot;a nuanced and holistic mental health care paradigm that intricately weaves together psychological science with the moral, spiritual, and pastoral traditions of the Catholic Church&quot;&lt;ref name=&quot;:13&quot;&gt;Perera, U. [https://www.researchgate.net/profile/Udeshini-Perera/publication/394095042_Catholic_Counselling_in_Sri_Lanka_Integrating_Faith_Psychology_and_Cultural_Healing/links/6889303af8031739e6098c79/Catholic-Counselling-in-Sri-Lanka-Integrating-Faith-Psychology-and-Cultural-Healing.pdf Catholic Counselling in Sri Lanka: Integrating Faith, Psychology, and Cultural Healing]. July 2025.&lt;/ref&gt; and aims to assimilate Catholic theology and evidence-based psychological treatment while including Sri Lankan cultural elements. This is achieved through emphasis on community cohesion and a locally-based understanding of &quot;personhood&quot;&lt;ref name=&quot;:13&quot; /&gt;. 

The origins of Catholic counseling trace back to the introduction of Roman Catholicism to the island in the 1600s, with the focus of the early Sri Lankan Catholic community being on &quot;[[w:Evangelism|evangelization]], education, and sacramental formation&quot;. Demand for counseling services in general increased due to the impacts of the Sri Lankan Civil War, where Catholic organizations (Caritas Sri Lanka, Seth Sarana, Subodhi Integral Centre (Piliyandala), etc.) established several Catholic-based trauma-informed programmes for victims of the Civil War. Programmes use group therapy, forgiveness rituals, and narrative repairs to alleviate war trauma. 

Examples of integration of Catholic virtues and counseling can be seen in [[w:Cognitive_Behavioral_Therapy|Cognitive Behavioral Therapy]] (CBT), where &quot;hope&quot; and &quot;humility&quot; are used as the frameworks for creating spiritual resilience&lt;ref name=&quot;:13&quot; /&gt;. The general Christian call for &quot;agape love and acceptance&quot; is echoed by the concept of [[w:Unconditional_positive_regard|unconditional positive regard]]. ''[[w:Lectio_Divina|Lectio Divina]]'' (Catholic prayer and meditation) and ''Marian devotions'' are integrated into therapeutic practices to achieve emotional regulation and mindfulness.   

Senior Lecturer [https://www.researchgate.net/profile/Udeshini-Perera Udeshini Perera] of the University of Colombo articulates a critical role of Catholic counseling. She claims that secular counseling fails to address the &quot;spiritual roots of distress and moral confusion&quot;. Catholic counseling fills in this gap by integrating &quot;psychological insights with a transcendent orientation, supporting lasting transformation and integrity&quot;&lt;ref name=&quot;:13&quot; /&gt;.  

As of 2025, no formal accreditation or standardized training exists for [[w:Pastoral_counseling|pastoral counselors]] in Sri Lanka, hampering the legitimacy of Catholic counseling. Udeshini Perera remarks that mental health stigma, lack of standardized training, research regarding Catholic counseling effectiveness, and acceptance of the combination of religion and science in a professional setting present challenges for Catholic pastoral counseling in the country. Additionally, Catholic psychiatry in Sri Lanka appears to be under-researched, and evidence of its empirical effects on followers appears sparse. Further research is needed in assessing the empirical effects of Catholic counseling in Sri Lanka.  

=== Islam ===
The literature on the empirical effects of Islamic-based psychotherapy in Sri Lanka is limited. Research is limited to a 2012 case study of a 21-year-old Muslim woman experiencing episodic possession states. The patient ceased attending psychiatric services and opted for religious rituals. The patient reported, in a follow-up visit, that the possession states had been absent for 3 months since her switch to religious rituals. The woman and her family attributed the apparent improvement of her condition to religious rituals&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|last3=Yoosuf|first3=Alam|last4=Karunaratne|first4=Sanjeewani|last5=de Silva|first5=Pushpa|date=2012|title=Religious Beliefs, Possession States, and Spirits: Three Case Studies from Sri Lanka|url=http://www.hindawi.com/journals/crips/2012/232740/|journal=Case Reports in Psychiatry|language=en|volume=2012|pages=1–3|doi=10.1155/2012/232740|issn=2090-682X|pmc=3437272|pmid=22970398}}&lt;/ref&gt;.

Future recommendations would be to conduct research on the foundations of Islamic psychiatry in the country, and to observe the rituals implemented and their effects on patients. Studies have found that Islamic prayer can be an effective means of &quot;support and coping&quot;&lt;ref name=&quot;:15&quot; /&gt;. Seven world-wide case studies using Islamic-based psychotherapy on patients, consisting of religious rituals such as scriptural reading from the [[w:Quran|Quran]], teaching of fundamental Islamic concepts (such as ''[[w:Tawakkul|tawakkul]]''), and active implementation of contemplation (''[[w:Tadabbur|tadabbur]]''), have reported positive effects in decreasing cognitive and emotional symptoms associated with &quot;religious, obsessive-compulsive disorder, depression, agoraphobia, generalized anxiety disorder, grief, and substance use disorder.”&lt;ref&gt;{{Cite journal|last=Kurhade|first=Chhaya Shantaram|last2=Jagannathan|first2=Aarti|last3=Varambally|first3=Shivarama|last4=Shivanna|first4=Sushrutha|date=2022-01|title=Religion-based interventions for mental health disorders: A systematic review|url=https://journals.lww.com/10.4103/ijoyppp.ijoyppp_14_21|journal=Journal of Applied Consciousness Studies|language=en|volume=10|issue=1|pages=20–33|doi=10.4103/ijoyppp.ijoyppp_14_21|issn=2949-6993}}&lt;/ref&gt; Additionally, a community-based study of elderly patients in Bangalore, India receiving Islamic-based psychotherapy observed decreased exhibitions of sleep disorders, eating disorders, and emotional distress&lt;ref&gt;{{Cite journal|last=Hafeez|first=Nimin|last2=Sanjay|first2=Thittamaranahalli Varadappa|last3=Puthussery|first3=Yannick Poulose|last4=Madhusudan|first4=Muralidhar|last5=Kariyappa|first5=Poornima Muddaiah|last6=Kulkarni|first6=Sridevi|last7=Raj|first7=Lavanya|date=2023-12-31|title=Spiritual practices among elderly, prevalence, pattern and associated factors: a community-based study from rural Bengaluru, India|url=https://jccpsl.sljol.info/articles/10.4038/jccpsl.v29i4.8610|journal=Journal of the College of Community Physicians of Sri Lanka|language=en|volume=29|issue=4|doi=10.4038/jccpsl.v29i4.8610|issn=1391-3174}}&lt;/ref&gt;. 

=== Hinduism ===
Despite Hindus being 12.6% of the population of Sri Lanka, the research on Hinduism-based therapy in the country is limited. Ayurvedic medicine, a form of medicine originating from ancient India, predominated the Sri Lankan medical landscape for over 2,000 years and even had a symbiotic relationship with Sinhalese medicine, which also played a significant and influential role in the country's medical framework&lt;ref name=&quot;:0&quot; /&gt;&lt;ref&gt;{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/article/10.4038/sljss.v44i1.7990/|journal=Sri Lanka Journal of Social Sciences|volume=44|issue=1|pages=33|doi=10.4038/sljss.v44i1.7990|issn=2478-1169}}&lt;/ref&gt;. Despite its historical dominance, Ayurvedic medicine has been challenged against modern evidence-based medical standards&lt;ref&gt;{{Cite book|url=https://philarchive.org/rec/DOMAAT|title=Ayurveda: Ancient Tradition or Pseudoscientific Practice? A Philosophical Inquiry|last=Dominic|first=Shubham K.}}&lt;/ref&gt;.

=== Comparative synthesis ===
Taking an overarching review of the role of religion in Sri Lanka, methods to improve mental well-being are practiced by adherents of Buddhism, Hinduism, Islam, and Christianity. These practices are implemented in traditionally-oriented mental health care, which has been reportedly preferred over psychiatric care at times. These rituals practiced across these religions indicate a common theme of psychologically integrated aspects of well-being. Interpretation of trauma is a central use in religion, with religious principles, such as karma and ''tawakkul'', serve as psychologically analogous mechanisms during times of distress.  

In terms of methodological comparisons to the studies described, qualitative interviews have documented Buddhist practices and principles, like Bodhipuja and the belief in karma, in response to traumatic events, while case studies found religious practices by other religious groups, such as a Muslim patient reading Islamic scripture and observing prayer, to reduce emotional distress. Peer-reviewed sources have documented Catholic practices and principles, such as ''Lectio Divina'' and unconditional positive regard, in improving mindfulness and emotional regulation. The paper acknowledges limitations in the evaluation of certain findings, such as in Islam and Hinduism. These shortcomings, however, are a reflection of the existing literature and its deficiencies. Empirical findings indicate mental health practices are complex and are multifaceted in their effects. 

Evidently, religion serves a parallel role to psychiatric services in improving mental health. Despite its perceived benefits, the findings surrounding religions' role in mental health suffer from conflicting, and sometimes contradictory, results. Additionally, a disproportionate amount of empirical findings seem to be Buddhist-predominant, while other religions are underrepresented in the research. Regarding research barriers, the methodological approaches implemented to study the practices of religious followers vary, though much of the research was brought from qualitative or case-based studies, impeding generalizability. Another noteworthy issue is that many studies do not utilize standardized, psychiatric measures.

== Future Outlook ==
Despite significant changes to the mental health environment in Sri Lanka, the current legal framework shaping mental health in the country has not been updated since 1956. A Cambridge University Press article detailed many limitations of the Mental Disease Ordinance of 1956, including discrepancies between the legal provisions of involuntary admissions and modern practices, potential exposure to trauma through extra-legal detentions of the mentally ill, and an absence of legal guidelines addressing the restraint of violent patients&lt;ref name=&quot;:6&quot; /&gt;. Participants from Sri Lanka reported in a comparative legislative questionnaire that they felt the mental health laws were &quot;outdated&quot; and descriptions of clinical roles remained ambiguous&lt;ref name=&quot;:16&quot; /&gt;. A draft mental health legislation from 2007 included provisions for human rights, but due to &quot;bureaucratic processes&quot; and a &quot;lack of consensus&quot;, the draft has not been officially approved.

These limitations pose challenges to the standardization of mental healthcare admissions and may impact the rights of detained patients. Detained patients may have their human rights violated due to a lack of updated legal framework, thereby impeding the identification of such violations. Additionally, with the lack of clarity on clinical roles, clinical responsibilities may not be routinely recognized and observed, leading to role confusion and potential legal ramifications&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|last=Dey|first=Sangeeta|last2=Mellsop|first2=Graham|last3=Diesfeld|first3=Kate|last4=Dharmawardene|first4=Vajira|last5=Mendis|first5=Susitha|last6=Chaudhuri|first6=Sreemanti|last7=Deb|first7=Aniruddha|last8=Huq|first8=Nafisa|last9=Ahmed|first9=Helal Uddin|date=2019-10-24|title=Comparing legislation for involuntary admission and treatment of mental illness in four South Asian countries|url=https://ijmhs.biomedcentral.com/articles/10.1186/s13033-019-0322-7|journal=International Journal of Mental Health Systems|volume=13|issue=1|pages=67|doi=10.1186/s13033-019-0322-7|issn=1752-4458|pmc=6813093|pmid=31666805}}&lt;/ref&gt;. Lastly, current efforts should ideally move beyond just addressing poverty-centered matters, and expand efforts to domestic violence victims and children with disabilities, as shelters and specialized services are limited&lt;ref name=&quot;:82&quot;&gt;{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}&lt;/ref&gt;.

Stagnation in policy development leaves Sri Lanka without a practical, up-to-date, and comprehensive mental health framework, which could put both clinicians and patients at risk. Future reforms should include clarification on the treatment and detention process of involuntary admissions of patients and a clear delineation of clinical roles and their responsibilities. Without the necessary reforms to advance Sri Lankan mental health legislation, clinicians and vulnerable patients may suffer from a lack of comprehensive oversight. 

==Acknowledgements==
No acknowledgments have been made.

==References==
{{reflist|35em}}

[[Category:Mental health]]
[[Category:Sri Lanka]]</text>
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| journal = WikiJournal of Medicine      &lt;!-- WikiJournal of Medicine, Science, or Humanities --&gt;
| last1 = Azeez
| orcid1 = 0009-0007-9202-4614
| first1 = Aaqib
| last2 = 
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| first3 = 
| last4 = 
| first4 = &lt;!-- up to 9 authors can be added in this above format --&gt;
| et_al = &lt;!-- if there are &gt;9 authors, hyperlink to the list here --&gt;
| affiliation1 = Old Dominion University
| correspondence1 = aaqib.azeez@yahoo.com
| affiliations = institutes / affiliations
| correspondence = email@address.com
| keywords = &lt;!-- up to 6 keywords --&gt;
| license = &lt;!-- default is CC-BY --&gt;
| abstract = Mental health issues continue to be a significant problem in Sri Lanka, with 2022 suicide rates in the country reporting 15 suicides per 100,000 people, above the global average of 10.5 suicides per 100,000 people. The barriers to mental healthcare on the island are multi-faceted and are best understood with historical context. This narrative review covers the historical developments of mental healthcare,  mental health impacts of historical events within the last 100 years, current challenges affecting mental health outcomes, the role of the island's major religions in mental health and mental healthcare, and recommendations for improving future mental healthcare. The author uses peer-reviewed journal articles, relevant books, historical documents, and governmental/non-governmental reports to support clinical and historical claims, though non-peer-reviewed sources were used to contextualize historical and non-clinical claims. The narrative review concludes that outdated legislation, impacts from recent conflicts or disasters, stigma surrounding mental health, and economic vulnerability contribute to mental health issues and the inefficiency of mental healthcare services. The author recommends updating legal frameworks, expanding services, and raising awareness to mitigate social stigma.
}}

== Introduction ==
Mental health continues to be a critically relevant topic as the island nation has experienced decades of [[w:Black_July|violent ethnic conflict]], terrorist attacks, alleged war crimes, and economic disruptions. Sri Lanka continues to recover from a [[w:Sri_Lankan_economic_crisis_(2019–2024)|severe economic crisis (2019 - 2024)]], a [[w:Sri_Lankan_civil_war|nearly 30-year civil war ending in 2009]], a [[w:2019_Sri_Lanka_Easter_bombings|2019 terrorist attack]], and the [[w:2004_Boxing_Day_tsunami|2004 Boxing Day tsunami]]. The exact effect these major events have had on mental health in the country is &quot;unknown&quot;, but the statistics remain concerning despite a declining trend in the overall suicide rate.

Suicide rates in the country during the mid-1990s were the second-highest in the world, with ingesting toxic products being the main suicide method. Despite the decline in suicide numbers since then—possibly attributed to Sri Lanka's ban on toxic products—evidence from a 2023 study reports an upward trend in suicide through hanging from 2016 to 2021—independent of the [[w:COVID-19_pandemic_in_Sri_Lanka|COVID-19 pandemic]]. Several risk factors for suicide, such as poverty and economic instability, are still prevalent and even increasing in the country&lt;ref&gt;{{Cite journal|last=Rajapakse|first=Thilini|last2=Silva|first2=Tharuka|last3=Hettiarachchi|first3=Nirosha Madhuwanthi|last4=Gunnell|first4=David|last5=Metcalfe|first5=Chris|last6=Spittal|first6=Matthew J.|last7=Knipe|first7=Duleeka|date=2023-01-19|title=The Impact of the COVID-19 Pandemic and Lockdowns on Self-Poisoning and Suicide in Sri Lanka: An Interrupted Time Series Analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC9914278/|journal=International Journal of Environmental Research and Public Health|volume=20|issue=3|pages=1833|doi=10.3390/ijerph20031833|issn=1660-4601|pmc=9914278|pmid=36767200}}&lt;/ref&gt;.

== Methods ==
A narrative review was conducted on mental health in Sri Lanka. Sources used included peer-reviewed journal articles, relevant books, historical documents, and governmental/non-governmental reports. These sources were found on Google Scholar, PubMed/PMC, Sri Lankan journals, and official Sri Lankan governmental websites showing relevant statistics/reports. Keywords used to conduct searches included, but were not limited to: &quot;Sri Lanka mental health&quot;, &quot;Sri Lanka civil war trauma&quot;, &quot;Sri Lanka suicide&quot;, &quot;Sri Lanka mental health ordinances&quot;, &quot;Sri Lanka religion and mental health&quot;, &quot;Sri Lanka public mental healthcare&quot;, and &quot;Sri Lanka poverty/economic crisis mental health impact.&quot; Literature that was included were relevant to the topic (Sri Lanka, South Asian mental health law, suicide, public mental health, conflict/disaster trauma, or cultural/religious practice), had full text available, and were in the English language. Non-peer-reviewed sources were primarily used to explain historical claims or contextualize non-clinical claims. 

==Historical Development of Mental Health Services==

Records attest to the care of the mentally ill through established hospitals in the island since the 4th century.&lt;ref name=&quot;:17&quot; /&gt; Prior to the incarceration of the mentally ill by the European colonizing forces, the mentally ill were regarded as ''Pissowetitch'', or people who had &quot;the spirit of the Gods within him&quot; and &quot;whatsoever he pronounceth, is looked upon as spoken by God himself, and the people will speak to him, as if it were the very person of God&quot;&lt;ref&gt;{{Cite web|url=https://www.gutenberg.org/files/14346/14346-h/14346-h.htm|title=An Historical Relation Of the Island Ceylon, in the East-Indies: Together, With an Account of the Detaining in Captivity the Author and divers other Englishmen now Living there, and of the Author’s Miraculous Escape.|last=Knox|first=Robert|website=www.gutenberg.org|language=en-us|access-date=2026-06-29}}&lt;/ref&gt;. With this religious understanding, Lucien de Alwis reasoned that the mentally ill in Sri Lanka were &quot;placed... at a higher social status than the mentally ill in the Western world&quot;, with this understanding correlating with the unsurprising absence of evidence of any &quot;large scale segregation[s] of [the] mentally ill from society&quot;&lt;ref name=&quot;:17&quot; /&gt;. 

In the 1800s, established care for mental health began shifting primarily from indigenous practices, mainly derived from [[w:Ayurveda|Ayurveda medicine]], [[w:Siddha_medicine|Siddha medicine]], and [[w:Unani_medicine|Unani medicine]], to a Western model by the British&lt;ref name=&quot;:17&quot; /&gt;&lt;ref name=&quot;:0&quot;&gt;Gambheera, H. (2011). [https://www.saarcpsychiatry.com/viewText?chapter=c6 The evolution of psychiatric services in Sri Lanka]. South Asian Journal of Psychiatry, 2(1), 25–27.&lt;/ref&gt;&lt;ref name=&quot;:15&quot;&gt;{{Cite book|url=https://doi.org/10.1007/978-981-96-8078-8_7|title=Social Psychiatry in Sri Lanka|last=Baminiwatta|first=Anuradha|last2=Williams|first2=Shehan|date=2025|publisher=Springer Nature|isbn=978-981-96-8078-8|editor-last=Arafat|editor-first=S. M. Yasir|location=Singapore|pages=141–158|language=en|doi=10.1007/978-981-96-8078-8_7|editor-last2=Singh|editor-first2=Amit|editor-last3=Kar|editor-first3=Sujita Kumar}}&lt;/ref&gt;. 

=== Adoption of a Western-based mental healthcare model and ordinances ===
In 1839, [[w:James_Alexander_Stewart-Mackenzie|James Alexander Stewart-Mackenzie]], the 7th Governor of British Ceylon, released the Lunacy Ordinance, authorizing municipal authorities to create lunatic asylums for the mentally ill&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=article&amp;id=6&amp;Itemid=125&amp;lang=en|title=History - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-10}}&lt;/ref&gt;. The ordinance was concerned with the legal frameworks of detaining individuals considered dangerous to others or individuals falsely presenting themselves as mentally ill, and not on medical treatments to alleviate the conditions of detained individuals. UK psychiatrist [[w:Edward_Mapother|Edward Mapother]] critiqued the ordinance during his 1937 inspection of British Ceylon's mental health institutions in a series of reports titled ''A Disgrace to a Civilised Community'', remarking that the ordinance &quot;[did] not seem to have contemplated treatment as a contingency to be considered&quot;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|title=Permeable walls: historical perspectives on hospital and asylum visiting|date=2009|publisher=Rodopi|isbn=978-90-420-2599-8|editor-last=Mooney|editor-first=Graham|series=Clio medica|location=Amsterdam New York, NY|editor-last2=Reinarz|editor-first2=Jonathan}}&lt;/ref&gt;.

The 1839 Ordinance was repealed and replaced by the 1840 Ordinance, which removed two requirements from the previous Ordinance: the requirement for official medical diagnoses of the mentally ill and the mandate to maintain adequate staff-to-patient ratios within lunatic asylums&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Alwis|first=L. A. P. de|last2=Seneviratne|first2=V. L.|last3=Mendis|first3=T. S. S.|last4=Abhayanayaka|first4=C.|date=2024-12-31|title=The development of laws related to the disposal of forensic patients in Sri Lanka: A historical review|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v15i2.8569|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=15|issue=2|doi=10.4038/sljpsyc.v15i2.8569|issn=2012-6883}}&lt;/ref&gt;.

In 1873, a third Ordinance was released. It included linguistic changes, where the term, &quot;insane&quot;, was replaced with &quot;of unsound mind&quot;. The Ordinance also gave more power to medical professionals in determining insanity diagnoses, and more power to detainees in appealing their commitment to the mental asylum. Despite the increased granted authority, the legal frameworks behind the detainment of the criminally insane were left identical to previous ordinances&lt;ref name=&quot;:3&quot; /&gt;.

=== Development of mental asylums ===
At the time the 1839 ordinance was released, mentally ill patients were placed either in prisons throughout the country or leprosy hospitals, such as the [[w:Hendala_Leprosy_Hospital|Hendala Leprosy Hospital]] in the Gampaha district&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;. After the creation of the first mental asylum in Borella in 1846, patients from the Hendala Leprosy Hospital were transferred to Borella. Overcrowding soon became an issue, which led to patients being sent to prisons. [[File:Edward Mapother.jpg|thumb|A portrait taken of Edward Mapother during his time working at [[w:Maudsley_Hospital|Maudsley Hospital]] in London.
]]

As medical institutions were being made to house the mentally ill, another mental asylum was created in the [[w:Cinnamon_Gardens|Cinnamon Gardens]] area of Colombo in 1884, though this mental asylum faced overcrowding issues in just one year&lt;ref name=&quot;:0&quot; /&gt;. Treatment in these asylums was limited to occupational and protection therapy, failing to provide treatment for the root causes.

In 1926, the Angoda Mental Hospital was established, marginally alleviating the severe overcrowding issues that were plaguing the preceding mental asylums. Despite the addition of 1,700 beds to the facility, treatment was still vastly limited and the patients were left in significantly poor conditions. 

=== Edward Mapother and his 1937 inspection of British Ceylon ===
Edward Mapother was born in Dublin, Ireland, on July 12, 1881 and moved to London when he was 7 years old&lt;ref&gt;{{Cite book|title=Madness to mental illness: a history of the Royal College of Psychiatrists|last=Bewley|first=Thomas|date=2008|publisher=RCPsych Publications ; Distributed in North America by Balogh International|isbn=978-1-904671-35-0|location=London : [S.l.]}}&lt;/ref&gt;. Mapother attained his M.D. in 1908. While Mapother was the Medical Superintendent of Maudsley Hospital in London, England, he was invited to inspect British Ceylon's mental health institutions by Dr S. T. Gunasekara, the first Medical Director of British Ceylon&lt;ref name=&quot;:1&quot; /&gt;.

In Mapother's visit, he commented that the Angoda Mental Hospital had the atmosphere of &quot;a prison that is neglected and dilapidated&quot;&lt;ref name=&quot;:1&quot; /&gt;. Overcrowding was still a major issue, with the institute hosting 3,000 patients—more than double the intended capacity. Patients were sleeping on mats and were clearly out of reach of adequate treatment. Mapother also noted that only 4% of public health expenditure in the country was being set for hospitals, drawing a stark comparison to London's 25%&lt;ref name=&quot;:1&quot; /&gt;. Mapother offered a vivid and grim account of the hospital in his reports:
&lt;blockquote&gt;
The floor, roof and walls of each cell consist alike of drab cement without any attempt at colouring or decoration. High up in one wall is a small window with stout iron bars. In the floor is a large hole into which the patient may pass his motion and urine. These cells are incompletely divided from one another by a partition which does not reach the roof so that the noise and stink from any one cell may reach at least all the others of the same row. Into these empty cells I was informed that the most noisy and troublesome patients in the hospital; were turned at night completely naked. The doors of the cell contain no observation window, and considering the violent character of many of these patients there is every ground for believing that the doors are rarely opened in the night by the solitary attendant on duty. It needs little imagination to picture the suffering of any patient in an early stage of bodily illness passing a night under such conditions, a situation which must frequently arise. I am told that the noise proceeding from this building is like that on a bad night in a menagerie&lt;ref name=&quot;:0&quot; /&gt;.&lt;/blockquote&gt;Mapother proposed a series of reinforcements to the legal, institutional, and medical frameworks of mental health care in British Ceylon. This included the decentralization of the psychiatric services, a reworking of the Lunacy Ordinance to incorporate treatment into the legal framework, and the establishment of a separate service of medical professionals dedicated to psychiatry. Mapother's recommendations led to several of the best local medical professionals to be sent to London for extensive training in psychiatry, while nurses from England were sent to British Ceylon to supervise hospital operations and train local staff&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;.

On August 25, 1938, the Executive Committee of Health approved the strategies proposed by Mapother, though the Government was unable to fully implement all of Mapother's interventions due to the 'heavy cost'. In fact, the Government decided to forego one of his proposals at the behest of the &quot;Visiting Committee&quot;, a committee that was tasked to &quot;meet at the hospital, carry out inspections, and make recommendations&quot; to the Executive Committee of Health&lt;ref name=&quot;:1&quot; /&gt;. The Government believed that deficiencies in their mental healthcare system could prove to be &quot;costly&quot; for their reputation, which enraged Mapother. Mapother intended to contact the Secretary of State regarding the &quot;distortion&quot; of his plans, but was interrupted by events preceding [[w:World_War_II|World War II]]&lt;ref name=&quot;:1&quot; /&gt;. Mapother passed away on March 20, 1940, without materializing his follow-up plans.

=== Post-Mapother developments and further innovations ===
[[File:Sri Lanka districts Colombo.svg|thumb|A map of Sri Lanka highlighting the Colombo District, where the capital is located. 
|right|250px]]Mapother's insights on the mental healthcare structure in British Ceylon proved to be the catalyst of significant renovations. In 1939, the first outpatient clinic was established in the [[w:National_Hospital_of_Sri_Lanka|National Hospital of Sri Lanka]] in Colombo. The first trained Ceylonese psychiatrists began practice in the 1940s, leading to the establishment of the first neuropsychiatric clinic in Colombo in 1943. Treatments for the mentally ill improved dramatically, as [[w:insulin_shock_therapy|insulin shock therapy]] and [[w:Electroconvulsive_therapy|cardiazol convulsive therapy]] were utilized&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Kathriarachchi|first=Samudra T.|last2=Seneviratne|first2=V. Lakmi|last3=Amarakoon|first3=Luckshika|date=2019-06|title=Development of Mental Health Care in Sri Lanka: Lessons Learned|url=https://journals.lww.com/tpsy/fulltext/2019/33020/development_of_mental_health_care_in_sri_lanka_.1.aspx|journal=Taiwanese Journal of Psychiatry|language=en-US|volume=33|issue=2|pages=55|doi=10.4103/TPSY.TPSY_15_19|issn=1028-3684}}&lt;/ref&gt;. Mapother's advocation for the decentralization of services were further honored through the 1947 establishment of a first child guidance clinic in Colombo General Hospital&lt;ref name=&quot;:0&quot; /&gt;. 

In 1948, British Ceylon was granted independence after the [[w:Sri_Lankan_independence_movement|Sri Lankan independence movement]]. Changes in the mental healthcare structure were not immediate following independence, but rapid expansions of mental healthcare services were continuing to actualize.  

The following decades saw positive institutional developments, such as the creation of a second hospital in [[w:Mulleriyawa|Mulleriyawa]] in 1957, and the creation of a psychiatric inpatient unit in Colombo General Hospital in 1967—effectively granting the city of Colombo the luxury of hosting the top psychiatric care in the country&lt;ref name=&quot;:5&quot;&gt;{{Cite book|url=http://link.springer.com/10.1007/978-1-4899-7999-5_4|title=Mental Health System Development in Sri Lanka|last=Minas|first=Harry|last2=Mendis|first2=Jayan|last3=Hall|first3=Teresa|date=2017|publisher=Springer US|isbn=978-1-4899-7997-1|editor-last=Minas|editor-first=Harry|location=Boston, MA|pages=59–77|language=en|doi=10.1007/978-1-4899-7999-5_4|editor-last2=Lewis|editor-first2=Milton}}&lt;/ref&gt;. The 1950s was also the start of psychopharmacological innovations, with the introduction of [[w:Lithium_(medication)|lithium]] and long-acting injectable antipsychotics ([[w:Depot_injection|depot]] [[w:Antipsychotic|neuroleptics]]) in the succeeding years&lt;ref name=&quot;:4&quot; /&gt;. Additionally, the number of public psychiatrist positions increased by 400% from 1953 to 1967&lt;ref name=&quot;:5&quot; /&gt;.

After 1960, mental health services were expanded from beyond the capital to other cities in the country&lt;ref name=&quot;:2&quot; /&gt;.

In 1980, the [[w:Postgraduate_Institute_of_Medicine|Postgraduate Institute of Medicine]] initiated a program where students would enroll in a 5-year medical course and attain an MD in psychiatry, curbing the need for Sri Lankan medical students to be sent abroad to complete their training. Many of the medical students sent abroad for training never returned to Sri Lanka to practice, resulting in a &quot;1:500,000 to 1000,000&quot; ratio of psychiatrists to patients on &quot;most occasions&quot;&lt;ref name=&quot;:0&quot; /&gt;.

=== Mental Disease Ordinance of 1956 ===
In 1956, the 1873 Ordinance was revised a second time. The Mental Disease Ordinance of 1956 featured another linguistic development, as &quot;lunacy&quot; was replaced with &quot;mental disease&quot;&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|last=Hapangama|first=Aruni|last2=Mendis|first2=Jayan|last3=Kuruppuarachchi|first3=K. a. L. A.|date=2023-02|title=Why are we still living in the past? Sri Lanka needs urgent and timely reforms of its archaic mental health laws|url=https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4|journal=BJPsych International|language=en|volume=20|issue=1|pages=4–6|doi=10.1192/bji.2022.26|issn=2056-4740|pmc=9909436|pmid=36812028}}&lt;/ref&gt;. The Ordinance paved way for community-based services to be delivered to patients closer to their residences, rather than strictly allocating services to just hospitals. This led to the creation of a [[w:WHO|WHO]]-backed community clinic near the [[w:University_of_Colombo|University of Colombo]] in the 1970s, where the focus was to eventually ease patients in the Angoda Mental Hospital back into the general population&lt;ref name=&quot;:5&quot; /&gt;.

=== Developments from the 1990s ===
The 1990s and onwards saw further positive developments in framing the mental healthcare system, including the establishment of the [https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=featured&amp;Itemid=101&amp;lang=en Directorate of Mental Health] in 1998. The Directorate of Mental Health is a part of the [[w:Ministry_of_Health_(Sri_Lanka)|Ministry of Health]] and is responsible for the monitoring and implementation of mental health programs across the country&lt;ref&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?lang=en|title=Home - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}&lt;/ref&gt;.  As of 2025, the current director of the Directorate of Mental Health is Dr. Chithramalee de Silva&lt;ref name=&quot;:2&quot; /&gt;.

On November 11, 2005, the Mental Health Policy was approved by the Government of Sri Lanka, advocating for establishments of more de-centralized, community-based mental health services across the country. The policy aimed to concisely define the rigorous standards needed to be met for each respected medical professional, including psychiatrists and clinical psychologists&lt;ref&gt;{{Cite journal|last=Rajapakshe|first=Onali Bimalka Wickramaseckara|last2=Mohan|first2=Mohapradeep|last3=Singh|first3=Swaran Preet|date=2023-05|title=Development of adolescent mental health services in Sri Lanka|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC10895478/|journal=BJPsych international|volume=20|issue=2|pages=41–43|doi=10.1192/bji.2022.32|issn=2056-4740|pmc=10895478|pmid=38414998}}&lt;/ref&gt;. The policy also included a new position, the &quot;Medical Officer of Mental Health&quot;, tasked with overseeing and assisting in creating community-based mental health services&lt;ref name=&quot;:0&quot; /&gt;. In the same year, the Sri Lankan government began implementing psychological services in state institutions, such as the military&lt;ref name=&quot;:8&quot; /&gt;.  

In 2007, the National Mental Health Advisory Council (NMHAC) was created to serve as an 'advisory' board for the Ministry of Health&lt;ref name=&quot;:7&quot;&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=article&amp;id=9&amp;Itemid=220&amp;lang=en|title=Introduction - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}&lt;/ref&gt;. 

In 2008, the Angoda Mental Hospital was restructured and renamed as the National Institute of Mental Health (NIMH)&lt;ref name=&quot;:7&quot; /&gt;. 

=== Modern-day Sri Lanka ===
[[File:Feeding Children in Sri Lanka.jpg|left|thumb|Despite the noteworthy improvements in mental healthcare services in recent decades, mental health remains a significant issue due to rising poverty. ]]
As of 2025, the Mental Health Act (mental health legislation) has been undergoing development since 2005 and is currently awaiting to be considered for the final stage of approval. This is expected to replace the 1956 Mental Health Ordinance&lt;ref name=&quot;:7&quot; /&gt;.

Currently, there are 7 tertiary care hospitals, 61 adult patient units, 3 child inpatient units, and 1 forensic unit with over 100 psychiatrists all throughout the 22 districts&lt;ref name=&quot;:4&quot; /&gt;. The [[w:Lady_Ridgeway_Hospital_for_Children|Lady Ridgeway Hospital]] in Colombo and the Sirimavo Bandaranayke Specialized Children Hospital in Kandy are specialized in treating children with [[w:Learning_disability|SLD]], [[w:ADHD|ADHD]], [[w:Autism_Spectrum_Disorder|ASD]], and provides family support for patients. As of 2017, 22 rehabilitation centers exist through the country, including 7 alcohol rehab centers&lt;ref name=&quot;:7&quot; /&gt;.   

Despite the impressive advancements in mental healthcare in the last couple of decades, Sri Lanka still suffers significant mental health issues due to increasing poverty levels in the country. The [[w:World_Bank|World Bank]] reported that [https://www.wsws.org/en/articles/2024/04/08/eesc-a08.html the poverty levels in Sri Lanka increased from 11% in 2019 to 26% in 2024], with 60% of Sri Lankan households facing &quot;decreased incomes&quot;&lt;ref&gt;Lakhtakia, Shruti, Atapattu Mudiyanselage, Udahiruni Shashadari Atapat, Walker, Richard Ancrum. ''Sri Lanka Development Update - Bridge to Recovery (English).'' Washington, D.C.: World Bank Group. &lt;nowiki&gt;http://documents.worldbank.org/curated/en/099634104012434919&lt;/nowiki&gt;&lt;/ref&gt;. This was exacerbated by Sri Lanka's excessive foreign debt, economic troubles stemming from [[w:Gotabaya_Rajapaksa|Gotabaya Rajapaksa]]'s presidential term, the COVID-19 pandemic, and the [[w:Russian_invasion_of_Ukraine|ongoing invasion of Ukraine by Russia (2022)]].  

According to [[w:NYU|New York University]] graduate student [https://gc-cuny.academia.edu/NadiaAugustyniak Nadia Augustyniak] in her 2025 overview of Sri Lanka's public mental healthcare system, poverty-induced financial precarity remains a major obstacle to receiving access to mental healthcare services. Even though trauma from adverse weather and conflict is deleterious to mental health, issues originating from every-day struggles, especially struggles related to poverty, could arguably play a more significant role&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}&lt;/ref&gt;.  

== Impact of Conflicts, Terrorism, Political Instability &amp; Natural Disasters ==

=== Sri Lankan Civil War ===
The '''Sri Lankan Civil War''' was a domestic conflict between the Sri Lankan government and the Liberation Tigers of Tamil Eelam (abbreviated as the ''LTTE),'' a militant group formed in the 1970s as a byproduct of rising tensions between the majority Sinhalese and minority Tamil population. The group is considered a terrorist organization&lt;ref&gt;{{Cite web|url=https://www.start.umd.edu/baad/database/liberation-tigers-tamil-eelam-ltte-1998.html|title=BAAD - Liberation Tigers of Tamil Eelam (LTTE) - 1998 {{!}} START.umd.edu|website=www.start.umd.edu|access-date=2025-06-09}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.cfr.org/backgrounder/liberation-tigers-tamil-eelam-aka-tamil-tigers-sri-lanka-separatists|title=Liberation Tigers of Tamil Eelam (aka Tamil Tigers) (Sri Lanka, separatists) {{!}} Council on Foreign Relations|last=Bhattacharji|first=Preeti|website=www.cfr.org|language=en|access-date=2025-06-09}}&lt;/ref&gt;. The LTTE conducted decades of massacres, assassinations of political figures, and suicide bombings to achieve ''[[w:Tamil_Eelam|Tamil Eelam]],'' leading to civilian displacement, infrastructure collapse, and the reduction of mental health services available in the northern region.[[File:DFID-funded, UNHCR emergency shelter tents, in the IDP camp at Menik Farm, Sri Lanka (3694081492).jpg|thumb|350x350px|An IDP camp in Menik Farm, Sri Lanka in 2009 ([https://www.bbc.com/news/world-asia-19703826 now closed]). Suicide rates in IDP camps were three times the general population.]]The civil war mainly affected the northeastern portion of the country, including the [[w:Vanni_(Sri_Lanka)|Vanni region]]. The conflict caused mass destruction to local mental healthcare facilities. Local residents described the conflict as ''varthayal varnicca mudiyathavai'', roughly translating into English as 'beyond description by words'&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|language=en|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}&lt;/ref&gt;. In 2003, only two psychiatrists were found in the region, operating on extremely limited resources. This furthered long-term trauma and mental health deterioration in the population&lt;ref name=&quot;:5&quot; /&gt;. 

In 2002, the humanitarian organization [https://www.msf.org/ Médecins Sans Frontières] (MSF) conducted an investigation on mental health needs in the [[w:Vavuniya|Vavuniya]] area, the site of intense conflict during the civil war (including the [[w:1985_Vavuniya_massacre|1985 Vavuniya massacre]]), and found that many of the residents suffered from high suicide rates, alcohol abuse, domestic violence, grief, and a &quot;sense of ‘learnt helplessness’&quot;&lt;ref name=&quot;:5&quot; /&gt;. A team from the University of Konstanz in Germany found that 92% of grade school children in the region were exposed to &quot;combat, shelling, and witnessing the death of loved ones&quot;&lt;ref name=&quot;:9&quot; /&gt;. 
[[File:Tractors. Jan 2009 displacement in the Vanni.jpg|left|thumb|350x350px|Displaced civilians evacuating from the Kilinochchi and Mullaitivu Districts due to military campaigns initiated by the Sri Lankan military (January 2009).]]
Additionally, accusations of war crimes have been made against [[w:War_crimes_during_the_final_stages_of_the_Sri_Lankan_civil_war|the Sri Lankan government]]&lt;ref&gt;See also [[w:Sexual violence in the Sri Lankan civil war]].&lt;/ref&gt;. A 2009 HRW report alleged that the Sri Lankan government considered the native Tamil population residing in war zones to be &quot;siding with the LTTE and [therefore, were] treated as combatants&quot;, and that the government conducted numerous shellings of &quot;areas crowded with civilians&quot;&lt;ref&gt;{{Cite journal|date=2009-02-19|title=War on the Displaced|url=https://www.hrw.org/report/2009/02/19/war-displaced/sri-lankan-army-and-ltte-abuses-against-civilians-vanni|journal=Human Rights Watch|language=en}}&lt;/ref&gt;. Furthermore, the LTTE conducted recruitment campaigns on the Vanni population where recruited men, women, and even children with minimal training, were recruited for war efforts.  

Over 200,000 Tamil civilians were moved into [[w:Internally_displaced_persons_in_Sri_Lanka|designated displacement camps during the war]], where conditions were poor&lt;ref&gt;{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000}}&lt;/ref&gt;. The suicide rate in these displacement camps was three times the community-level (2002), with a ratio of 103.5 suicides per 10,000 persons, compared to the general population's rate of 37.5 suicides per 10,000 persons. Almost all suicide attempts involved poisonous substances. Other forms of violence included domestic violence and child abuse.  

Local health officials in Vavuniya admitted that mental health concerns were a major problem, but were unable to address these concerns due to a lack of resources and support from the government. During the [[wikipedia:Sri_Lankan_civil_war#2002_peace_process_(2002%E2%80%932006)|brief 2002 ceasefire]], the MSF implemented a &quot;community-based programme&quot; which included &quot;increasing awareness, community strengthening, reinforcing coping-strategies for long-term war-affected communities, and counselling&quot;. The MSF also advocated for restrictions of poisonous substances due its means for suicide attempts, and stressed that &quot;much more [than resettlement]&quot; would need to be done to help alleviate the psychological pain the northern population had faced due to the war&lt;ref&gt;{{Cite journal|last=de Jong|first=Kaz|last2=Mulhern|first2=Maureen|last3=Ford|first3=Nathan|last4=Simpson|first4=Isabel|last5=Swan|first5=Alison|last6=van der Kam|first6=Saskia|date=2002-04|title=Psychological trauma of the civil war in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S0140673602084209|journal=The Lancet|language=en|volume=359|issue=9316|pages=1517–1518|doi=10.1016/S0140-6736(02)08420-9}}&lt;/ref&gt;. The ceasefire ended in 2006 and led to the [[w:Eelam_War_IV|final phase of the civil war]], eventually ending in 2009 with the [[w:https://en.wikipedia.org/wiki/Velupillai_Prabhakaran#Sri_Lankan_Army_Northern_offensive_and_death|death of the LTTE's leader]]. 

'''Post-war'''
[[File:Puttalam district.svg|left|thumb|Puttalam District, unlike its northern counterparts, was largely spared from the intense conflict, possibly explaining the lower rates of common mental disorders (CMDs).]]
The first district-wide cross-sectional multistage cluster sample survey was conducted in the [[w:Jaffna_District|Jaffna District]] shortly after the war ended in 2009. The study's sample included 1517 households and 2 internally displaced peoples camps. With a response rate of 92%, the study found that symptoms for PTSD were found in 7% of participants, symptoms of anxiety were found in 32.6% of participants, and symptoms of depression were found in 22.2% of participants. 2% of respondents were being placed in internally displaced peoples camps at the time of the study, 29.5% were freshly resettled from the internally displaced peoples camps, and the rest of the participants (68.5%) were never placed into camps. In comparison to residents who were never placed into camps, participants that were actively held in camps generally reported more symptoms of PTSD, anxiety, and depression. The researchers also found that women were especially vulnerable to deteriorating mental health conditions. This was explained by two factors: women having to assume the roles of both the father and the mother in the family setting after the, either voluntary or forced, departure of their husband to war, and sexist violence&lt;ref&gt;{{Cite journal|last=Husain|first=Farah|last2=Anderson|first2=Mark|last3=Lopes Cardozo|first3=Barbara|last4=Becknell|first4=Kristin|last5=Blanton|first5=Curtis|last6=Araki|first6=Diane|last7=Kottegoda Vithana|first7=Eeshara|date=2011-08-03|title=Prevalence of War-Related Mental Health Conditions and Association With Displacement Status in Postwar Jaffna District, Sri Lanka|url=https://doi.org/10.1001/jama.2011.1052|journal=JAMA|volume=306|issue=5|pages=522–531|doi=10.1001/jama.2011.1052|issn=0098-7484}}&lt;/ref&gt;. A 2013 study on adult patients in [https://www.ncbi.nlm.nih.gov/books/NBK232631/ primary care settings] (divisional hospitals, primary medical care units) found major depression to be significantly higher in females (5.1%) than males (3.6%), bolstering the findings from the 2009 study&lt;ref&gt;{{Cite journal|last=Senarath|first=Upul|last2=Wickramage|first2=Kolitha|last3=Peiris|first3=Sharika Lasanthi|date=2014-03-24|title=Prevalence of depression and its associated factors among patients attending primary care settings in the post-conflict Northern Province in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/1471-244X-14-85|journal=BMC Psychiatry|language=en|volume=14|issue=1|pages=85|doi=10.1186/1471-244X-14-85|issn=1471-244X|pmc=3987835|pmid=24661436}}&lt;/ref&gt;.

Muslims in Northern Sri Lanka also faced violence and discrimination during the conflict. Most notable incidents include [[w:Expulsion_of_Muslims_from_the_Northern_Province_of_Sri_Lanka|the October 1990 expulsion of Muslims from the North to the Puttalam District or Jaffna]] and the [[w:Kattankudy_mosque_massacre|1990 Kattankudy mosque massacre]]. The only study testing the displaced Muslim population post-civil war was completed in 2011, where a cross-sectional survey of 450 internally displaced people or people born into displacement (ages 18 - 65) revealed 18.8% of the sample suffering from common mental health disorders (CMD), including [[w:Somatoform_disorder|somatoform disorder]] (14%), &quot;other depressive syndromes&quot; (7.3%), major depression (5.1%), and anxiety disorder (2.8%). The percentages found in this study for somatoform disorder and major depression were &quot;considerably higher&quot; than the national percentages, though the researchers noted that the prevalence of CMD was lower in comparison to other countries marred with conflict, including Palestine (40.3%) and Ethiopia (27.8%). The researchers explained that the lower rate of CMD may be attributed to the [[w:Puttalam_District|serenity of the post-settlement destination]], as conflict was mainly centered in the North and East. In contrast to earlier findings, this study did not observe a higher prevalence of CMDs among women, although increased rates of somatoform disorders were noted (though the researchers did not reveal the data behind this)&lt;ref&gt;{{Cite journal|last=Siriwardhana|first=Chesmal|last2=Adikari|first2=Anushka|last3=Pannala|first3=Gayani|last4=Siribaddana|first4=Sisira|last5=Abas|first5=Melanie|last6=Sumathipala|first6=Athula|last7=Stewart|first7=Robert|date=2013-05-22|title=Prolonged Internal Displacement and Common Mental Disorders in Sri Lanka: The COMRAID Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0064742|journal=PLOS ONE|language=en|volume=8|issue=5|pages=e64742|doi=10.1371/journal.pone.0064742|issn=1932-6203|pmc=3661540|pmid=23717656}}&lt;/ref&gt;.

Research on the mental state of combatants has been limited, but a post-war 2009 study done between soldiers of the [[w:Sri_Lanka_Army_Special_Forces_Regiment|Special Forces]] and regular soldiers showed higher levels of exposure to traumatic events for units of the Special Forces, yet the former exhibited significantly less symptoms of CMDs compared to the latter. The authors of this study, [https://scholar.google.co.uk/citations?user=cVKEBdwAAAAJ&amp;hl=en&amp;oi=ao Raveen Hanwella] and [https://scholar.google.co.uk/citations?user=ZRj74qMAAAAJ&amp;hl=en&amp;oi=sra Varuni de Silva], offered the camaraderie of the military unit as an explanation for the discrepancy&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|date=2012-08|title=Mental health of Special Forces personnel deployed in battle|url=https://pubmed.ncbi.nlm.nih.gov/22038567|journal=Social Psychiatry and Psychiatric Epidemiology|volume=47|issue=8|pages=1343–1351|doi=10.1007/s00127-011-0442-0|issn=1433-9285|pmid=22038567}}&lt;/ref&gt;. A follow-up study was completed by the pair (with the addition of former Director-General of the Health Services of the Sri Lanka Navy [[w:Nicholas_Jayasekera|Nicholas Jayasekera]]), where the findings were similar, though the statistically significant bridge between the two cohorts in the previous study evaporated in the follow-up study. This may be due to the significant decline in mental health problems observed in the regular unit forces, potentially reflecting resilience in the aftermath of the conflict&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=Jayasekera|first2=Nicholas E. L. W.|last3=Silva|first3=Varuni A. de|date=2014-09-25|title=Mental Health Status of Sri Lanka Navy Personnel Three Years after End of Combat Operations: A Follow Up Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0108113|journal=PLOS ONE|language=en|volume=9|issue=9|pages=e108113|doi=10.1371/journal.pone.0108113|issn=1932-6203|pmc=4177866|pmid=25254557}}&lt;/ref&gt;. Amputees or soldiers with spinal injuries exhibited drastically different numbers, with approximately 40% of nearly 100 male-veterans in a post-war 2009 study displaying PTSD-like symptoms&lt;ref&gt;{{Cite journal|last=Abeyasinghe|first=N. L.|last2=de Zoysa|first2=P.|last3=Bandara|first3=K.M.K.C.|last4=Bartholameuz|first4=N. A.|last5=Bandara|first5=J. M.U.J.|date=2012-05-01|title=The prevalence of symptoms of Post-Traumatic Stress Disorder among soldiers with amputation of a limb or spinal injury: A report from a rehabilitation centre in Sri Lanka|url=https://doi.org/10.1080/13548506.2011.608805|journal=Psychology, Health &amp; Medicine|volume=17|issue=3|pages=376–381|doi=10.1080/13548506.2011.608805|issn=1354-8506|pmid=21942815}}&lt;/ref&gt;.

About a decade after the conflict ceased, a few notable studies have emerged to help guide understanding on the longer-term mental health effects on victims of the civil war. 

From July 2019 to October 2020, a study conducted on 585 local adolescents (ages 12-19) in the Vavuniya district revealed that despite 15.6% of the statistic having faced one or more war-related events, only 3.9% of the participants had moderate to severe depression. In addition to considerably low depression rates, only 5.7% of participants age 17+ were found to have moderate to severe hopelessness&lt;ref&gt;{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000|pmc=10472617|pmid=37653394}}&lt;/ref&gt;. The authors referenced a 2010 observation by psychiatrist [https://us.sagepub.com/en-us/nam/author/daya-somasundaram Daya Somasundaram], who noted that many Tamil IDPs presented &quot;remarkable resilience and post-traumatic growth&quot; after the civil war—an outcome he attributed to the close-knit, family-centered nature of Tamil communities&lt;ref&gt;{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}&lt;/ref&gt;.  

However, findings originating from a 2019 study, undertook by several faculty members from the University of Kelaniya, the University of Jaffna, the [[w:Gampaha_Wickramarachchi_University_of_Indigenous_Medicine|Gampaha Wickramarachchi University of Indigenous Medicine]], and the [https://onur.gov.lk/ Office for National Unity and Reconciliation (ONUR)] in Jaffna, found contrasting results. Out of 336 participants from districts which faced significant ramifications of the conflict (Jaffna, Kilinochchi, Mullaithivu, Vavuniya, and Mannar districts), 50.5% had extreme anxiety symptoms and 36.5% exhibited &quot;extremely severe&quot; symptoms of depression. 92.5% of families in the sample experienced suicidal ideation, with an observed negative correlation between trauma exposure and life satisfaction with families. Drug abuse (86.2%) and alcohol abuse (84.5%) were the two highest problematic behaviors recorded on a community-level, suggesting that the negative consequences of the civil war still persist, possibly on a substantial scale than previously recognized, in Tamil communities residing in the North&lt;ref&gt;{{Cite journal|last=Thamotharampillai|first=Umaharan|last2=Perera|first2=Ruwanthi|last3=Wickremasinghe|first3=Rajitha|last4=Williams|first4=Shehan|last5=Vijayasangar|first5=Thedsanamoorthy|last6=Sivatharsan|first6=Balasubramaniam|last7=Hilbert|first7=Vanceline|last8=Somasundaram|first8=Daya|date=2025-05-06|title=Collective Trauma- Psychosocial consequences of war in northern Sri Lanka 10 years on, a mixed methods study|url=https://www.sciencedirect.com/science/article/pii/S2666560325000696|journal=SSM - Mental Health|pages=100457|doi=10.1016/j.ssmmh.2025.100457|issn=2666-5603}}&lt;/ref&gt;. Further research should be conducted on Northern Tamil populations to assess the extent of mental health issues stemming from the conflict. 

In 2019, [https://www.researchgate.net/scientific-contributions/R-M-M-Monaragala-2087692299 Dr. R. M. M. Monaragala] conducted a study on 1,845 soldiers with combat experience, finding that 3.9% of the sample suffered from PTSD. Dr. Monaragala noted that &quot;probable depression, fatigue, aggression, and family history of mental disorder&quot; were correlative of PTSD presence. He suggested that &quot;screening and psychosocial intervention[s]&quot; could alleviate CMDs of former combatants&lt;ref&gt;{{Cite journal|last=Monaragala|first=R. M. M.|date=2024-04-19|title=Exploring the effects of the past civil war in terms of the prevalence and associating factors of PTSD|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v14i2.8465|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=14|issue=2|doi=10.4038/sljpsyc.v14i2.8465|issn=2012-6883}}&lt;/ref&gt;. 

=== 2004 Boxing Day Tsunami ===
The '''2004 Boxing Day Tsunami''' was a natural disaster where a tsunami spawned off a 9.2–9.3 magnitude earthquake off the coast of Aceh in Indonesia on December 26. The tsunami greatly affected the coastlines of the country, with the death toll reaching to around 35,000 deaths. In addition, 90,000 houses were destroyed and 516,000 people were forced to migrate due to severe infrastructural damage&lt;ref name=&quot;:5&quot; /&gt;. It stands as the [http://www.china.org.cn/english/features/tsunami_relief/119821.htm worst natural disaster to have ever hit Sri Lanka].  
[[File:Tsunami relief 2004 02.jpg|thumb|300x300px|Volunteers from [[w:Royal_College,_Colombo|Royal College in Colombo]] assisting in tsunami relief efforts (Sarvodaya Headquarters, Moratuwa).]]
A survey conducted on schoolchildren (ages 8-14) in Manadkadu (a Tamil-majority village in the northern coast), [[w:Kosgoda|Kosgoda]] (western coast), and [[w:Galle|Galle]] (southern coast), just a few weeks after the tsunami hit Sri Lanka, revealed that 33.8%, 13.9%, and 38.8% of children interviewed exhibited signs of PTSD (according to the DSM-IV's criteria), respectively (minus the time criteria, as the DSM-IV does not permit diagnosis of PTSD within 4 weeks of a traumatic incident). The loss of family members and exposure to previously traumatic incidents appeared to be highly correlate with PTSD development&lt;ref&gt;{{Cite journal|last=Neuner|first=Frank|last2=Schauer|first2=Elisabeth|last3=Catani|first3=Claudia|last4=Ruf|first4=Martina|last5=Elbert|first5=Thomas|date=2006|title=Post-tsunami stress: A study of posttraumatic stress disorder in children living in three severely affected regions in Sri Lanka|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/jts.20121|journal=Journal of Traumatic Stress|language=en|volume=19|issue=3|pages=339–347|doi=10.1002/jts.20121|issn=1573-6598}}&lt;/ref&gt;. 

Many victims in the Jaffna area suffered with &quot;[https://www.psychiatry.org/patients-families/prolonged-grief-disorder pathological grief], phobias, depression and PTSD&quot; post-tsunami. Schizophrenia in the Jaffna Tamil community, which had already suffered elevated prevalence of PTSD prior to the tsunami, had worsened—highlighting the need for specialized care in response to cumulative exposures to chronic and acute traumas. In a study published in ''International Psychiatry'' (2006), Jaffna-based researchers noted that, contrary to their initial inclinations, there was not a &quot;large[r] (than expected) rise in [the] number of people&quot; seeking mental health support 3 months after the tsunami. However, 10 months after the disaster, the researchers anticipated that &quot;more psychiatric disorders&quot; would emerge due to &quot;very little rebuilding [efforts]&quot; and an apparent &quot;unfairness in the aid system&quot;.&lt;ref&gt;{{Cite journal|last=Somasundaram|first=D. J.|last2=Yoganathan|first2=S.|last3=Ganesvaran|first3=T.|date=1993-09|title=Schizophrenia in northern Sri Lanka|url=https://pubmed.ncbi.nlm.nih.gov/7828234|journal=The Ceylon Medical Journal..|volume=38|issue=3|pages=131–135|issn=0009-0875|pmid=7828234}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Danvers|first=K.|last2=Sivayokan|first2=S.|last3=Somasundaram|first3=D. J.|last4=Sivashankar|first4=R.|date=2006-07|title=Ten months on: qualitative assessment of psychosocial issues in northern Sri Lanka following the tsunami|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6734678/|journal=International Psychiatry: Bulletin of the Board of International Affairs of the Royal College of Psychiatrists|volume=3|issue=3|pages=5–8|issn=1749-3676|pmc=6734678|pmid=31507850}}&lt;/ref&gt; 

At the February 2005 ''After the Tsunami: Mental Health Challenges to the Community for Today and Tomorrow'' conference in Thailand, [https://www.researchgate.net/profile/Chandanie-Hewage Dr. Chandanie Hewage] of the [[w:University_of_Ruhuna|University of Ruhuna]] commentated that measures taken to assist the affected were &quot;not coordinated&quot; due to poor &quot;communication systems and road [conditions].&quot; Regardless, efforts were continued by the government and health professionals to alleviate the struggles the victims were facing, including the psychological ramifications of the disaster. 

Several issues in the delivery of these services were highlighted by Dr. Hewage, including poor maintenance of health records, lack of awareness on drug consumption by the patients themselves, and shortages of health professionals. Dr. Hewage points out that personnel had &quot;little&quot; mental health training prior to the disaster, suggesting increased &quot;research&quot; and adequate &quot;provision[ing] and training of staff&quot; for the long-term&lt;ref&gt;{{Cite journal|last=Davidson|first=Jonathan R. T.|date=2006|title=Foreword. After the tsunami: mental health challenges to the community for today and tomorrow|url=https://pubmed.ncbi.nlm.nih.gov/16602809|journal=The Journal of Clinical Psychiatry|volume=67 Suppl 2|pages=3–8|issn=0160-6689|pmid=16602809}}&lt;/ref&gt;. With inadequate documentation, no systematic procedures in place, and insufficient personnel, tsunami victims with mental health concerns may not receive the services they need, further compacting neuropsychological ailments.  

In 2008 (about 3-4 years after the tsunami), researchers in the hard-hit village of [[w:Peraliya|Peraliya]] (Galle District) found that from a sample of approximately 90 adults, 25% suffered from moderate–severe PTSD, with women scoring &quot;above the cut-off for anxiety&quot; and reporting more &quot;somatic symptoms&quot;, though researchers inferred that the PTSD rate found in the study may be influenced by other factors, including war or economic hardship&lt;ref&gt;{{Cite journal|last=Hollifield|first=Michael|last2=Hewage|first2=Chandanie|last3=Gunawardena|first3=Charlotte N.|last4=Kodituwakku|first4=Piyadasa|last5=Bopagoda|first5=Kalum|last6=Weerarathnege|first6=Krishantha|last7=Group|first7=International Post-Tsunami Study|date=2008-01|title=Symptoms and coping in Sri Lanka 20–21 months after the 2004 tsunami|url=https://www.cambridge.org/core/journals/the-british-journal-of-psychiatry/article/symptoms-and-coping-in-sri-lanka-2021-months-after-the-2004-tsunami/CB33752239AF362A0BFD55B3668D60B0|journal=The British Journal of Psychiatry|language=en|volume=192|issue=1|pages=39–44|doi=10.1192/bjp.bp.107.038422|issn=0007-1250}}&lt;/ref&gt;.  

=== 2019 Easter Bombings ===
The '''2019 Easter Bombings''' were a series of coordinated attacks perpetrated by the Islamic extremist group, [[w:National_Thowheeth_Jama'ath|National Thowheeth Jama'ath]], on April 21, 2019. The attack targeted three churches and three hotels in the Colombo area, killing nearly 300 people and injuring over 500. The attacks were also attributed to the incompetency of the Sri Lankan government, who ignored [https://www.bbc.com/news/world-asia-48044636 multiple warnings preceding the attacks]. The attacks negatively affected the Sri Lankan Catholic community and further weakened relations between the major religious groups&lt;ref&gt;{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}&lt;/ref&gt;.  

In the aftermath of the attacks, professionals in the [[w:Gampaha_District|Gampaha District]] resorted to &quot;low-cost methodologies&quot; for children and adolescents affected by the attack, as a &quot;severe shortage&quot; of children and adolescent mental health experts were exposed&lt;ref&gt;{{Cite journal|last=Chandradasa|first=Miyuru|last2=Rathnayake|first2=Layani C|last3=Rowel|first3=Madushi|last4=Fernando|first4=Lalin|date=2020-06-01|title=Early phase child and adolescent psychiatry response after mass trauma: Lessons learned from the Easter Sunday attack in Sri Lanka|url=https://doi.org/10.1177/0020764020913314|journal=International Journal of Social Psychiatry|language=EN|volume=66|issue=4|pages=331–334|doi=10.1177/0020764020913314|issn=0020-7640}}&lt;/ref&gt;. In a qualitative study of 8 survivors of the attacks receiving grief counseling, [[w:University_of_Ruhuna|University of Ruhuna]] assistant professor [https://www.researchgate.net/profile/Virasha-Godakanda Virasha Godakanda] observed that 70% of the sample size expressed a lack of confidence in adequate mental health interventions from the government, reducing the quality of such services. Professor Godakanda strongly endorsed for &quot;culturally-sensitive&quot; programs, a diversity in therapeutic approaches (including nature-based therapy), and &quot;prolonged investigations&quot; to track developments in mental health resources and impacts of implemented interventions&lt;ref&gt;{{Cite journal|last=Godakanda|first=Virasha|date=2025-01-29|title=A GRIEF COUNSELING INTERVENTION AFTER THE MASS TRAUMA: LESSONS LEARNED FROM THE VICTIMS OF THE EASTER SUNDAY ATTACK IN SRI LANKA|url=https://kjmr.com.pk/kjmr/article/view/216|journal=Kashf Journal of Multidisciplinary Research|language=en|volume=2|issue=01|pages=13–32|doi=10.71146/kjmr216|issn=3007-200X}}&lt;/ref&gt;. 

A few weeks following the attacks, Muslims in Sri Lanka were subjected to [[w:2019_anti-Muslim_riots_in_Sri_Lanka|violent, coordinated riots]] masterminded by Sinhalese national forces&lt;ref&gt;{{Cite journal|last=Mujahidin|first=Muhammad Saekul|date=2023-07-03|title=Extremism and Islamophobia Against the Muslim Minority in Sri Lanka|url=https://www.ajis.org/|journal=American Journal of Islam and Society|language=en|volume=40|issue=1-2|pages=213–241|doi=10.35632/ajis.v40i1-2.3135|issn=2690-3741}}&lt;/ref&gt;. Riots were mainly centered in the [[w:Kurunegala_District|Kurunegala]], Gampaha, and [[w:Kandy_District|Kandy]] Districts. At least [https://www.aljazeera.com/news/2019/5/21/in-sri-lanka-muslims-say-sinhala-neighbours-turned-against-them one confirmed death was reported]. Calls for vague ''niqab'' and ''burqa'' bans were increasingly prominent, eventually leading to the 2021 burqa ban by the Sri Lankan government. Pakistani and Afghani refugees fleeing religious persecution in Negombo were forced to be &quot;made refugees again&quot; after local protests were orchestrated against their settlement. Anti-Muslim sentiment was &quot;unleashed online, in the law, and on the street&quot;&lt;ref&gt;{{Cite book|title=CARTOGRAPHIC JOURNEY OF RACE, GENDER AND POWER: global identity|date=2021|publisher=CAMBRIDGE SCHOLARS PUBLIS|isbn=978-1-5275-6965-2|location=S.l.}}&lt;/ref&gt;. Albeit its relevancy to the attacks, no in-depth mental health studies have took place on the minority Muslim population following the Easter bombings. Further research is imperative in exploring the sustained psychological effects of Islamophobia and its effect on the Muslim minority community in the aftermath of the 2019 Easter attacks. 

Literature on the impact of the 2019 Easter Bombings on mental health is limited and further research should be conducted.  

=== 2019-2024 Economic Crisis ===
The '''2019-2024 Economic Crisis''' refers to a 5 year period where the Sri Lankan economy experienced significant inflation and an abrupt hike in prices on basic, everyday items. It is the worse economic crisis the country has faced since the Sri Lankans were granted independence in 1948. Schools in Sri Lanka were forced to postpone examinations due to paper shortages. Gas shortages led to long lines at gas stations, some lasting for days, throughout the island. Shortages in electricity, cooking gas, and aviation feul were additional consequences of the economic crisis.

Healthcare workers faced a barrage of impediments in their line of work during the crisis, including a lopsided work-life balance due to unprecedented demand, increased stress and mental fatigue from a lack of resources and personnel, unhealthy coping mechanisms, job dissatisfaction, and a reduction in work quality. Such effects perpetuated a self-enforcing cycle of psychologically distressed mental healthcare workers providing subpar services, affecting patients and amplifying mental health issues experienced by both the workforce and their patients&lt;ref&gt;{{Cite journal|last=Dilogini|first=S.|last2=Grace|first2=H. H.|last3=Thasika|first3=T.|date=2024|title=Exploring The Mental Health and Well-Being of Public Healthcare Workers (HCWs) Amid Economic Crisis in Sri Lanka|url=http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/11092|language=en|publisher=Chartered Institute of Personnel Management}}&lt;/ref&gt;. 

Medical students from the Faculty of Medicine at the University of Colombo reported that the economic crisis forced abrupt changes in dietary consumption, increased hopelessness in the future, increased stress and anxiety, and a decrease in interest in pursuing a &quot;clinical post-graduate career&quot;&lt;ref&gt;{{Cite journal|last=Adikaranayake|first=Pesala Randika|last2=Perera|first2=Anusha Nimrod|last3=Nilaweera|first3=Akhila Imantha|last4=Fernando|first4=Desha Rajni|last5=Wijayaratne|first5=Dilushi Rowena|date=2025-07-01|title=Effects of Sri Lankan economic crisis on health, lifestyle and education of medical students in Faculty of Medicine, University of Colombo – an online survey|url=https://doi.org/10.1186/s12909-025-07506-y|journal=BMC Medical Education|language=en|volume=25|issue=1|pages=938|doi=10.1186/s12909-025-07506-y|issn=1472-6920|pmc=12211748}}&lt;/ref&gt;. 283 government-school teachers completed a web-based cross-sectional survey in April 2024, with majority of the participants reporting a severe reduction in monthly income &amp; 1/3 of participants exhibiting &quot;clinical levels of psychological distress&quot;&lt;ref&gt;{{Cite journal|last=Senevirathne|first=C. P.|last2=Senarathne|first2=D. L. P.|last3=Fernando|first3=M. S.|last4=Senevirathne|first4=S. P.|date=2025-05-28|title=Examining the economic burden and mental health distress among government school teachers in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/s40359-025-02921-8|journal=BMC Psychology|language=en|volume=13|issue=1|pages=572|doi=10.1186/s40359-025-02921-8|issn=2050-7283}}&lt;/ref&gt;. A study published in that same year reported that out of 261 nurses working in teaching hospitals, 91.6% were forced to allocate their finances to strictly &quot;general needs&quot;, while more than 50% looked into international opportunities for employment. Notably, the study reported an overall near &quot;twofold greater&quot; rate of depression, anxiety, and stress compared to previous studies on nurses in Sri Lanka&lt;ref&gt;{{Cite journal|last=Senevirathne|first=C.P|last2=Senarathne|first2=L.|last3=Fernando|first3=M.|date=2024-04-01|title=Exploring the Association Between Behavioural Modification in Response to the Prevailing Economic Crisis and Mental Health Outcomes of Nurses from Teaching Hospitals, Sri Lanka|url=https://doi.org/10.1177/23779608241272679|journal=SAGE Open Nursing|language=EN|volume=10|pages=23779608241272679|doi=10.1177/23779608241272679|issn=2377-9608|pmc=11311183}}&lt;/ref&gt;. 

The detrimental effects the crisis has had on the mental health sector reveal a concerning area of underappreciation and under compensation towards a critical sector for the well-being of the country. Adequate staffing, increased funding, and an improved work-life balance should be emphasized for the workers of health sector of the country.  

== Present-Day Challenges ==

=== Ethnic tension ===
Despite the ending of the Sri Lankan civil war and the introduction of pluralist policies (such as the [https://srilankaembassy.fr/sites/default/files/files/media/pdf/NationalPolicy-English.pdf 2017 National Policy on Reconciliation and Coexistence] under the Sirisena administration), tensions amongst members of the ethnic groups still persist. Evidence of these tensions was found in a 2022 study conducted in the Ratnapura district, where religious leaders expressed skepticism through semi-structured interviews on &quot;conflict transformation&quot;. A Tamil citizen of the Ratnapura community recounted that they were forced to &quot;hide in jungles&quot; and consume &quot;dirty water in drainage[s]&quot; due to scarcity of food and drinkable water as a result of the conflict. In certain personal accounts, ethnic conflicts appear to affect the social behavior and identity of the majority ethnic group. One Sinhala participant recounted his objection to the war-time retaliatory destruction of a shop run by a Tamil shopkeeper was met with interrogative questions about &quot;whether [he was] Sinhalese or not&quot;. Both accounts convey interethnic tensions stemming from decade-long conflicts&lt;ref&gt;Jayathilaka, Aruna &amp; Gamage, Sayuri. (2024). Role of Buddhist and Hindu Religious Leaders Role of Buddhist and Hindu Religious Leaders in the Post-War Conflict Transformation Process: A Study Based on Rathnapura District in Srilanka. ''Retrieved from'' https://gandhimargjournal.org/wp-content/uploads/2024/09/Volume-46-Issue-1-April-June-2024.pdf#page=66&lt;/ref&gt;. 

Beyond individual accounts and the official end of the civil war, the minority groups in the country continue to feel ostracized. The Sri Lankan Tamil population remains dissatisfied with the Sri Lankan government due to their alleged lack of accountability of perpetrators of war crimes and lack of information on the whereabouts of [[w:Enforced_disappearances_in_Sri_Lanka|thousands of enforced disappearances]] that took place from the 1980s. Additionally, rising anti-Muslim sentiment in recent years has contributed to increased ethnic tensions, a stark contrast to the previous centuries of peaceful co-existence between the groups. 
[[File:Bodu Bala Sena symbol.svg|thumb|The symbol for Bodu Bala Sena, a nationalistic Sinhala Buddhist group criticized for catalyzing ethnic tensions in Sri Lanka.]]
Laws passed by the Sri Lankan government, such as the [[w:Prevention_of_Terrorism_Act_(Sri_Lanka)|Prevention of Terrorism Act]] and [[wikipedia:Anti-conversion_law#Sri_Lanka|anti-conversion laws]], have forced the United States Commission on International Religious Freedom to label Sri Lanka as a nation that &quot;[engages] or [tolerates] severe violations of religious freedom&quot; in their 2024 report. The government has been criticized by human rights organizations for &quot;disproportionately targeting religious minorities&quot;&lt;ref&gt;{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}&lt;/ref&gt;. Additionally, the implementation of the three dominant languages, English, Sinhala, and Tamil, across formal education and government services have been lackadaisical, narrowing opportunities of foundational social interactions between the groups. Persistent discrimination and prejudice towards minority groups can lead to an array of complex and self-deprecating mental health issues.   

Efforts to mitigate ethnic tensions include strategies like [[w:Community-based_participatory_research|community-based participatory research]] (CBPR), task-sharing, and securing online mental health services in order to expand mental health services. However, the implementation of evidence-based plans has been met with difficulty due to inaccessibility, high costs, and shortages of adequately-trained personnel.  

Movements aiming for improved intra group and inter group coexistences, such as the Jaffna People’s Forum for Coexistence, should be emphasized on a systematic and multi-level basis, including but not limited to education, public sectors, and within communities. Pluralistic values are encouraged to be emphasized across both private and public schools to foster cultural sensitivity and tolerance. Measures should be taken against groups criticized for promoting sectarian hostility, such as the [[w:Bodu_Bala_Sena|Bodu Bala Sena]]. 

=== Poverty ===
It has been proven that poverty significantly increases the chances of developing mental illnesses. This is further amplified by possible discrimination&lt;ref&gt;{{Cite journal|last=Knifton|first=Lee|last2=Inglis|first2=Greig|date=2020-10|title=Poverty and mental health: policy, practice and research implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC7525587/|journal=BJPsych bulletin|volume=44|issue=5|pages=193–196|doi=10.1192/bjb.2020.78|issn=2056-4694|pmc=7525587|pmid=32744210}}&lt;/ref&gt;. Poverty also affects the ability for individuals with mental health concerns to receive the treatment they need. Due to the repercussions of the economic crisis, clients in Sri Lanka could not attend further counseling sessions&lt;ref name=&quot;:8&quot; /&gt;. Poverty from 2021 to 2022 [https://databankfiles.worldbank.org/public/ddpext_download/poverty/987B9C90-CB9F-4D93-AE8C-750588BF00QA/current/Global_POVEQ_LKA.pdf reportedly doubled], with future forecasts predicting the poverty line to &quot;remain above 25 percent&quot;. Suicide has been empirically linked to economic hardships in previous studies&lt;ref&gt;{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}&lt;/ref&gt;. A 2013 study done on suicidal patients in [[w:Batticaloa_Teaching_Hospital|Batticaloa Teaching Hospital]] revealed 76% of patients who attempted suicide were from rural areas while 15% were from urban areas&lt;ref&gt;{{Cite book|url=http://ir.lib.seu.ac.lk/handle/123456789/1457|title=The influence of common risk factors for the patient with attempted suicide hospitalized at the teaching hospital, Batticaloa|last=Kisokanth|first=G.|last2=Najeem|first2=M. M.|last3=Karunakaran|first3=K. E.|date=2014-08-02|publisher=South Eastern University of Sri Lanka, University Park, Oluvil #32360, Sri Lanka|isbn=978-955-627-053-2|language=en-US}}&lt;/ref&gt;. The Sri Lankan government should consider the economical impacts that poverty has on mental health and implement ways to aid poverty-stricken individuals with mental health concerns.   

=== Stigmas ===
Stigma consists of the &quot;combined effect of prejudice, ignorance and discrimination.&quot;&lt;ref name=&quot;:10&quot;&gt;{{Cite web|url=http://www.researchgate.net/publication/233990797_The_Stigma_of_Mental_Illness_in_Sri_Lanka_The_Perspectives_of_Community_Mental_Health_Workers|title=(PDF) The Stigma of Mental Illness in Sri Lanka: The Perspectives of Community Mental Health Workers|website=ResearchGate|language=en|access-date=2025-07-25}}&lt;/ref&gt;.

A 2012 interview consisting of nine participants (two doctors, three nurses, one occupational therapist, one development worker, and two volunteers) revealed a number of concerning societal viewpoints on individuals with mental health concerns. The interviews revealed that negative judgements were not only levied against the individual with the mental illness, but also the family. Families hid mentally ill family members from the public to avoid &quot;shame&quot; and possible hinderances in marriage proposals. Views that mentally ill individuals were &quot;violent&quot; served as the motivating factor behind socially isolating those with mental illness from their communities. Interviewees mentioned that individuals dealing with mental health challenges would be attacked with stones and called &quot;derogatory names.&quot; A lack of community awareness regarding mental health and negative portrayals of mentally ill individuals in media exacerbates stigmatization, though the researchers commented that the media was &quot;improving&quot; in their depiction of mental illness. Beliefs that illnesses are caused by &quot;spirits&quot; can be problematic for individuals dealing with mental health issues and suggests poor mental health awareness. Mental health workers themselves believed that they were being stigmatized, as mental health was reportedly not taken as seriously as physical health. Despite the intriguing perspectives provided, the small sample size and usage of snow sampling raise questionable concerns regarding the generalizability of the results&lt;ref name=&quot;:10&quot; /&gt;. 

Improving media portrayal of subjects concerning mental health and involving community members in interventions dealing with mental health issues are ways that could destigmatize mental health amongst communities in Sri Lanka. Tying collaborations between allopathic services and traditional healers instead of having these two services work individually could enhance engagement between traditional medicine and Western medicine.

=== Suicide Trends &amp; Risk Factors ===
Suicide is defined as &quot;the act of killing oneself deliberately, initiated and performed by the person concerned in the full knowledge or expectation of its fatal outcome&quot;&lt;ref name=&quot;:11&quot;&gt;{{Cite book|title=The neuroscience of suicidal behavior|last=Heeringen|first=Kees van|date=2018|publisher=Cambridge University Press|isbn=978-1-316-60290-4|series=Cambridge fundamentals of neuroscience in psychology|location=Cambridge, United Kingdom New York, NY, USA Port Melbourne, VIC, Australia New Delhi, India Singapore}}&lt;/ref&gt;. Although Sri Lanka has seen a significant reduction in suicide rates from the mid 1990s, largely stemming from its ban on extremely toxic pesticide products, suicide and self harm remains a significant issue. The suicide rate per 100,000 people increased from 14.0 in 2019 to [https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide 15.0 in 2022] (according to WHO). On average, 27 males per 100,000 males and 5 females per 100,000 females committed suicide in 2022&lt;ref&gt;{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}&lt;/ref&gt;. Hanging appears to be the most used method for suicide for both males and females, with studies revealing a steady increase in recent years&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Bandara|first=Piumee|last2=Wickrama|first2=Prabath|last3=Sivayokan|first3=Sambasivamoorthy|last4=Knipe|first4=Duleeka|last5=Rajapakse|first5=Thilini|date=2024-04-17|title=Reflections on the trends of suicide in Sri Lanka, 1997–2022: The need for continued vigilance|url=https://journals.plos.org/globalpublichealth/article?id=10.1371/journal.pgph.0003054|journal=PLOS Global Public Health|language=en|volume=4|issue=4|pages=e0003054|doi=10.1371/journal.pgph.0003054|issn=2767-3375|pmc=11023397|pmid=38630779}}&lt;/ref&gt;. 

From 2023 to 2024, a group of researchers from the [[w:Eastern_University,_Sri_Lanka|Eastern University in Sri Lanka]] assessed 828 patients admitted to the Teaching Hospital in [[w:Batticaloa,_Sri_Lanka|Batticaloa, Sri Lanka]] for attempted suicide. They concluded that suicide prevention programs should be attuned to younger people (ages 15 to 35 in the study), emphasize the importance of education and reducing unemployment, and increase social support in the Tamil community. Despite accounting for other factors that could lead to suicidal ideation (ie, poverty), the results from this study suffer in external validity as 90% of the patients were Tamil and over 50% were between 16 and 25 years. In addition, correlations between suicide and unemployment rates have been questioned, with [[w:Austerity|austerity]] being a more reliable indicator of suicide rates than unemployment rates&lt;ref name=&quot;:11&quot; /&gt;. Further comprehensive studies on risk factors relating to suicide should be studied to examine correlations between unemployment rates and austerity measures.  

The WHO suggests implementing evidence-based suicide prevention programs, such as [https://www.who.int/initiatives/live-life-initiative-for-suicide-prevention LIVE LIFE], to reduce the national suicide rate&lt;ref&gt;{{Cite web|url=https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide|title=World Suicide Prevention day 2024 “Changing the Narrative on Suicide”|website=www.who.int|language=en|access-date=2025-07-29}}&lt;/ref&gt;. Media depictions of suicidal methods, such as hanging, can lead to sensationalism and the media should be cautious of such displays in movies and TV shows&lt;ref name=&quot;:12&quot; /&gt;. Awareness of depression and other mental health issues can serve as a safeguard against suicidal ideation in Sri Lankan men and women.  

== Role of Religion ==
According to the last demographic report (2012), 70.2% of Sri Lankans are Buddhist, 12.6% are Hindus, 9.7% are Muslims, and 7.4% are Christians. The Theravada Buddhist community makes up the majority in several provinces throughout the country&lt;ref&gt;{{Cite web|url=https://www.state.gov/reports/2022-report-on-international-religious-freedom/sri-lanka/|title=Sri Lanka|website=United States Department of State|language=en-US|access-date=2025-08-07}}&lt;/ref&gt;. Religion, especially Theravada Buddhism, has had a significant influence on not only the historical treatment of mental health in the country, but also everyday life&lt;ref name=&quot;:15&quot; /&gt;. The [[w:Mahāvaṃsa|''Mahāvaṃsa'']] details hospitals treating patients suffering from mental health issues as early as the 4th century BC. Additionally, the 1700s Nayaka king [[w:Kirti_Sri_Rajasinha|Kirthi Sri Rajasinghe]] detailed the implementation of Buddhist philosophy in psychiatry&lt;ref name=&quot;:4&quot; /&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Alwis|first=L. A. P. De|date=2017-12-05|title=Development of civil commitment statutes (laws of involuntary detention and treatment) in Sri Lanka: a historical review|url=https://mljsl.sljol.info/articles/10.4038/mljsl.v5i1.7351|journal=Medico-Legal Journal of Sri Lanka|language=en|volume=5|issue=1|doi=10.4038/mljsl.v5i1.7351|issn=2012-8231}}&lt;/ref&gt;.

Modern-day empirical studies have attested to the usefulness of religion in mitigating stress and elevating mental health&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-94-007-4276-5_22|title=Religion and Mental Health|last=Schieman|first=Scott|last2=Bierman|first2=Alex|last3=Ellison|first3=Christopher G.|date=2013|publisher=Springer Netherlands|isbn=978-94-007-4276-5|editor-last=Aneshensel|editor-first=Carol S.|location=Dordrecht|pages=457–478|language=en|doi=10.1007/978-94-007-4276-5_22|editor-last2=Phelan|editor-first2=Jo C.|editor-last3=Bierman|editor-first3=Alex}}&lt;/ref&gt;. Religion has been found to be positively correlated with improved mental health, and more religious patients were concluded to have &quot;better mental health and adapt[ed] more quickly to health problems&quot; versus patients who weren't religious&lt;ref&gt;{{Cite journal|last=Koenig|first=Harold G.|date=2012|title=Religion, spirituality, and health: the research and clinical implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3671693/|journal=ISRN psychiatry|volume=2012|pages=278730|doi=10.5402/2012/278730|issn=2090-7966|pmc=3671693|pmid=23762764}}&lt;/ref&gt;. [https://www.researchgate.net/scientific-contributions/T-N-Wickramarathna-2247724082 Dr. Wickramarathna] of the University Psychiatry Unit (UPU) at the National Hospital of Sri Lanka (NHSL) argues that psychiatrists must strive for a balance in their approach to patients and &quot;make positive use of religion in [their] practice[s]&quot;&lt;ref&gt;{{Cite journal|last=Wickramarathna|first=T. N.|date=2022-12-31|title=Psychiatrists should stand far from the shrine: why and why not we should separate religion from psychiatry|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v13i2.8397|journal=Sri Lanka Journal of Psychiatry|language=en|volume=13|issue=2|doi=10.4038/sljpsyc.v13i2.8397|issn=2012-6883}}&lt;/ref&gt;.

=== Buddhism ===
27 Sinhalese Buddhists from four Buddhist temples were selected for a series of 70-minute interviews and focus group discussions with the aim of learning the Sinhala Buddhist understanding and experience of spiritual well-being and psychological well-being. The interviewees held spiritual wellness to be the &quot;center&quot; of overall wellness, the &quot;precondition for a successful life&quot;&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/articles/10.4038/sljss.v44i1.7990|journal=Sri Lanka Journal of Social Sciences|language=en-US|volume=44|issue=1|doi=10.4038/sljss.v44i1.7990|issn=0258-9710}}&lt;/ref&gt;. Sinhala Buddhists believe that wellness cannot be achieved without spiritual tranquility. The report states that participants emphasized that spirituality &quot;cannot be directly intervened&quot; and can only be seen through &quot;[interactions] with society&quot;&lt;ref name=&quot;:14&quot; /&gt;. Despite the ''athmaya'' (soul) being &quot;unreachable&quot;, it can be &quot;intervened&quot;, or treated, through the actions of the mind and body with society&lt;ref name=&quot;:14&quot; /&gt;. One being &quot;psychologically ill&quot; can affect one's spiritual being, as the participants reported in their interviews, and can be impaired through &quot;lifestyle stressors, environmental and socio-cultural causes, non-human related causes and bad-karma in the past lives&quot;&lt;ref name=&quot;:14&quot; /&gt;.  

The researchers concluded that despite Sinhala Buddhists not being able to articulately decipher the discrepancies between psychological well-being and spiritual well-being, they are able to conceptualize and maintain a culturally embedded understanding between the two, serving as reputable evidence of the integration of mental health in Sinhala Buddhist practices. However, it is important to note that these results come from a very small sample size and cannot be generalized to all Sri Lankan Buddhists.   

In addition, a 2009 study found that a belief in karma was correlated with poor health. However, an earlier study found a positive correlation between the reliance on the [[w:Karma_in_Buddhism|Buddhist concept of karma]] and trauma, inferencing Buddhist karma being a prevalent response to trauma&lt;ref&gt;{{Cite journal|last=Levy|first=Becca R.|last2=Slade|first2=Martin D.|last3=Ranasinghe|first3=Padmini|date=2009-03|title=Causal thinking after a tsunami wave: karma beliefs, pessimistic explanatory style and health among Sri Lankan survivors|url=https://pubmed.ncbi.nlm.nih.gov/19229624|journal=Journal of Religion and Health|volume=48|issue=1|pages=38–45|doi=10.1007/s10943-008-9162-5|issn=1573-6571|pmid=19229624}}&lt;/ref&gt;. Overall, the effectiveness of karma as a coping mechanism appears to be conflicted.   

Studies indicate that other practices of Buddhism seem to be utilized by individuals affected by the war. 40% of Sri Lankan Buddhists affected by the 2004 tsunami found the Buddhist ritual ''Bodhipuja'' to be helpful in dealing with traumatic experiences&lt;ref&gt;{{Cite web|url=https://jmvh.org/article/mental-health-and-the-role-of-cultural-and-religious-support-in-the-assistance-of-disabled-veterans-in-sri-lanka/|title=Mental Health and the Role of Cultural and Religious Support in the Assistance of Disabled Veterans in Sri Lanka|website=JMVH|language=en-US|access-date=2025-08-12}}&lt;/ref&gt;.    

=== Catholicism ===
Catholic counseling refers to &quot;a nuanced and holistic mental health care paradigm that intricately weaves together psychological science with the moral, spiritual, and pastoral traditions of the Catholic Church&quot;&lt;ref name=&quot;:13&quot;&gt;Perera, U. [https://www.researchgate.net/profile/Udeshini-Perera/publication/394095042_Catholic_Counselling_in_Sri_Lanka_Integrating_Faith_Psychology_and_Cultural_Healing/links/6889303af8031739e6098c79/Catholic-Counselling-in-Sri-Lanka-Integrating-Faith-Psychology-and-Cultural-Healing.pdf Catholic Counselling in Sri Lanka: Integrating Faith, Psychology, and Cultural Healing]. July 2025.&lt;/ref&gt; and aims to assimilate Catholic theology and evidence-based psychological treatment while including Sri Lankan cultural elements. This is achieved through emphasis on community cohesion and a locally-based understanding of &quot;personhood&quot;&lt;ref name=&quot;:13&quot; /&gt;. 

The origins of Catholic counseling trace back to the introduction of Roman Catholicism to the island in the 1600s, with the focus of the early Sri Lankan Catholic community being on &quot;[[w:Evangelism|evangelization]], education, and sacramental formation&quot;. Demand for counseling services in general increased due to the impacts of the Sri Lankan Civil War, where Catholic organizations (Caritas Sri Lanka, Seth Sarana, Subodhi Integral Centre (Piliyandala), etc.) established several Catholic-based trauma-informed programmes for victims of the Civil War. Programmes use group therapy, forgiveness rituals, and narrative repairs to alleviate war trauma. 

Examples of integration of Catholic virtues and counseling can be seen in [[w:Cognitive_Behavioral_Therapy|Cognitive Behavioral Therapy]] (CBT), where &quot;hope&quot; and &quot;humility&quot; are used as the frameworks for creating spiritual resilience&lt;ref name=&quot;:13&quot; /&gt;. The general Christian call for &quot;agape love and acceptance&quot; is echoed by the concept of [[w:Unconditional_positive_regard|unconditional positive regard]]. ''[[w:Lectio_Divina|Lectio Divina]]'' (Catholic prayer and meditation) and ''Marian devotions'' are integrated into therapeutic practices to achieve emotional regulation and mindfulness.   

Senior Lecturer [https://www.researchgate.net/profile/Udeshini-Perera Udeshini Perera] of the University of Colombo articulates a critical role of Catholic counseling. She claims that secular counseling fails to address the &quot;spiritual roots of distress and moral confusion&quot;. Catholic counseling fills in this gap by integrating &quot;psychological insights with a transcendent orientation, supporting lasting transformation and integrity&quot;&lt;ref name=&quot;:13&quot; /&gt;.  

As of 2025, no formal accreditation or standardized training exists for [[w:Pastoral_counseling|pastoral counselors]] in Sri Lanka, hampering the legitimacy of Catholic counseling. Udeshini Perera remarks that mental health stigma, lack of standardized training, research regarding Catholic counseling effectiveness, and acceptance of the combination of religion and science in a professional setting present challenges for Catholic pastoral counseling in the country. Additionally, Catholic psychiatry in Sri Lanka appears to be under-researched, and evidence of its empirical effects on followers appears sparse. Further research is needed in assessing the empirical effects of Catholic counseling in Sri Lanka.  

=== Islam ===
The literature on the empirical effects of Islamic-based psychotherapy in Sri Lanka is limited. Research is limited to a 2012 case study of a 21-year-old Muslim woman experiencing episodic possession states. The patient ceased attending psychiatric services and opted for religious rituals. The patient reported, in a follow-up visit, that the possession states had been absent for 3 months since her switch to religious rituals. The woman and her family attributed the apparent improvement of her condition to religious rituals&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|last3=Yoosuf|first3=Alam|last4=Karunaratne|first4=Sanjeewani|last5=de Silva|first5=Pushpa|date=2012|title=Religious Beliefs, Possession States, and Spirits: Three Case Studies from Sri Lanka|url=http://www.hindawi.com/journals/crips/2012/232740/|journal=Case Reports in Psychiatry|language=en|volume=2012|pages=1–3|doi=10.1155/2012/232740|issn=2090-682X|pmc=3437272|pmid=22970398}}&lt;/ref&gt;.

Future recommendations would be to conduct research on the foundations of Islamic psychiatry in the country, and to observe the rituals implemented and their effects on patients. Studies have found that Islamic prayer can be an effective means of &quot;support and coping&quot;&lt;ref name=&quot;:15&quot; /&gt;. Seven world-wide case studies using Islamic-based psychotherapy on patients, consisting of religious rituals such as scriptural reading from the [[w:Quran|Quran]], teaching of fundamental Islamic concepts (such as ''[[w:Tawakkul|tawakkul]]''), and active implementation of contemplation (''[[w:Tadabbur|tadabbur]]''), have reported positive effects in decreasing cognitive and emotional symptoms associated with &quot;religious, obsessive-compulsive disorder, depression, agoraphobia, generalized anxiety disorder, grief, and substance use disorder.”&lt;ref&gt;{{Cite journal|last=Kurhade|first=Chhaya Shantaram|last2=Jagannathan|first2=Aarti|last3=Varambally|first3=Shivarama|last4=Shivanna|first4=Sushrutha|date=2022-01|title=Religion-based interventions for mental health disorders: A systematic review|url=https://journals.lww.com/10.4103/ijoyppp.ijoyppp_14_21|journal=Journal of Applied Consciousness Studies|language=en|volume=10|issue=1|pages=20–33|doi=10.4103/ijoyppp.ijoyppp_14_21|issn=2949-6993}}&lt;/ref&gt; Additionally, a community-based study of elderly patients in Bangalore, India receiving Islamic-based psychotherapy observed decreased exhibitions of sleep disorders, eating disorders, and emotional distress&lt;ref&gt;{{Cite journal|last=Hafeez|first=Nimin|last2=Sanjay|first2=Thittamaranahalli Varadappa|last3=Puthussery|first3=Yannick Poulose|last4=Madhusudan|first4=Muralidhar|last5=Kariyappa|first5=Poornima Muddaiah|last6=Kulkarni|first6=Sridevi|last7=Raj|first7=Lavanya|date=2023-12-31|title=Spiritual practices among elderly, prevalence, pattern and associated factors: a community-based study from rural Bengaluru, India|url=https://jccpsl.sljol.info/articles/10.4038/jccpsl.v29i4.8610|journal=Journal of the College of Community Physicians of Sri Lanka|language=en|volume=29|issue=4|doi=10.4038/jccpsl.v29i4.8610|issn=1391-3174}}&lt;/ref&gt;. 

=== Hinduism ===
Despite Hindus being 12.6% of the population of Sri Lanka, the research on Hinduism-based therapy in the country is limited. Ayurvedic medicine, a form of medicine originating from ancient India, predominated the Sri Lankan medical landscape for over 2,000 years and even had a symbiotic relationship with Sinhalese medicine, which also played a significant and influential role in the country's medical framework&lt;ref name=&quot;:0&quot; /&gt;&lt;ref&gt;{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/article/10.4038/sljss.v44i1.7990/|journal=Sri Lanka Journal of Social Sciences|volume=44|issue=1|pages=33|doi=10.4038/sljss.v44i1.7990|issn=2478-1169}}&lt;/ref&gt;. Despite its historical dominance, Ayurvedic medicine has been challenged against modern evidence-based medical standards&lt;ref&gt;{{Cite book|url=https://philarchive.org/rec/DOMAAT|title=Ayurveda: Ancient Tradition or Pseudoscientific Practice? A Philosophical Inquiry|last=Dominic|first=Shubham K.}}&lt;/ref&gt;.

=== Comparative synthesis ===
Taking an overarching review of the role of religion in Sri Lanka, methods to improve mental well-being are practiced by adherents of Buddhism, Hinduism, Islam, and Christianity. These practices are implemented in traditionally-oriented mental health care, which has been reportedly preferred over psychiatric care at times. These rituals practiced across these religions indicate a common theme of psychologically integrated aspects of well-being. Interpretation of trauma is a central use in religion, with religious principles, such as karma and ''tawakkul'', serve as psychologically analogous mechanisms during times of distress.  

In terms of methodological comparisons to the studies described, qualitative interviews have documented Buddhist practices and principles, like Bodhipuja and the belief in karma, in response to traumatic events, while case studies found religious practices by other religious groups, such as a Muslim patient reading Islamic scripture and observing prayer, to reduce emotional distress. Peer-reviewed sources have documented Catholic practices and principles, such as ''Lectio Divina'' and unconditional positive regard, in improving mindfulness and emotional regulation. The paper acknowledges limitations in the evaluation of certain findings, such as in Islam and Hinduism. These shortcomings, however, are a reflection of the existing literature and its deficiencies. Empirical findings indicate mental health practices are complex and are multifaceted in their effects. 

Evidently, religion serves a parallel role to psychiatric services in improving mental health. Despite its perceived benefits, the findings surrounding religions' role in mental health suffer from conflicting, and sometimes contradictory, results. Additionally, a disproportionate amount of empirical findings seem to be Buddhist-predominant, while other religions are underrepresented in the research. Regarding research barriers, the methodological approaches implemented to study the practices of religious followers vary, though much of the research was brought from qualitative or case-based studies, impeding generalizability. Another noteworthy issue is that many studies do not utilize standardized, psychiatric measures.

== Future Outlook ==
Despite significant changes to the mental health environment in Sri Lanka, the current legal framework shaping mental health in the country has not been updated since 1956. A Cambridge University Press article detailed many limitations of the Mental Disease Ordinance of 1956, including discrepancies between the legal provisions of involuntary admissions and modern practices, potential exposure to trauma through extra-legal detentions of the mentally ill, and an absence of legal guidelines addressing the restraint of violent patients&lt;ref name=&quot;:6&quot; /&gt;. Participants from Sri Lanka reported in a comparative legislative questionnaire that they felt the mental health laws were &quot;outdated&quot; and descriptions of clinical roles remained ambiguous&lt;ref name=&quot;:16&quot; /&gt;. A draft mental health legislation from 2007 included provisions for human rights, but due to &quot;bureaucratic processes&quot; and a &quot;lack of consensus&quot;, the draft has not been officially approved.

These limitations pose challenges to the standardization of mental healthcare admissions and may impact the rights of detained patients. Detained patients may have their human rights violated due to a lack of updated legal framework, thereby impeding the identification of such violations. Additionally, with the lack of clarity on clinical roles, clinical responsibilities may not be routinely recognized and observed, leading to role confusion and potential legal ramifications&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|last=Dey|first=Sangeeta|last2=Mellsop|first2=Graham|last3=Diesfeld|first3=Kate|last4=Dharmawardene|first4=Vajira|last5=Mendis|first5=Susitha|last6=Chaudhuri|first6=Sreemanti|last7=Deb|first7=Aniruddha|last8=Huq|first8=Nafisa|last9=Ahmed|first9=Helal Uddin|date=2019-10-24|title=Comparing legislation for involuntary admission and treatment of mental illness in four South Asian countries|url=https://ijmhs.biomedcentral.com/articles/10.1186/s13033-019-0322-7|journal=International Journal of Mental Health Systems|volume=13|issue=1|pages=67|doi=10.1186/s13033-019-0322-7|issn=1752-4458|pmc=6813093|pmid=31666805}}&lt;/ref&gt;. Lastly, current efforts should ideally move beyond just addressing poverty-centered matters, and expand efforts to domestic violence victims and children with disabilities, as shelters and specialized services are limited&lt;ref name=&quot;:82&quot;&gt;{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}&lt;/ref&gt;.

Stagnation in policy development leaves Sri Lanka without a practical, up-to-date, and comprehensive mental health framework, which could put both clinicians and patients at risk. Future reforms should include clarification on the treatment and detention process of involuntary admissions of patients and a clear delineation of clinical roles and their responsibilities. Without the necessary reforms to advance Sri Lankan mental health legislation, clinicians and vulnerable patients may suffer from a lack of comprehensive oversight. 

==Acknowledgements==
No acknowledgments have been made.

==References==
{{reflist|35em}}

[[Category:Mental health]]
[[Category:Sri Lanka]]</text>
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| journal = WikiJournal of Medicine      &lt;!-- WikiJournal of Medicine, Science, or Humanities --&gt;
| last1 = Azeez
| orcid1 = 0009-0007-9202-4614
| first1 = Aaqib
| last2 = 
| first2 = 
| last3 = 
| first3 = 
| last4 = 
| first4 = &lt;!-- up to 9 authors can be added in this above format --&gt;
| et_al = &lt;!-- if there are &gt;9 authors, hyperlink to the list here --&gt;
| affiliation1 = Old Dominion University
| correspondence1 = aaqib.azeez@yahoo.com
| affiliations = institutes / affiliations
| correspondence = email@address.com
| keywords = &lt;!-- up to 6 keywords --&gt;
| license = &lt;!-- default is CC-BY --&gt;
| abstract = Mental health issues continue to be a significant problem in Sri Lanka, with 2022 suicide rates in the country reporting 15 suicides per 100,000 people, above the global average of 10.5 suicides per 100,000 people. The barriers to mental healthcare on the island are multi-faceted and are best understood with historical context. This narrative review covers the historical developments of mental healthcare,  mental health impacts of historical events within the last 100 years, current challenges affecting mental health outcomes, the role of the island's major religions in mental health and mental healthcare, and recommendations for improving future mental healthcare. The author uses peer-reviewed journal articles, relevant books, historical documents, and governmental/non-governmental reports to support clinical and historical claims, though non-peer-reviewed sources were used to contextualize historical and non-clinical claims. The narrative review concludes that outdated legislation, impacts from recent conflicts or disasters, stigma surrounding mental health, and economic vulnerability contribute to mental health issues and the inefficiency of mental healthcare services. The author recommends updating legal frameworks, expanding services, and raising awareness to mitigate social stigma.
}}

== Introduction ==
Mental health continues to be a critically relevant topic as the island nation has experienced decades of [[w:Black_July|violent ethnic conflict]], terrorist attacks, alleged war crimes, and economic disruptions. Sri Lanka continues to recover from a [[w:Sri_Lankan_economic_crisis_(2019–2024)|severe economic crisis (2019 - 2024)]], a [[w:Sri_Lankan_civil_war|nearly 30-year civil war ending in 2009]], a [[w:2019_Sri_Lanka_Easter_bombings|2019 terrorist attack]], and the [[w:2004_Boxing_Day_tsunami|2004 Boxing Day tsunami]]. The exact effect these major events have had on mental health in the country is &quot;unknown&quot;, but the statistics remain concerning despite a declining trend in the overall suicide rate.

Suicide rates in the country during the mid-1990s were the second-highest in the world, with ingesting toxic products being the main suicide method. Despite the decline in suicide numbers since then—possibly attributed to Sri Lanka's ban on toxic products—evidence from a 2023 study reports an upward trend in suicide through hanging from 2016 to 2021—independent of the [[w:COVID-19_pandemic_in_Sri_Lanka|COVID-19 pandemic]]. Several risk factors for suicide, such as poverty and economic instability, are still prevalent and even increasing in the country&lt;ref&gt;{{Cite journal|last=Rajapakse|first=Thilini|last2=Silva|first2=Tharuka|last3=Hettiarachchi|first3=Nirosha Madhuwanthi|last4=Gunnell|first4=David|last5=Metcalfe|first5=Chris|last6=Spittal|first6=Matthew J.|last7=Knipe|first7=Duleeka|date=2023-01-19|title=The Impact of the COVID-19 Pandemic and Lockdowns on Self-Poisoning and Suicide in Sri Lanka: An Interrupted Time Series Analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC9914278/|journal=International Journal of Environmental Research and Public Health|volume=20|issue=3|pages=1833|doi=10.3390/ijerph20031833|issn=1660-4601|pmc=9914278|pmid=36767200}}&lt;/ref&gt;.

== Methods ==
A narrative review was conducted on mental health in Sri Lanka. Sources used included peer-reviewed journal articles, relevant books, historical documents, and governmental/non-governmental reports. These sources were found on Google Scholar, PubMed/PMC, Sri Lankan journals, and official Sri Lankan governmental websites showing relevant statistics/reports. Keywords used to conduct searches included, but were not limited to: &quot;Sri Lanka mental health&quot;, &quot;Sri Lanka civil war trauma&quot;, &quot;Sri Lanka suicide&quot;, &quot;Sri Lanka mental health ordinances&quot;, &quot;Sri Lanka religion and mental health&quot;, &quot;Sri Lanka public mental healthcare&quot;, and &quot;Sri Lanka poverty/economic crisis mental health impact.&quot; Literature included were relevant to the topic (Sri Lanka, South Asian mental health law, suicide, public mental health, conflict/disaster trauma, or cultural/religious practice), had full text available, and were in the English language. Non-peer-reviewed sources were primarily used to explain historical claims or contextualize non-clinical claims. 

==Historical Development of Mental Health Services==

Records attest to the care of the mentally ill through established hospitals in the island since the 4th century.&lt;ref name=&quot;:17&quot; /&gt; Prior to the incarceration of the mentally ill by the European colonizing forces, the mentally ill were regarded as ''Pissowetitch'', or people who had &quot;the spirit of the Gods within him&quot; and &quot;whatsoever he pronounceth, is looked upon as spoken by God himself, and the people will speak to him, as if it were the very person of God&quot;&lt;ref&gt;{{Cite web|url=https://www.gutenberg.org/files/14346/14346-h/14346-h.htm|title=An Historical Relation Of the Island Ceylon, in the East-Indies: Together, With an Account of the Detaining in Captivity the Author and divers other Englishmen now Living there, and of the Author’s Miraculous Escape.|last=Knox|first=Robert|website=www.gutenberg.org|language=en-us|access-date=2026-06-29}}&lt;/ref&gt;. With this religious understanding, Lucien de Alwis reasoned that the mentally ill in Sri Lanka were &quot;placed... at a higher social status than the mentally ill in the Western world&quot;, with this understanding correlating with the unsurprising absence of evidence of any &quot;large scale segregation[s] of [the] mentally ill from society&quot;&lt;ref name=&quot;:17&quot; /&gt;. 

In the 1800s, established care for mental health began shifting primarily from indigenous practices, mainly derived from [[w:Ayurveda|Ayurveda medicine]], [[w:Siddha_medicine|Siddha medicine]], and [[w:Unani_medicine|Unani medicine]], to a Western model by the British&lt;ref name=&quot;:17&quot; /&gt;&lt;ref name=&quot;:0&quot;&gt;Gambheera, H. (2011). [https://www.saarcpsychiatry.com/viewText?chapter=c6 The evolution of psychiatric services in Sri Lanka]. South Asian Journal of Psychiatry, 2(1), 25–27.&lt;/ref&gt;&lt;ref name=&quot;:15&quot;&gt;{{Cite book|url=https://doi.org/10.1007/978-981-96-8078-8_7|title=Social Psychiatry in Sri Lanka|last=Baminiwatta|first=Anuradha|last2=Williams|first2=Shehan|date=2025|publisher=Springer Nature|isbn=978-981-96-8078-8|editor-last=Arafat|editor-first=S. M. Yasir|location=Singapore|pages=141–158|language=en|doi=10.1007/978-981-96-8078-8_7|editor-last2=Singh|editor-first2=Amit|editor-last3=Kar|editor-first3=Sujita Kumar}}&lt;/ref&gt;. 

=== Adoption of a Western-based mental healthcare model and ordinances ===
In 1839, [[w:James_Alexander_Stewart-Mackenzie|James Alexander Stewart-Mackenzie]], the 7th Governor of British Ceylon, released the Lunacy Ordinance, authorizing municipal authorities to create lunatic asylums for the mentally ill&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=article&amp;id=6&amp;Itemid=125&amp;lang=en|title=History - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-10}}&lt;/ref&gt;. The ordinance was concerned with the legal frameworks of detaining individuals considered dangerous to others or individuals falsely presenting themselves as mentally ill, and not on medical treatments to alleviate the conditions of detained individuals. UK psychiatrist [[w:Edward_Mapother|Edward Mapother]] critiqued the ordinance during his 1937 inspection of British Ceylon's mental health institutions in a series of reports titled ''A Disgrace to a Civilised Community'', remarking that the ordinance &quot;[did] not seem to have contemplated treatment as a contingency to be considered&quot;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|title=Permeable walls: historical perspectives on hospital and asylum visiting|date=2009|publisher=Rodopi|isbn=978-90-420-2599-8|editor-last=Mooney|editor-first=Graham|series=Clio medica|location=Amsterdam New York, NY|editor-last2=Reinarz|editor-first2=Jonathan}}&lt;/ref&gt;.

The 1839 Ordinance was repealed and replaced by the 1840 Ordinance, which removed two requirements from the previous Ordinance: the requirement for official medical diagnoses of the mentally ill and the mandate to maintain adequate staff-to-patient ratios within lunatic asylums&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Alwis|first=L. A. P. de|last2=Seneviratne|first2=V. L.|last3=Mendis|first3=T. S. S.|last4=Abhayanayaka|first4=C.|date=2024-12-31|title=The development of laws related to the disposal of forensic patients in Sri Lanka: A historical review|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v15i2.8569|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=15|issue=2|doi=10.4038/sljpsyc.v15i2.8569|issn=2012-6883}}&lt;/ref&gt;.

In 1873, a third Ordinance was released. It included linguistic changes, where the term, &quot;insane&quot;, was replaced with &quot;of unsound mind&quot;. The Ordinance also gave more power to medical professionals in determining insanity diagnoses, and more power to detainees in appealing their commitment to the mental asylum. Despite the increased granted authority, the legal frameworks behind the detainment of the criminally insane were left identical to previous ordinances&lt;ref name=&quot;:3&quot; /&gt;.

=== Development of mental asylums ===
At the time the 1839 ordinance was released, mentally ill patients were placed either in prisons throughout the country or leprosy hospitals, such as the [[w:Hendala_Leprosy_Hospital|Hendala Leprosy Hospital]] in the Gampaha district&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;. After the creation of the first mental asylum in Borella in 1846, patients from the Hendala Leprosy Hospital were transferred to Borella. Overcrowding soon became an issue, which led to patients being sent to prisons. [[File:Edward Mapother.jpg|thumb|A portrait taken of Edward Mapother during his time working at [[w:Maudsley_Hospital|Maudsley Hospital]] in London.
]]

As medical institutions were being made to house the mentally ill, another mental asylum was created in the [[w:Cinnamon_Gardens|Cinnamon Gardens]] area of Colombo in 1884, though this mental asylum faced overcrowding issues in just one year&lt;ref name=&quot;:0&quot; /&gt;. Treatment in these asylums was limited to occupational and protection therapy, failing to provide treatment for the root causes.

In 1926, the Angoda Mental Hospital was established, marginally alleviating the severe overcrowding issues that were plaguing the preceding mental asylums. Despite the addition of 1,700 beds to the facility, treatment was still vastly limited and the patients were left in significantly poor conditions. 

=== Edward Mapother and his 1937 inspection of British Ceylon ===
Edward Mapother was born in Dublin, Ireland, on July 12, 1881 and moved to London when he was 7 years old&lt;ref&gt;{{Cite book|title=Madness to mental illness: a history of the Royal College of Psychiatrists|last=Bewley|first=Thomas|date=2008|publisher=RCPsych Publications ; Distributed in North America by Balogh International|isbn=978-1-904671-35-0|location=London : [S.l.]}}&lt;/ref&gt;. Mapother attained his M.D. in 1908. While Mapother was the Medical Superintendent of Maudsley Hospital in London, England, he was invited to inspect British Ceylon's mental health institutions by Dr S. T. Gunasekara, the first Medical Director of British Ceylon&lt;ref name=&quot;:1&quot; /&gt;.

In Mapother's visit, he commented that the Angoda Mental Hospital had the atmosphere of &quot;a prison that is neglected and dilapidated&quot;&lt;ref name=&quot;:1&quot; /&gt;. Overcrowding was still a major issue, with the institute hosting 3,000 patients—more than double the intended capacity. Patients were sleeping on mats and were clearly out of reach of adequate treatment. Mapother also noted that only 4% of public health expenditure in the country was being set for hospitals, drawing a stark comparison to London's 25%&lt;ref name=&quot;:1&quot; /&gt;. Mapother offered a vivid and grim account of the hospital in his reports:
&lt;blockquote&gt;
The floor, roof and walls of each cell consist alike of drab cement without any attempt at colouring or decoration. High up in one wall is a small window with stout iron bars. In the floor is a large hole into which the patient may pass his motion and urine. These cells are incompletely divided from one another by a partition which does not reach the roof so that the noise and stink from any one cell may reach at least all the others of the same row. Into these empty cells I was informed that the most noisy and troublesome patients in the hospital; were turned at night completely naked. The doors of the cell contain no observation window, and considering the violent character of many of these patients there is every ground for believing that the doors are rarely opened in the night by the solitary attendant on duty. It needs little imagination to picture the suffering of any patient in an early stage of bodily illness passing a night under such conditions, a situation which must frequently arise. I am told that the noise proceeding from this building is like that on a bad night in a menagerie&lt;ref name=&quot;:0&quot; /&gt;.&lt;/blockquote&gt;Mapother proposed a series of reinforcements to the legal, institutional, and medical frameworks of mental health care in British Ceylon. This included the decentralization of the psychiatric services, a reworking of the Lunacy Ordinance to incorporate treatment into the legal framework, and the establishment of a separate service of medical professionals dedicated to psychiatry. Mapother's recommendations led to several of the best local medical professionals to be sent to London for extensive training in psychiatry, while nurses from England were sent to British Ceylon to supervise hospital operations and train local staff&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;.

On August 25, 1938, the Executive Committee of Health approved the strategies proposed by Mapother, though the Government was unable to fully implement all of Mapother's interventions due to the 'heavy cost'. In fact, the Government decided to forego one of his proposals at the behest of the &quot;Visiting Committee&quot;, a committee that was tasked to &quot;meet at the hospital, carry out inspections, and make recommendations&quot; to the Executive Committee of Health&lt;ref name=&quot;:1&quot; /&gt;. The Government believed that deficiencies in their mental healthcare system could prove to be &quot;costly&quot; for their reputation, which enraged Mapother. Mapother intended to contact the Secretary of State regarding the &quot;distortion&quot; of his plans, but was interrupted by events preceding [[w:World_War_II|World War II]]&lt;ref name=&quot;:1&quot; /&gt;. Mapother passed away on March 20, 1940, without materializing his follow-up plans.

=== Post-Mapother developments and further innovations ===
[[File:Sri Lanka districts Colombo.svg|thumb|A map of Sri Lanka highlighting the Colombo District, where the capital is located. 
|right|250px]]Mapother's insights on the mental healthcare structure in British Ceylon proved to be the catalyst of significant renovations. In 1939, the first outpatient clinic was established in the [[w:National_Hospital_of_Sri_Lanka|National Hospital of Sri Lanka]] in Colombo. The first trained Ceylonese psychiatrists began practice in the 1940s, leading to the establishment of the first neuropsychiatric clinic in Colombo in 1943. Treatments for the mentally ill improved dramatically, as [[w:insulin_shock_therapy|insulin shock therapy]] and [[w:Electroconvulsive_therapy|cardiazol convulsive therapy]] were utilized&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Kathriarachchi|first=Samudra T.|last2=Seneviratne|first2=V. Lakmi|last3=Amarakoon|first3=Luckshika|date=2019-06|title=Development of Mental Health Care in Sri Lanka: Lessons Learned|url=https://journals.lww.com/tpsy/fulltext/2019/33020/development_of_mental_health_care_in_sri_lanka_.1.aspx|journal=Taiwanese Journal of Psychiatry|language=en-US|volume=33|issue=2|pages=55|doi=10.4103/TPSY.TPSY_15_19|issn=1028-3684}}&lt;/ref&gt;. Mapother's advocation for the decentralization of services were further honored through the 1947 establishment of a first child guidance clinic in Colombo General Hospital&lt;ref name=&quot;:0&quot; /&gt;. 

In 1948, British Ceylon was granted independence after the [[w:Sri_Lankan_independence_movement|Sri Lankan independence movement]]. Changes in the mental healthcare structure were not immediate following independence, but rapid expansions of mental healthcare services were continuing to actualize.  

The following decades saw positive institutional developments, such as the creation of a second hospital in [[w:Mulleriyawa|Mulleriyawa]] in 1957, and the creation of a psychiatric inpatient unit in Colombo General Hospital in 1967—effectively granting the city of Colombo the luxury of hosting the top psychiatric care in the country&lt;ref name=&quot;:5&quot;&gt;{{Cite book|url=http://link.springer.com/10.1007/978-1-4899-7999-5_4|title=Mental Health System Development in Sri Lanka|last=Minas|first=Harry|last2=Mendis|first2=Jayan|last3=Hall|first3=Teresa|date=2017|publisher=Springer US|isbn=978-1-4899-7997-1|editor-last=Minas|editor-first=Harry|location=Boston, MA|pages=59–77|language=en|doi=10.1007/978-1-4899-7999-5_4|editor-last2=Lewis|editor-first2=Milton}}&lt;/ref&gt;. The 1950s was also the start of psychopharmacological innovations, with the introduction of [[w:Lithium_(medication)|lithium]] and long-acting injectable antipsychotics ([[w:Depot_injection|depot]] [[w:Antipsychotic|neuroleptics]]) in the succeeding years&lt;ref name=&quot;:4&quot; /&gt;. Additionally, the number of public psychiatrist positions increased by 400% from 1953 to 1967&lt;ref name=&quot;:5&quot; /&gt;.

After 1960, mental health services were expanded from beyond the capital to other cities in the country&lt;ref name=&quot;:2&quot; /&gt;.

In 1980, the [[w:Postgraduate_Institute_of_Medicine|Postgraduate Institute of Medicine]] initiated a program where students would enroll in a 5-year medical course and attain an MD in psychiatry, curbing the need for Sri Lankan medical students to be sent abroad to complete their training. Many of the medical students sent abroad for training never returned to Sri Lanka to practice, resulting in a &quot;1:500,000 to 1000,000&quot; ratio of psychiatrists to patients on &quot;most occasions&quot;&lt;ref name=&quot;:0&quot; /&gt;.

=== Mental Disease Ordinance of 1956 ===
In 1956, the 1873 Ordinance was revised a second time. The Mental Disease Ordinance of 1956 featured another linguistic development, as &quot;lunacy&quot; was replaced with &quot;mental disease&quot;&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:6&quot;&gt;{{Cite journal|last=Hapangama|first=Aruni|last2=Mendis|first2=Jayan|last3=Kuruppuarachchi|first3=K. a. L. A.|date=2023-02|title=Why are we still living in the past? Sri Lanka needs urgent and timely reforms of its archaic mental health laws|url=https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4|journal=BJPsych International|language=en|volume=20|issue=1|pages=4–6|doi=10.1192/bji.2022.26|issn=2056-4740|pmc=9909436|pmid=36812028}}&lt;/ref&gt;. The Ordinance paved way for community-based services to be delivered to patients closer to their residences, rather than strictly allocating services to just hospitals. This led to the creation of a [[w:WHO|WHO]]-backed community clinic near the [[w:University_of_Colombo|University of Colombo]] in the 1970s, where the focus was to eventually ease patients in the Angoda Mental Hospital back into the general population&lt;ref name=&quot;:5&quot; /&gt;.

=== Developments from the 1990s ===
The 1990s and onwards saw further positive developments in framing the mental healthcare system, including the establishment of the [https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=featured&amp;Itemid=101&amp;lang=en Directorate of Mental Health] in 1998. The Directorate of Mental Health is a part of the [[w:Ministry_of_Health_(Sri_Lanka)|Ministry of Health]] and is responsible for the monitoring and implementation of mental health programs across the country&lt;ref&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?lang=en|title=Home - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}&lt;/ref&gt;.  As of 2025, the current director of the Directorate of Mental Health is Dr. Chithramalee de Silva&lt;ref name=&quot;:2&quot; /&gt;.

On November 11, 2005, the Mental Health Policy was approved by the Government of Sri Lanka, advocating for establishments of more de-centralized, community-based mental health services across the country. The policy aimed to concisely define the rigorous standards needed to be met for each respected medical professional, including psychiatrists and clinical psychologists&lt;ref&gt;{{Cite journal|last=Rajapakshe|first=Onali Bimalka Wickramaseckara|last2=Mohan|first2=Mohapradeep|last3=Singh|first3=Swaran Preet|date=2023-05|title=Development of adolescent mental health services in Sri Lanka|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC10895478/|journal=BJPsych international|volume=20|issue=2|pages=41–43|doi=10.1192/bji.2022.32|issn=2056-4740|pmc=10895478|pmid=38414998}}&lt;/ref&gt;. The policy also included a new position, the &quot;Medical Officer of Mental Health&quot;, tasked with overseeing and assisting in creating community-based mental health services&lt;ref name=&quot;:0&quot; /&gt;. In the same year, the Sri Lankan government began implementing psychological services in state institutions, such as the military&lt;ref name=&quot;:8&quot; /&gt;.  

In 2007, the National Mental Health Advisory Council (NMHAC) was created to serve as an 'advisory' board for the Ministry of Health&lt;ref name=&quot;:7&quot;&gt;{{Cite web|url=https://mentalhealth.health.gov.lk/index.php?option=com_content&amp;view=article&amp;id=9&amp;Itemid=220&amp;lang=en|title=Introduction - Directorate of Mental Health|website=mentalhealth.health.gov.lk|access-date=2025-05-12}}&lt;/ref&gt;. 

In 2008, the Angoda Mental Hospital was restructured and renamed as the National Institute of Mental Health (NIMH)&lt;ref name=&quot;:7&quot; /&gt;. 

=== Modern-day Sri Lanka ===
[[File:Feeding Children in Sri Lanka.jpg|left|thumb|Despite the noteworthy improvements in mental healthcare services in recent decades, mental health remains a significant issue due to rising poverty. ]]
As of 2025, the Mental Health Act (mental health legislation) has been undergoing development since 2005 and is currently awaiting to be considered for the final stage of approval. This is expected to replace the 1956 Mental Health Ordinance&lt;ref name=&quot;:7&quot; /&gt;.

Currently, there are 7 tertiary care hospitals, 61 adult patient units, 3 child inpatient units, and 1 forensic unit with over 100 psychiatrists all throughout the 22 districts&lt;ref name=&quot;:4&quot; /&gt;. The [[w:Lady_Ridgeway_Hospital_for_Children|Lady Ridgeway Hospital]] in Colombo and the Sirimavo Bandaranayke Specialized Children Hospital in Kandy are specialized in treating children with [[w:Learning_disability|SLD]], [[w:ADHD|ADHD]], [[w:Autism_Spectrum_Disorder|ASD]], and provides family support for patients. As of 2017, 22 rehabilitation centers exist through the country, including 7 alcohol rehab centers&lt;ref name=&quot;:7&quot; /&gt;.   

Despite the impressive advancements in mental healthcare in the last couple of decades, Sri Lanka still suffers significant mental health issues due to increasing poverty levels in the country. The [[w:World_Bank|World Bank]] reported that [https://www.wsws.org/en/articles/2024/04/08/eesc-a08.html the poverty levels in Sri Lanka increased from 11% in 2019 to 26% in 2024], with 60% of Sri Lankan households facing &quot;decreased incomes&quot;&lt;ref&gt;Lakhtakia, Shruti, Atapattu Mudiyanselage, Udahiruni Shashadari Atapat, Walker, Richard Ancrum. ''Sri Lanka Development Update - Bridge to Recovery (English).'' Washington, D.C.: World Bank Group. &lt;nowiki&gt;http://documents.worldbank.org/curated/en/099634104012434919&lt;/nowiki&gt;&lt;/ref&gt;. This was exacerbated by Sri Lanka's excessive foreign debt, economic troubles stemming from [[w:Gotabaya_Rajapaksa|Gotabaya Rajapaksa]]'s presidential term, the COVID-19 pandemic, and the [[w:Russian_invasion_of_Ukraine|ongoing invasion of Ukraine by Russia (2022)]].  

According to [[w:NYU|New York University]] graduate student [https://gc-cuny.academia.edu/NadiaAugustyniak Nadia Augustyniak] in her 2025 overview of Sri Lanka's public mental healthcare system, poverty-induced financial precarity remains a major obstacle to receiving access to mental healthcare services. Even though trauma from adverse weather and conflict is deleterious to mental health, issues originating from every-day struggles, especially struggles related to poverty, could arguably play a more significant role&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}&lt;/ref&gt;.  

== Impact of Conflicts, Terrorism, Political Instability &amp; Natural Disasters ==

=== Sri Lankan Civil War ===
The '''Sri Lankan Civil War''' was a domestic conflict between the Sri Lankan government and the Liberation Tigers of Tamil Eelam (abbreviated as the ''LTTE),'' a militant group formed in the 1970s as a byproduct of rising tensions between the majority Sinhalese and minority Tamil population. The group is considered a terrorist organization&lt;ref&gt;{{Cite web|url=https://www.start.umd.edu/baad/database/liberation-tigers-tamil-eelam-ltte-1998.html|title=BAAD - Liberation Tigers of Tamil Eelam (LTTE) - 1998 {{!}} START.umd.edu|website=www.start.umd.edu|access-date=2025-06-09}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://www.cfr.org/backgrounder/liberation-tigers-tamil-eelam-aka-tamil-tigers-sri-lanka-separatists|title=Liberation Tigers of Tamil Eelam (aka Tamil Tigers) (Sri Lanka, separatists) {{!}} Council on Foreign Relations|last=Bhattacharji|first=Preeti|website=www.cfr.org|language=en|access-date=2025-06-09}}&lt;/ref&gt;. The LTTE conducted decades of massacres, assassinations of political figures, and suicide bombings to achieve ''[[w:Tamil_Eelam|Tamil Eelam]],'' leading to civilian displacement, infrastructure collapse, and the reduction of mental health services available in the northern region.[[File:DFID-funded, UNHCR emergency shelter tents, in the IDP camp at Menik Farm, Sri Lanka (3694081492).jpg|thumb|350x350px|An IDP camp in Menik Farm, Sri Lanka in 2009 ([https://www.bbc.com/news/world-asia-19703826 now closed]). Suicide rates in IDP camps were three times the general population.]]The civil war mainly affected the northeastern portion of the country, including the [[w:Vanni_(Sri_Lanka)|Vanni region]]. The conflict caused mass destruction to local mental healthcare facilities. Local residents described the conflict as ''varthayal varnicca mudiyathavai'', roughly translating into English as 'beyond description by words'&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|language=en|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}&lt;/ref&gt;. In 2003, only two psychiatrists were found in the region, operating on extremely limited resources. This furthered long-term trauma and mental health deterioration in the population&lt;ref name=&quot;:5&quot; /&gt;. 

In 2002, the humanitarian organization [https://www.msf.org/ Médecins Sans Frontières] (MSF) conducted an investigation on mental health needs in the [[w:Vavuniya|Vavuniya]] area, the site of intense conflict during the civil war (including the [[w:1985_Vavuniya_massacre|1985 Vavuniya massacre]]), and found that many of the residents suffered from high suicide rates, alcohol abuse, domestic violence, grief, and a &quot;sense of ‘learnt helplessness’&quot;&lt;ref name=&quot;:5&quot; /&gt;. A team from the University of Konstanz in Germany found that 92% of grade school children in the region were exposed to &quot;combat, shelling, and witnessing the death of loved ones&quot;&lt;ref name=&quot;:9&quot; /&gt;. 
[[File:Tractors. Jan 2009 displacement in the Vanni.jpg|left|thumb|350x350px|Displaced civilians evacuating from the Kilinochchi and Mullaitivu Districts due to military campaigns initiated by the Sri Lankan military (January 2009).]]
Additionally, accusations of war crimes have been made against [[w:War_crimes_during_the_final_stages_of_the_Sri_Lankan_civil_war|the Sri Lankan government]]&lt;ref&gt;See also [[w:Sexual violence in the Sri Lankan civil war]].&lt;/ref&gt;. A 2009 HRW report alleged that the Sri Lankan government considered the native Tamil population residing in war zones to be &quot;siding with the LTTE and [therefore, were] treated as combatants&quot;, and that the government conducted numerous shellings of &quot;areas crowded with civilians&quot;&lt;ref&gt;{{Cite journal|date=2009-02-19|title=War on the Displaced|url=https://www.hrw.org/report/2009/02/19/war-displaced/sri-lankan-army-and-ltte-abuses-against-civilians-vanni|journal=Human Rights Watch|language=en}}&lt;/ref&gt;. Furthermore, the LTTE conducted recruitment campaigns on the Vanni population where recruited men, women, and even children with minimal training, were recruited for war efforts.  

Over 200,000 Tamil civilians were moved into [[w:Internally_displaced_persons_in_Sri_Lanka|designated displacement camps during the war]], where conditions were poor&lt;ref&gt;{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000}}&lt;/ref&gt;. The suicide rate in these displacement camps was three times the community-level (2002), with a ratio of 103.5 suicides per 10,000 persons, compared to the general population's rate of 37.5 suicides per 10,000 persons. Almost all suicide attempts involved poisonous substances. Other forms of violence included domestic violence and child abuse.  

Local health officials in Vavuniya admitted that mental health concerns were a major problem, but were unable to address these concerns due to a lack of resources and support from the government. During the [[wikipedia:Sri_Lankan_civil_war#2002_peace_process_(2002%E2%80%932006)|brief 2002 ceasefire]], the MSF implemented a &quot;community-based programme&quot; which included &quot;increasing awareness, community strengthening, reinforcing coping-strategies for long-term war-affected communities, and counselling&quot;. The MSF also advocated for restrictions of poisonous substances due its means for suicide attempts, and stressed that &quot;much more [than resettlement]&quot; would need to be done to help alleviate the psychological pain the northern population had faced due to the war&lt;ref&gt;{{Cite journal|last=de Jong|first=Kaz|last2=Mulhern|first2=Maureen|last3=Ford|first3=Nathan|last4=Simpson|first4=Isabel|last5=Swan|first5=Alison|last6=van der Kam|first6=Saskia|date=2002-04|title=Psychological trauma of the civil war in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S0140673602084209|journal=The Lancet|language=en|volume=359|issue=9316|pages=1517–1518|doi=10.1016/S0140-6736(02)08420-9}}&lt;/ref&gt;. The ceasefire ended in 2006 and led to the [[w:Eelam_War_IV|final phase of the civil war]], eventually ending in 2009 with the [[w:https://en.wikipedia.org/wiki/Velupillai_Prabhakaran#Sri_Lankan_Army_Northern_offensive_and_death|death of the LTTE's leader]]. 

'''Post-war'''
[[File:Puttalam district.svg|left|thumb|Puttalam District, unlike its northern counterparts, was largely spared from the intense conflict, possibly explaining the lower rates of common mental disorders (CMDs).]]
The first district-wide cross-sectional multistage cluster sample survey was conducted in the [[w:Jaffna_District|Jaffna District]] shortly after the war ended in 2009. The study's sample included 1517 households and 2 internally displaced peoples camps. With a response rate of 92%, the study found that symptoms for PTSD were found in 7% of participants, symptoms of anxiety were found in 32.6% of participants, and symptoms of depression were found in 22.2% of participants. 2% of respondents were being placed in internally displaced peoples camps at the time of the study, 29.5% were freshly resettled from the internally displaced peoples camps, and the rest of the participants (68.5%) were never placed into camps. In comparison to residents who were never placed into camps, participants that were actively held in camps generally reported more symptoms of PTSD, anxiety, and depression. The researchers also found that women were especially vulnerable to deteriorating mental health conditions. This was explained by two factors: women having to assume the roles of both the father and the mother in the family setting after the, either voluntary or forced, departure of their husband to war, and sexist violence&lt;ref&gt;{{Cite journal|last=Husain|first=Farah|last2=Anderson|first2=Mark|last3=Lopes Cardozo|first3=Barbara|last4=Becknell|first4=Kristin|last5=Blanton|first5=Curtis|last6=Araki|first6=Diane|last7=Kottegoda Vithana|first7=Eeshara|date=2011-08-03|title=Prevalence of War-Related Mental Health Conditions and Association With Displacement Status in Postwar Jaffna District, Sri Lanka|url=https://doi.org/10.1001/jama.2011.1052|journal=JAMA|volume=306|issue=5|pages=522–531|doi=10.1001/jama.2011.1052|issn=0098-7484}}&lt;/ref&gt;. A 2013 study on adult patients in [https://www.ncbi.nlm.nih.gov/books/NBK232631/ primary care settings] (divisional hospitals, primary medical care units) found major depression to be significantly higher in females (5.1%) than males (3.6%), bolstering the findings from the 2009 study&lt;ref&gt;{{Cite journal|last=Senarath|first=Upul|last2=Wickramage|first2=Kolitha|last3=Peiris|first3=Sharika Lasanthi|date=2014-03-24|title=Prevalence of depression and its associated factors among patients attending primary care settings in the post-conflict Northern Province in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/1471-244X-14-85|journal=BMC Psychiatry|language=en|volume=14|issue=1|pages=85|doi=10.1186/1471-244X-14-85|issn=1471-244X|pmc=3987835|pmid=24661436}}&lt;/ref&gt;.

Muslims in Northern Sri Lanka also faced violence and discrimination during the conflict. Most notable incidents include [[w:Expulsion_of_Muslims_from_the_Northern_Province_of_Sri_Lanka|the October 1990 expulsion of Muslims from the North to the Puttalam District or Jaffna]] and the [[w:Kattankudy_mosque_massacre|1990 Kattankudy mosque massacre]]. The only study testing the displaced Muslim population post-civil war was completed in 2011, where a cross-sectional survey of 450 internally displaced people or people born into displacement (ages 18 - 65) revealed 18.8% of the sample suffering from common mental health disorders (CMD), including [[w:Somatoform_disorder|somatoform disorder]] (14%), &quot;other depressive syndromes&quot; (7.3%), major depression (5.1%), and anxiety disorder (2.8%). The percentages found in this study for somatoform disorder and major depression were &quot;considerably higher&quot; than the national percentages, though the researchers noted that the prevalence of CMD was lower in comparison to other countries marred with conflict, including Palestine (40.3%) and Ethiopia (27.8%). The researchers explained that the lower rate of CMD may be attributed to the [[w:Puttalam_District|serenity of the post-settlement destination]], as conflict was mainly centered in the North and East. In contrast to earlier findings, this study did not observe a higher prevalence of CMDs among women, although increased rates of somatoform disorders were noted (though the researchers did not reveal the data behind this)&lt;ref&gt;{{Cite journal|last=Siriwardhana|first=Chesmal|last2=Adikari|first2=Anushka|last3=Pannala|first3=Gayani|last4=Siribaddana|first4=Sisira|last5=Abas|first5=Melanie|last6=Sumathipala|first6=Athula|last7=Stewart|first7=Robert|date=2013-05-22|title=Prolonged Internal Displacement and Common Mental Disorders in Sri Lanka: The COMRAID Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0064742|journal=PLOS ONE|language=en|volume=8|issue=5|pages=e64742|doi=10.1371/journal.pone.0064742|issn=1932-6203|pmc=3661540|pmid=23717656}}&lt;/ref&gt;.

Research on the mental state of combatants has been limited, but a post-war 2009 study done between soldiers of the [[w:Sri_Lanka_Army_Special_Forces_Regiment|Special Forces]] and regular soldiers showed higher levels of exposure to traumatic events for units of the Special Forces, yet the former exhibited significantly less symptoms of CMDs compared to the latter. The authors of this study, [https://scholar.google.co.uk/citations?user=cVKEBdwAAAAJ&amp;hl=en&amp;oi=ao Raveen Hanwella] and [https://scholar.google.co.uk/citations?user=ZRj74qMAAAAJ&amp;hl=en&amp;oi=sra Varuni de Silva], offered the camaraderie of the military unit as an explanation for the discrepancy&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|date=2012-08|title=Mental health of Special Forces personnel deployed in battle|url=https://pubmed.ncbi.nlm.nih.gov/22038567|journal=Social Psychiatry and Psychiatric Epidemiology|volume=47|issue=8|pages=1343–1351|doi=10.1007/s00127-011-0442-0|issn=1433-9285|pmid=22038567}}&lt;/ref&gt;. A follow-up study was completed by the pair (with the addition of former Director-General of the Health Services of the Sri Lanka Navy [[w:Nicholas_Jayasekera|Nicholas Jayasekera]]), where the findings were similar, though the statistically significant bridge between the two cohorts in the previous study evaporated in the follow-up study. This may be due to the significant decline in mental health problems observed in the regular unit forces, potentially reflecting resilience in the aftermath of the conflict&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=Jayasekera|first2=Nicholas E. L. W.|last3=Silva|first3=Varuni A. de|date=2014-09-25|title=Mental Health Status of Sri Lanka Navy Personnel Three Years after End of Combat Operations: A Follow Up Study|url=https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0108113|journal=PLOS ONE|language=en|volume=9|issue=9|pages=e108113|doi=10.1371/journal.pone.0108113|issn=1932-6203|pmc=4177866|pmid=25254557}}&lt;/ref&gt;. Amputees or soldiers with spinal injuries exhibited drastically different numbers, with approximately 40% of nearly 100 male-veterans in a post-war 2009 study displaying PTSD-like symptoms&lt;ref&gt;{{Cite journal|last=Abeyasinghe|first=N. L.|last2=de Zoysa|first2=P.|last3=Bandara|first3=K.M.K.C.|last4=Bartholameuz|first4=N. A.|last5=Bandara|first5=J. M.U.J.|date=2012-05-01|title=The prevalence of symptoms of Post-Traumatic Stress Disorder among soldiers with amputation of a limb or spinal injury: A report from a rehabilitation centre in Sri Lanka|url=https://doi.org/10.1080/13548506.2011.608805|journal=Psychology, Health &amp; Medicine|volume=17|issue=3|pages=376–381|doi=10.1080/13548506.2011.608805|issn=1354-8506|pmid=21942815}}&lt;/ref&gt;.

About a decade after the conflict ceased, a few notable studies have emerged to help guide understanding on the longer-term mental health effects on victims of the civil war. 

From July 2019 to October 2020, a study conducted on 585 local adolescents (ages 12-19) in the Vavuniya district revealed that despite 15.6% of the statistic having faced one or more war-related events, only 3.9% of the participants had moderate to severe depression. In addition to considerably low depression rates, only 5.7% of participants age 17+ were found to have moderate to severe hopelessness&lt;ref&gt;{{Cite journal|last=Dissanayake|first=Lasith|last2=Jabir|first2=Sameeha|last3=Shepherd|first3=Thomas|last4=Helliwell|first4=Toby|last5=Selvaratnam|first5=Lavan|last6=Jayaweera|first6=Kaushalya|last7=Abeysinghe|first7=Nihal|last8=Mallen|first8=Christian|last9=Sumathipala|first9=Athula|date=2023-08-31|title=The aftermath of war; mental health, substance use and their correlates with social support and resilience among adolescents in a post-conflict region of Sri Lanka|url=https://doi.org/10.1186/s13034-023-00648-1|journal=Child and Adolescent Psychiatry and Mental Health|language=en|volume=17|issue=1|pages=101|doi=10.1186/s13034-023-00648-1|issn=1753-2000|pmc=10472617|pmid=37653394}}&lt;/ref&gt;. The authors referenced a 2010 observation by psychiatrist [https://us.sagepub.com/en-us/nam/author/daya-somasundaram Daya Somasundaram], who noted that many Tamil IDPs presented &quot;remarkable resilience and post-traumatic growth&quot; after the civil war—an outcome he attributed to the close-knit, family-centered nature of Tamil communities&lt;ref&gt;{{Cite journal|last=Somasundaram|first=Daya|date=2010-07-28|title=Collective trauma in the Vanni- a qualitative inquiry into the mental health of the internally displaced due to the civil war in Sri Lanka|url=https://doi.org/10.1186/1752-4458-4-22|journal=International Journal of Mental Health Systems|volume=4|issue=1|pages=22|doi=10.1186/1752-4458-4-22|issn=1752-4458|pmc=2923106|pmid=20667090}}&lt;/ref&gt;.  

However, findings originating from a 2019 study, undertook by several faculty members from the University of Kelaniya, the University of Jaffna, the [[w:Gampaha_Wickramarachchi_University_of_Indigenous_Medicine|Gampaha Wickramarachchi University of Indigenous Medicine]], and the [https://onur.gov.lk/ Office for National Unity and Reconciliation (ONUR)] in Jaffna, found contrasting results. Out of 336 participants from districts which faced significant ramifications of the conflict (Jaffna, Kilinochchi, Mullaithivu, Vavuniya, and Mannar districts), 50.5% had extreme anxiety symptoms and 36.5% exhibited &quot;extremely severe&quot; symptoms of depression. 92.5% of families in the sample experienced suicidal ideation, with an observed negative correlation between trauma exposure and life satisfaction with families. Drug abuse (86.2%) and alcohol abuse (84.5%) were the two highest problematic behaviors recorded on a community-level, suggesting that the negative consequences of the civil war still persist, possibly on a substantial scale than previously recognized, in Tamil communities residing in the North&lt;ref&gt;{{Cite journal|last=Thamotharampillai|first=Umaharan|last2=Perera|first2=Ruwanthi|last3=Wickremasinghe|first3=Rajitha|last4=Williams|first4=Shehan|last5=Vijayasangar|first5=Thedsanamoorthy|last6=Sivatharsan|first6=Balasubramaniam|last7=Hilbert|first7=Vanceline|last8=Somasundaram|first8=Daya|date=2025-05-06|title=Collective Trauma- Psychosocial consequences of war in northern Sri Lanka 10 years on, a mixed methods study|url=https://www.sciencedirect.com/science/article/pii/S2666560325000696|journal=SSM - Mental Health|pages=100457|doi=10.1016/j.ssmmh.2025.100457|issn=2666-5603}}&lt;/ref&gt;. Further research should be conducted on Northern Tamil populations to assess the extent of mental health issues stemming from the conflict. 

In 2019, [https://www.researchgate.net/scientific-contributions/R-M-M-Monaragala-2087692299 Dr. R. M. M. Monaragala] conducted a study on 1,845 soldiers with combat experience, finding that 3.9% of the sample suffered from PTSD. Dr. Monaragala noted that &quot;probable depression, fatigue, aggression, and family history of mental disorder&quot; were correlative of PTSD presence. He suggested that &quot;screening and psychosocial intervention[s]&quot; could alleviate CMDs of former combatants&lt;ref&gt;{{Cite journal|last=Monaragala|first=R. M. M.|date=2024-04-19|title=Exploring the effects of the past civil war in terms of the prevalence and associating factors of PTSD|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v14i2.8465|journal=Sri Lanka Journal of Psychiatry|language=en-US|volume=14|issue=2|doi=10.4038/sljpsyc.v14i2.8465|issn=2012-6883}}&lt;/ref&gt;. 

=== 2004 Boxing Day Tsunami ===
The '''2004 Boxing Day Tsunami''' was a natural disaster where a tsunami spawned off a 9.2–9.3 magnitude earthquake off the coast of Aceh in Indonesia on December 26. The tsunami greatly affected the coastlines of the country, with the death toll reaching to around 35,000 deaths. In addition, 90,000 houses were destroyed and 516,000 people were forced to migrate due to severe infrastructural damage&lt;ref name=&quot;:5&quot; /&gt;. It stands as the [http://www.china.org.cn/english/features/tsunami_relief/119821.htm worst natural disaster to have ever hit Sri Lanka].  
[[File:Tsunami relief 2004 02.jpg|thumb|300x300px|Volunteers from [[w:Royal_College,_Colombo|Royal College in Colombo]] assisting in tsunami relief efforts (Sarvodaya Headquarters, Moratuwa).]]
A survey conducted on schoolchildren (ages 8-14) in Manadkadu (a Tamil-majority village in the northern coast), [[w:Kosgoda|Kosgoda]] (western coast), and [[w:Galle|Galle]] (southern coast), just a few weeks after the tsunami hit Sri Lanka, revealed that 33.8%, 13.9%, and 38.8% of children interviewed exhibited signs of PTSD (according to the DSM-IV's criteria), respectively (minus the time criteria, as the DSM-IV does not permit diagnosis of PTSD within 4 weeks of a traumatic incident). The loss of family members and exposure to previously traumatic incidents appeared to be highly correlate with PTSD development&lt;ref&gt;{{Cite journal|last=Neuner|first=Frank|last2=Schauer|first2=Elisabeth|last3=Catani|first3=Claudia|last4=Ruf|first4=Martina|last5=Elbert|first5=Thomas|date=2006|title=Post-tsunami stress: A study of posttraumatic stress disorder in children living in three severely affected regions in Sri Lanka|url=https://onlinelibrary.wiley.com/doi/abs/10.1002/jts.20121|journal=Journal of Traumatic Stress|language=en|volume=19|issue=3|pages=339–347|doi=10.1002/jts.20121|issn=1573-6598}}&lt;/ref&gt;. 

Many victims in the Jaffna area suffered with &quot;[https://www.psychiatry.org/patients-families/prolonged-grief-disorder pathological grief], phobias, depression and PTSD&quot; post-tsunami. Schizophrenia in the Jaffna Tamil community, which had already suffered elevated prevalence of PTSD prior to the tsunami, had worsened—highlighting the need for specialized care in response to cumulative exposures to chronic and acute traumas. In a study published in ''International Psychiatry'' (2006), Jaffna-based researchers noted that, contrary to their initial inclinations, there was not a &quot;large[r] (than expected) rise in [the] number of people&quot; seeking mental health support 3 months after the tsunami. However, 10 months after the disaster, the researchers anticipated that &quot;more psychiatric disorders&quot; would emerge due to &quot;very little rebuilding [efforts]&quot; and an apparent &quot;unfairness in the aid system&quot;.&lt;ref&gt;{{Cite journal|last=Somasundaram|first=D. J.|last2=Yoganathan|first2=S.|last3=Ganesvaran|first3=T.|date=1993-09|title=Schizophrenia in northern Sri Lanka|url=https://pubmed.ncbi.nlm.nih.gov/7828234|journal=The Ceylon Medical Journal..|volume=38|issue=3|pages=131–135|issn=0009-0875|pmid=7828234}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|last=Danvers|first=K.|last2=Sivayokan|first2=S.|last3=Somasundaram|first3=D. J.|last4=Sivashankar|first4=R.|date=2006-07|title=Ten months on: qualitative assessment of psychosocial issues in northern Sri Lanka following the tsunami|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6734678/|journal=International Psychiatry: Bulletin of the Board of International Affairs of the Royal College of Psychiatrists|volume=3|issue=3|pages=5–8|issn=1749-3676|pmc=6734678|pmid=31507850}}&lt;/ref&gt; 

At the February 2005 ''After the Tsunami: Mental Health Challenges to the Community for Today and Tomorrow'' conference in Thailand, [https://www.researchgate.net/profile/Chandanie-Hewage Dr. Chandanie Hewage] of the [[w:University_of_Ruhuna|University of Ruhuna]] commentated that measures taken to assist the affected were &quot;not coordinated&quot; due to poor &quot;communication systems and road [conditions].&quot; Regardless, efforts were continued by the government and health professionals to alleviate the struggles the victims were facing, including the psychological ramifications of the disaster. 

Several issues in the delivery of these services were highlighted by Dr. Hewage, including poor maintenance of health records, lack of awareness on drug consumption by the patients themselves, and shortages of health professionals. Dr. Hewage points out that personnel had &quot;little&quot; mental health training prior to the disaster, suggesting increased &quot;research&quot; and adequate &quot;provision[ing] and training of staff&quot; for the long-term&lt;ref&gt;{{Cite journal|last=Davidson|first=Jonathan R. T.|date=2006|title=Foreword. After the tsunami: mental health challenges to the community for today and tomorrow|url=https://pubmed.ncbi.nlm.nih.gov/16602809|journal=The Journal of Clinical Psychiatry|volume=67 Suppl 2|pages=3–8|issn=0160-6689|pmid=16602809}}&lt;/ref&gt;. With inadequate documentation, no systematic procedures in place, and insufficient personnel, tsunami victims with mental health concerns may not receive the services they need, further compacting neuropsychological ailments.  

In 2008 (about 3-4 years after the tsunami), researchers in the hard-hit village of [[w:Peraliya|Peraliya]] (Galle District) found that from a sample of approximately 90 adults, 25% suffered from moderate–severe PTSD, with women scoring &quot;above the cut-off for anxiety&quot; and reporting more &quot;somatic symptoms&quot;, though researchers inferred that the PTSD rate found in the study may be influenced by other factors, including war or economic hardship&lt;ref&gt;{{Cite journal|last=Hollifield|first=Michael|last2=Hewage|first2=Chandanie|last3=Gunawardena|first3=Charlotte N.|last4=Kodituwakku|first4=Piyadasa|last5=Bopagoda|first5=Kalum|last6=Weerarathnege|first6=Krishantha|last7=Group|first7=International Post-Tsunami Study|date=2008-01|title=Symptoms and coping in Sri Lanka 20–21 months after the 2004 tsunami|url=https://www.cambridge.org/core/journals/the-british-journal-of-psychiatry/article/symptoms-and-coping-in-sri-lanka-2021-months-after-the-2004-tsunami/CB33752239AF362A0BFD55B3668D60B0|journal=The British Journal of Psychiatry|language=en|volume=192|issue=1|pages=39–44|doi=10.1192/bjp.bp.107.038422|issn=0007-1250}}&lt;/ref&gt;.  

=== 2019 Easter Bombings ===
The '''2019 Easter Bombings''' were a series of coordinated attacks perpetrated by the Islamic extremist group, [[w:National_Thowheeth_Jama'ath|National Thowheeth Jama'ath]], on April 21, 2019. The attack targeted three churches and three hotels in the Colombo area, killing nearly 300 people and injuring over 500. The attacks were also attributed to the incompetency of the Sri Lankan government, who ignored [https://www.bbc.com/news/world-asia-48044636 multiple warnings preceding the attacks]. The attacks negatively affected the Sri Lankan Catholic community and further weakened relations between the major religious groups&lt;ref&gt;{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}&lt;/ref&gt;.  

In the aftermath of the attacks, professionals in the [[w:Gampaha_District|Gampaha District]] resorted to &quot;low-cost methodologies&quot; for children and adolescents affected by the attack, as a &quot;severe shortage&quot; of children and adolescent mental health experts were exposed&lt;ref&gt;{{Cite journal|last=Chandradasa|first=Miyuru|last2=Rathnayake|first2=Layani C|last3=Rowel|first3=Madushi|last4=Fernando|first4=Lalin|date=2020-06-01|title=Early phase child and adolescent psychiatry response after mass trauma: Lessons learned from the Easter Sunday attack in Sri Lanka|url=https://doi.org/10.1177/0020764020913314|journal=International Journal of Social Psychiatry|language=EN|volume=66|issue=4|pages=331–334|doi=10.1177/0020764020913314|issn=0020-7640}}&lt;/ref&gt;. In a qualitative study of 8 survivors of the attacks receiving grief counseling, [[w:University_of_Ruhuna|University of Ruhuna]] assistant professor [https://www.researchgate.net/profile/Virasha-Godakanda Virasha Godakanda] observed that 70% of the sample size expressed a lack of confidence in adequate mental health interventions from the government, reducing the quality of such services. Professor Godakanda strongly endorsed for &quot;culturally-sensitive&quot; programs, a diversity in therapeutic approaches (including nature-based therapy), and &quot;prolonged investigations&quot; to track developments in mental health resources and impacts of implemented interventions&lt;ref&gt;{{Cite journal|last=Godakanda|first=Virasha|date=2025-01-29|title=A GRIEF COUNSELING INTERVENTION AFTER THE MASS TRAUMA: LESSONS LEARNED FROM THE VICTIMS OF THE EASTER SUNDAY ATTACK IN SRI LANKA|url=https://kjmr.com.pk/kjmr/article/view/216|journal=Kashf Journal of Multidisciplinary Research|language=en|volume=2|issue=01|pages=13–32|doi=10.71146/kjmr216|issn=3007-200X}}&lt;/ref&gt;. 

A few weeks following the attacks, Muslims in Sri Lanka were subjected to [[w:2019_anti-Muslim_riots_in_Sri_Lanka|violent, coordinated riots]] masterminded by Sinhalese national forces&lt;ref&gt;{{Cite journal|last=Mujahidin|first=Muhammad Saekul|date=2023-07-03|title=Extremism and Islamophobia Against the Muslim Minority in Sri Lanka|url=https://www.ajis.org/|journal=American Journal of Islam and Society|language=en|volume=40|issue=1-2|pages=213–241|doi=10.35632/ajis.v40i1-2.3135|issn=2690-3741}}&lt;/ref&gt;. Riots were mainly centered in the [[w:Kurunegala_District|Kurunegala]], Gampaha, and [[w:Kandy_District|Kandy]] Districts. At least [https://www.aljazeera.com/news/2019/5/21/in-sri-lanka-muslims-say-sinhala-neighbours-turned-against-them one confirmed death was reported]. Calls for vague ''niqab'' and ''burqa'' bans were increasingly prominent, eventually leading to the 2021 burqa ban by the Sri Lankan government. Pakistani and Afghani refugees fleeing religious persecution in Negombo were forced to be &quot;made refugees again&quot; after local protests were orchestrated against their settlement. Anti-Muslim sentiment was &quot;unleashed online, in the law, and on the street&quot;&lt;ref&gt;{{Cite book|title=CARTOGRAPHIC JOURNEY OF RACE, GENDER AND POWER: global identity|date=2021|publisher=CAMBRIDGE SCHOLARS PUBLIS|isbn=978-1-5275-6965-2|location=S.l.}}&lt;/ref&gt;. Albeit its relevancy to the attacks, no in-depth mental health studies have took place on the minority Muslim population following the Easter bombings. Further research is imperative in exploring the sustained psychological effects of Islamophobia and its effect on the Muslim minority community in the aftermath of the 2019 Easter attacks. 

Literature on the impact of the 2019 Easter Bombings on mental health is limited and further research should be conducted.  

=== 2019-2024 Economic Crisis ===
The '''2019-2024 Economic Crisis''' refers to a 5 year period where the Sri Lankan economy experienced significant inflation and an abrupt hike in prices on basic, everyday items. It is the worse economic crisis the country has faced since the Sri Lankans were granted independence in 1948. Schools in Sri Lanka were forced to postpone examinations due to paper shortages. Gas shortages led to long lines at gas stations, some lasting for days, throughout the island. Shortages in electricity, cooking gas, and aviation feul were additional consequences of the economic crisis.

Healthcare workers faced a barrage of impediments in their line of work during the crisis, including a lopsided work-life balance due to unprecedented demand, increased stress and mental fatigue from a lack of resources and personnel, unhealthy coping mechanisms, job dissatisfaction, and a reduction in work quality. Such effects perpetuated a self-enforcing cycle of psychologically distressed mental healthcare workers providing subpar services, affecting patients and amplifying mental health issues experienced by both the workforce and their patients&lt;ref&gt;{{Cite journal|last=Dilogini|first=S.|last2=Grace|first2=H. H.|last3=Thasika|first3=T.|date=2024|title=Exploring The Mental Health and Well-Being of Public Healthcare Workers (HCWs) Amid Economic Crisis in Sri Lanka|url=http://repo.lib.jfn.ac.lk/ujrr/handle/123456789/11092|language=en|publisher=Chartered Institute of Personnel Management}}&lt;/ref&gt;. 

Medical students from the Faculty of Medicine at the University of Colombo reported that the economic crisis forced abrupt changes in dietary consumption, increased hopelessness in the future, increased stress and anxiety, and a decrease in interest in pursuing a &quot;clinical post-graduate career&quot;&lt;ref&gt;{{Cite journal|last=Adikaranayake|first=Pesala Randika|last2=Perera|first2=Anusha Nimrod|last3=Nilaweera|first3=Akhila Imantha|last4=Fernando|first4=Desha Rajni|last5=Wijayaratne|first5=Dilushi Rowena|date=2025-07-01|title=Effects of Sri Lankan economic crisis on health, lifestyle and education of medical students in Faculty of Medicine, University of Colombo – an online survey|url=https://doi.org/10.1186/s12909-025-07506-y|journal=BMC Medical Education|language=en|volume=25|issue=1|pages=938|doi=10.1186/s12909-025-07506-y|issn=1472-6920|pmc=12211748}}&lt;/ref&gt;. 283 government-school teachers completed a web-based cross-sectional survey in April 2024, with majority of the participants reporting a severe reduction in monthly income &amp; 1/3 of participants exhibiting &quot;clinical levels of psychological distress&quot;&lt;ref&gt;{{Cite journal|last=Senevirathne|first=C. P.|last2=Senarathne|first2=D. L. P.|last3=Fernando|first3=M. S.|last4=Senevirathne|first4=S. P.|date=2025-05-28|title=Examining the economic burden and mental health distress among government school teachers in Sri Lanka: a cross-sectional study|url=https://doi.org/10.1186/s40359-025-02921-8|journal=BMC Psychology|language=en|volume=13|issue=1|pages=572|doi=10.1186/s40359-025-02921-8|issn=2050-7283}}&lt;/ref&gt;. A study published in that same year reported that out of 261 nurses working in teaching hospitals, 91.6% were forced to allocate their finances to strictly &quot;general needs&quot;, while more than 50% looked into international opportunities for employment. Notably, the study reported an overall near &quot;twofold greater&quot; rate of depression, anxiety, and stress compared to previous studies on nurses in Sri Lanka&lt;ref&gt;{{Cite journal|last=Senevirathne|first=C.P|last2=Senarathne|first2=L.|last3=Fernando|first3=M.|date=2024-04-01|title=Exploring the Association Between Behavioural Modification in Response to the Prevailing Economic Crisis and Mental Health Outcomes of Nurses from Teaching Hospitals, Sri Lanka|url=https://doi.org/10.1177/23779608241272679|journal=SAGE Open Nursing|language=EN|volume=10|pages=23779608241272679|doi=10.1177/23779608241272679|issn=2377-9608|pmc=11311183}}&lt;/ref&gt;. 

The detrimental effects the crisis has had on the mental health sector reveal a concerning area of underappreciation and under compensation towards a critical sector for the well-being of the country. Adequate staffing, increased funding, and an improved work-life balance should be emphasized for the workers of health sector of the country.  

== Present-Day Challenges ==

=== Ethnic tension ===
Despite the ending of the Sri Lankan civil war and the introduction of pluralist policies (such as the [https://srilankaembassy.fr/sites/default/files/files/media/pdf/NationalPolicy-English.pdf 2017 National Policy on Reconciliation and Coexistence] under the Sirisena administration), tensions amongst members of the ethnic groups still persist. Evidence of these tensions was found in a 2022 study conducted in the Ratnapura district, where religious leaders expressed skepticism through semi-structured interviews on &quot;conflict transformation&quot;. A Tamil citizen of the Ratnapura community recounted that they were forced to &quot;hide in jungles&quot; and consume &quot;dirty water in drainage[s]&quot; due to scarcity of food and drinkable water as a result of the conflict. In certain personal accounts, ethnic conflicts appear to affect the social behavior and identity of the majority ethnic group. One Sinhala participant recounted his objection to the war-time retaliatory destruction of a shop run by a Tamil shopkeeper was met with interrogative questions about &quot;whether [he was] Sinhalese or not&quot;. Both accounts convey interethnic tensions stemming from decade-long conflicts&lt;ref&gt;Jayathilaka, Aruna &amp; Gamage, Sayuri. (2024). Role of Buddhist and Hindu Religious Leaders Role of Buddhist and Hindu Religious Leaders in the Post-War Conflict Transformation Process: A Study Based on Rathnapura District in Srilanka. ''Retrieved from'' https://gandhimargjournal.org/wp-content/uploads/2024/09/Volume-46-Issue-1-April-June-2024.pdf#page=66&lt;/ref&gt;. 

Beyond individual accounts and the official end of the civil war, the minority groups in the country continue to feel ostracized. The Sri Lankan Tamil population remains dissatisfied with the Sri Lankan government due to their alleged lack of accountability of perpetrators of war crimes and lack of information on the whereabouts of [[w:Enforced_disappearances_in_Sri_Lanka|thousands of enforced disappearances]] that took place from the 1980s. Additionally, rising anti-Muslim sentiment in recent years has contributed to increased ethnic tensions, a stark contrast to the previous centuries of peaceful co-existence between the groups. 
[[File:Bodu Bala Sena symbol.svg|thumb|The symbol for Bodu Bala Sena, a nationalistic Sinhala Buddhist group criticized for catalyzing ethnic tensions in Sri Lanka.]]
Laws passed by the Sri Lankan government, such as the [[w:Prevention_of_Terrorism_Act_(Sri_Lanka)|Prevention of Terrorism Act]] and [[wikipedia:Anti-conversion_law#Sri_Lanka|anti-conversion laws]], have forced the United States Commission on International Religious Freedom to label Sri Lanka as a nation that &quot;[engages] or [tolerates] severe violations of religious freedom&quot; in their 2024 report. The government has been criticized by human rights organizations for &quot;disproportionately targeting religious minorities&quot;&lt;ref&gt;{{Cite journal|last=Jayawickreme|first=Nuwan|last2=Jayawickreme|first2=Eranda|last3=McCaffrey|first3=Amy Z.|last4=Thiruvarangan|first4=Mahendran|date=2025-06-01|title=Mental health futures in post-war Sri Lanka: Resilience, relational pluralism, and implementation pathways|url=https://www.sciencedirect.com/science/article/pii/S2666560325000775|journal=SSM - Mental Health|volume=7|pages=100465|doi=10.1016/j.ssmmh.2025.100465|issn=2666-5603}}&lt;/ref&gt;. Additionally, the implementation of the three dominant languages, English, Sinhala, and Tamil, across formal education and government services have been lackadaisical, narrowing opportunities of foundational social interactions between the groups. Persistent discrimination and prejudice towards minority groups can lead to an array of complex and self-deprecating mental health issues.   

Efforts to mitigate ethnic tensions include strategies like [[w:Community-based_participatory_research|community-based participatory research]] (CBPR), task-sharing, and securing online mental health services in order to expand mental health services. However, the implementation of evidence-based plans has been met with difficulty due to inaccessibility, high costs, and shortages of adequately-trained personnel.  

Movements aiming for improved intra group and inter group coexistences, such as the Jaffna People’s Forum for Coexistence, should be emphasized on a systematic and multi-level basis, including but not limited to education, public sectors, and within communities. Pluralistic values are encouraged to be emphasized across both private and public schools to foster cultural sensitivity and tolerance. Measures should be taken against groups criticized for promoting sectarian hostility, such as the [[w:Bodu_Bala_Sena|Bodu Bala Sena]]. 

=== Poverty ===
It has been proven that poverty significantly increases the chances of developing mental illnesses. This is further amplified by possible discrimination&lt;ref&gt;{{Cite journal|last=Knifton|first=Lee|last2=Inglis|first2=Greig|date=2020-10|title=Poverty and mental health: policy, practice and research implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC7525587/|journal=BJPsych bulletin|volume=44|issue=5|pages=193–196|doi=10.1192/bjb.2020.78|issn=2056-4694|pmc=7525587|pmid=32744210}}&lt;/ref&gt;. Poverty also affects the ability for individuals with mental health concerns to receive the treatment they need. Due to the repercussions of the economic crisis, clients in Sri Lanka could not attend further counseling sessions&lt;ref name=&quot;:8&quot; /&gt;. Poverty from 2021 to 2022 [https://databankfiles.worldbank.org/public/ddpext_download/poverty/987B9C90-CB9F-4D93-AE8C-750588BF00QA/current/Global_POVEQ_LKA.pdf reportedly doubled], with future forecasts predicting the poverty line to &quot;remain above 25 percent&quot;. Suicide has been empirically linked to economic hardships in previous studies&lt;ref&gt;{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}&lt;/ref&gt;. A 2013 study done on suicidal patients in [[w:Batticaloa_Teaching_Hospital|Batticaloa Teaching Hospital]] revealed 76% of patients who attempted suicide were from rural areas while 15% were from urban areas&lt;ref&gt;{{Cite book|url=http://ir.lib.seu.ac.lk/handle/123456789/1457|title=The influence of common risk factors for the patient with attempted suicide hospitalized at the teaching hospital, Batticaloa|last=Kisokanth|first=G.|last2=Najeem|first2=M. M.|last3=Karunakaran|first3=K. E.|date=2014-08-02|publisher=South Eastern University of Sri Lanka, University Park, Oluvil #32360, Sri Lanka|isbn=978-955-627-053-2|language=en-US}}&lt;/ref&gt;. The Sri Lankan government should consider the economical impacts that poverty has on mental health and implement ways to aid poverty-stricken individuals with mental health concerns.   

=== Stigmas ===
Stigma consists of the &quot;combined effect of prejudice, ignorance and discrimination.&quot;&lt;ref name=&quot;:10&quot;&gt;{{Cite web|url=http://www.researchgate.net/publication/233990797_The_Stigma_of_Mental_Illness_in_Sri_Lanka_The_Perspectives_of_Community_Mental_Health_Workers|title=(PDF) The Stigma of Mental Illness in Sri Lanka: The Perspectives of Community Mental Health Workers|website=ResearchGate|language=en|access-date=2025-07-25}}&lt;/ref&gt;.

A 2012 interview consisting of nine participants (two doctors, three nurses, one occupational therapist, one development worker, and two volunteers) revealed a number of concerning societal viewpoints on individuals with mental health concerns. The interviews revealed that negative judgements were not only levied against the individual with the mental illness, but also the family. Families hid mentally ill family members from the public to avoid &quot;shame&quot; and possible hinderances in marriage proposals. Views that mentally ill individuals were &quot;violent&quot; served as the motivating factor behind socially isolating those with mental illness from their communities. Interviewees mentioned that individuals dealing with mental health challenges would be attacked with stones and called &quot;derogatory names.&quot; A lack of community awareness regarding mental health and negative portrayals of mentally ill individuals in media exacerbates stigmatization, though the researchers commented that the media was &quot;improving&quot; in their depiction of mental illness. Beliefs that illnesses are caused by &quot;spirits&quot; can be problematic for individuals dealing with mental health issues and suggests poor mental health awareness. Mental health workers themselves believed that they were being stigmatized, as mental health was reportedly not taken as seriously as physical health. Despite the intriguing perspectives provided, the small sample size and usage of snow sampling raise questionable concerns regarding the generalizability of the results&lt;ref name=&quot;:10&quot; /&gt;. 

Improving media portrayal of subjects concerning mental health and involving community members in interventions dealing with mental health issues are ways that could destigmatize mental health amongst communities in Sri Lanka. Tying collaborations between allopathic services and traditional healers instead of having these two services work individually could enhance engagement between traditional medicine and Western medicine.

=== Suicide Trends &amp; Risk Factors ===
Suicide is defined as &quot;the act of killing oneself deliberately, initiated and performed by the person concerned in the full knowledge or expectation of its fatal outcome&quot;&lt;ref name=&quot;:11&quot;&gt;{{Cite book|title=The neuroscience of suicidal behavior|last=Heeringen|first=Kees van|date=2018|publisher=Cambridge University Press|isbn=978-1-316-60290-4|series=Cambridge fundamentals of neuroscience in psychology|location=Cambridge, United Kingdom New York, NY, USA Port Melbourne, VIC, Australia New Delhi, India Singapore}}&lt;/ref&gt;. Although Sri Lanka has seen a significant reduction in suicide rates from the mid 1990s, largely stemming from its ban on extremely toxic pesticide products, suicide and self harm remains a significant issue. The suicide rate per 100,000 people increased from 14.0 in 2019 to [https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide 15.0 in 2022] (according to WHO). On average, 27 males per 100,000 males and 5 females per 100,000 females committed suicide in 2022&lt;ref&gt;{{Cite journal|last=Kithulagoda|first=A. S.|last2=Gunasinghe|first2=U. C. M.|last3=Senevirathna|first3=J. M. M. S.|last4=Nufail|first4=A. L. M.|last5=Alahakoon|first5=A. M. S. S.|date=2025-07-16|title=An Analysis of Attempted Suicide Cases Registered at Teaching Hospital Batticaloa, Sri Lanka|url=https://bmj.sljol.info/articles/10.4038/bmj.v19i1.67|journal=Batticaloa Medical Journal|language=en-US|volume=19|issue=1|doi=10.4038/bmj.v19i1.67|issn=1800-4903}}&lt;/ref&gt;. Hanging appears to be the most used method for suicide for both males and females, with studies revealing a steady increase in recent years&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|last=Bandara|first=Piumee|last2=Wickrama|first2=Prabath|last3=Sivayokan|first3=Sambasivamoorthy|last4=Knipe|first4=Duleeka|last5=Rajapakse|first5=Thilini|date=2024-04-17|title=Reflections on the trends of suicide in Sri Lanka, 1997–2022: The need for continued vigilance|url=https://journals.plos.org/globalpublichealth/article?id=10.1371/journal.pgph.0003054|journal=PLOS Global Public Health|language=en|volume=4|issue=4|pages=e0003054|doi=10.1371/journal.pgph.0003054|issn=2767-3375|pmc=11023397|pmid=38630779}}&lt;/ref&gt;. 

From 2023 to 2024, a group of researchers from the [[w:Eastern_University,_Sri_Lanka|Eastern University in Sri Lanka]] assessed 828 patients admitted to the Teaching Hospital in [[w:Batticaloa,_Sri_Lanka|Batticaloa, Sri Lanka]] for attempted suicide. They concluded that suicide prevention programs should be attuned to younger people (ages 15 to 35 in the study), emphasize the importance of education and reducing unemployment, and increase social support in the Tamil community. Despite accounting for other factors that could lead to suicidal ideation (ie, poverty), the results from this study suffer in external validity as 90% of the patients were Tamil and over 50% were between 16 and 25 years. In addition, correlations between suicide and unemployment rates have been questioned, with [[w:Austerity|austerity]] being a more reliable indicator of suicide rates than unemployment rates&lt;ref name=&quot;:11&quot; /&gt;. Further comprehensive studies on risk factors relating to suicide should be studied to examine correlations between unemployment rates and austerity measures.  

The WHO suggests implementing evidence-based suicide prevention programs, such as [https://www.who.int/initiatives/live-life-initiative-for-suicide-prevention LIVE LIFE], to reduce the national suicide rate&lt;ref&gt;{{Cite web|url=https://www.who.int/srilanka/news/detail/06-09-2024-world-suicide-prevention-day-2024--changing-the-narrative-on-suicide|title=World Suicide Prevention day 2024 “Changing the Narrative on Suicide”|website=www.who.int|language=en|access-date=2025-07-29}}&lt;/ref&gt;. Media depictions of suicidal methods, such as hanging, can lead to sensationalism and the media should be cautious of such displays in movies and TV shows&lt;ref name=&quot;:12&quot; /&gt;. Awareness of depression and other mental health issues can serve as a safeguard against suicidal ideation in Sri Lankan men and women.  

== Role of Religion ==
According to the last demographic report (2012), 70.2% of Sri Lankans are Buddhist, 12.6% are Hindus, 9.7% are Muslims, and 7.4% are Christians. The Theravada Buddhist community makes up the majority in several provinces throughout the country&lt;ref&gt;{{Cite web|url=https://www.state.gov/reports/2022-report-on-international-religious-freedom/sri-lanka/|title=Sri Lanka|website=United States Department of State|language=en-US|access-date=2025-08-07}}&lt;/ref&gt;. Religion, especially Theravada Buddhism, has had a significant influence on not only the historical treatment of mental health in the country, but also everyday life&lt;ref name=&quot;:15&quot; /&gt;. The [[w:Mahāvaṃsa|''Mahāvaṃsa'']] details hospitals treating patients suffering from mental health issues as early as the 4th century BC. Additionally, the 1700s Nayaka king [[w:Kirti_Sri_Rajasinha|Kirthi Sri Rajasinghe]] detailed the implementation of Buddhist philosophy in psychiatry&lt;ref name=&quot;:4&quot; /&gt;&lt;ref name=&quot;:17&quot;&gt;{{Cite journal|last=Alwis|first=L. A. P. De|date=2017-12-05|title=Development of civil commitment statutes (laws of involuntary detention and treatment) in Sri Lanka: a historical review|url=https://mljsl.sljol.info/articles/10.4038/mljsl.v5i1.7351|journal=Medico-Legal Journal of Sri Lanka|language=en|volume=5|issue=1|doi=10.4038/mljsl.v5i1.7351|issn=2012-8231}}&lt;/ref&gt;.

Modern-day empirical studies have attested to the usefulness of religion in mitigating stress and elevating mental health&lt;ref&gt;{{Cite book|url=https://doi.org/10.1007/978-94-007-4276-5_22|title=Religion and Mental Health|last=Schieman|first=Scott|last2=Bierman|first2=Alex|last3=Ellison|first3=Christopher G.|date=2013|publisher=Springer Netherlands|isbn=978-94-007-4276-5|editor-last=Aneshensel|editor-first=Carol S.|location=Dordrecht|pages=457–478|language=en|doi=10.1007/978-94-007-4276-5_22|editor-last2=Phelan|editor-first2=Jo C.|editor-last3=Bierman|editor-first3=Alex}}&lt;/ref&gt;. Religion has been found to be positively correlated with improved mental health, and more religious patients were concluded to have &quot;better mental health and adapt[ed] more quickly to health problems&quot; versus patients who weren't religious&lt;ref&gt;{{Cite journal|last=Koenig|first=Harold G.|date=2012|title=Religion, spirituality, and health: the research and clinical implications|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC3671693/|journal=ISRN psychiatry|volume=2012|pages=278730|doi=10.5402/2012/278730|issn=2090-7966|pmc=3671693|pmid=23762764}}&lt;/ref&gt;. [https://www.researchgate.net/scientific-contributions/T-N-Wickramarathna-2247724082 Dr. Wickramarathna] of the University Psychiatry Unit (UPU) at the National Hospital of Sri Lanka (NHSL) argues that psychiatrists must strive for a balance in their approach to patients and &quot;make positive use of religion in [their] practice[s]&quot;&lt;ref&gt;{{Cite journal|last=Wickramarathna|first=T. N.|date=2022-12-31|title=Psychiatrists should stand far from the shrine: why and why not we should separate religion from psychiatry|url=https://sljpsyc.sljol.info/articles/10.4038/sljpsyc.v13i2.8397|journal=Sri Lanka Journal of Psychiatry|language=en|volume=13|issue=2|doi=10.4038/sljpsyc.v13i2.8397|issn=2012-6883}}&lt;/ref&gt;.

=== Buddhism ===
27 Sinhalese Buddhists from four Buddhist temples were selected for a series of 70-minute interviews and focus group discussions with the aim of learning the Sinhala Buddhist understanding and experience of spiritual well-being and psychological well-being. The interviewees held spiritual wellness to be the &quot;center&quot; of overall wellness, the &quot;precondition for a successful life&quot;&lt;ref name=&quot;:14&quot;&gt;{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/articles/10.4038/sljss.v44i1.7990|journal=Sri Lanka Journal of Social Sciences|language=en-US|volume=44|issue=1|doi=10.4038/sljss.v44i1.7990|issn=0258-9710}}&lt;/ref&gt;. Sinhala Buddhists believe that wellness cannot be achieved without spiritual tranquility. The report states that participants emphasized that spirituality &quot;cannot be directly intervened&quot; and can only be seen through &quot;[interactions] with society&quot;&lt;ref name=&quot;:14&quot; /&gt;. Despite the ''athmaya'' (soul) being &quot;unreachable&quot;, it can be &quot;intervened&quot;, or treated, through the actions of the mind and body with society&lt;ref name=&quot;:14&quot; /&gt;. One being &quot;psychologically ill&quot; can affect one's spiritual being, as the participants reported in their interviews, and can be impaired through &quot;lifestyle stressors, environmental and socio-cultural causes, non-human related causes and bad-karma in the past lives&quot;&lt;ref name=&quot;:14&quot; /&gt;.  

The researchers concluded that despite Sinhala Buddhists not being able to articulately decipher the discrepancies between psychological well-being and spiritual well-being, they are able to conceptualize and maintain a culturally embedded understanding between the two, serving as reputable evidence of the integration of mental health in Sinhala Buddhist practices. However, it is important to note that these results come from a very small sample size and cannot be generalized to all Sri Lankan Buddhists.   

In addition, a 2009 study found that a belief in karma was correlated with poor health. However, an earlier study found a positive correlation between the reliance on the [[w:Karma_in_Buddhism|Buddhist concept of karma]] and trauma, inferencing Buddhist karma being a prevalent response to trauma&lt;ref&gt;{{Cite journal|last=Levy|first=Becca R.|last2=Slade|first2=Martin D.|last3=Ranasinghe|first3=Padmini|date=2009-03|title=Causal thinking after a tsunami wave: karma beliefs, pessimistic explanatory style and health among Sri Lankan survivors|url=https://pubmed.ncbi.nlm.nih.gov/19229624|journal=Journal of Religion and Health|volume=48|issue=1|pages=38–45|doi=10.1007/s10943-008-9162-5|issn=1573-6571|pmid=19229624}}&lt;/ref&gt;. Overall, the effectiveness of karma as a coping mechanism appears to be conflicted.   

Studies indicate that other practices of Buddhism seem to be utilized by individuals affected by the war. 40% of Sri Lankan Buddhists affected by the 2004 tsunami found the Buddhist ritual ''Bodhipuja'' to be helpful in dealing with traumatic experiences&lt;ref&gt;{{Cite web|url=https://jmvh.org/article/mental-health-and-the-role-of-cultural-and-religious-support-in-the-assistance-of-disabled-veterans-in-sri-lanka/|title=Mental Health and the Role of Cultural and Religious Support in the Assistance of Disabled Veterans in Sri Lanka|website=JMVH|language=en-US|access-date=2025-08-12}}&lt;/ref&gt;.    

=== Catholicism ===
Catholic counseling refers to &quot;a nuanced and holistic mental health care paradigm that intricately weaves together psychological science with the moral, spiritual, and pastoral traditions of the Catholic Church&quot;&lt;ref name=&quot;:13&quot;&gt;Perera, U. [https://www.researchgate.net/profile/Udeshini-Perera/publication/394095042_Catholic_Counselling_in_Sri_Lanka_Integrating_Faith_Psychology_and_Cultural_Healing/links/6889303af8031739e6098c79/Catholic-Counselling-in-Sri-Lanka-Integrating-Faith-Psychology-and-Cultural-Healing.pdf Catholic Counselling in Sri Lanka: Integrating Faith, Psychology, and Cultural Healing]. July 2025.&lt;/ref&gt; and aims to assimilate Catholic theology and evidence-based psychological treatment while including Sri Lankan cultural elements. This is achieved through emphasis on community cohesion and a locally-based understanding of &quot;personhood&quot;&lt;ref name=&quot;:13&quot; /&gt;. 

The origins of Catholic counseling trace back to the introduction of Roman Catholicism to the island in the 1600s, with the focus of the early Sri Lankan Catholic community being on &quot;[[w:Evangelism|evangelization]], education, and sacramental formation&quot;. Demand for counseling services in general increased due to the impacts of the Sri Lankan Civil War, where Catholic organizations (Caritas Sri Lanka, Seth Sarana, Subodhi Integral Centre (Piliyandala), etc.) established several Catholic-based trauma-informed programmes for victims of the Civil War. Programmes use group therapy, forgiveness rituals, and narrative repairs to alleviate war trauma. 

Examples of integration of Catholic virtues and counseling can be seen in [[w:Cognitive_Behavioral_Therapy|Cognitive Behavioral Therapy]] (CBT), where &quot;hope&quot; and &quot;humility&quot; are used as the frameworks for creating spiritual resilience&lt;ref name=&quot;:13&quot; /&gt;. The general Christian call for &quot;agape love and acceptance&quot; is echoed by the concept of [[w:Unconditional_positive_regard|unconditional positive regard]]. ''[[w:Lectio_Divina|Lectio Divina]]'' (Catholic prayer and meditation) and ''Marian devotions'' are integrated into therapeutic practices to achieve emotional regulation and mindfulness.   

Senior Lecturer [https://www.researchgate.net/profile/Udeshini-Perera Udeshini Perera] of the University of Colombo articulates a critical role of Catholic counseling. She claims that secular counseling fails to address the &quot;spiritual roots of distress and moral confusion&quot;. Catholic counseling fills in this gap by integrating &quot;psychological insights with a transcendent orientation, supporting lasting transformation and integrity&quot;&lt;ref name=&quot;:13&quot; /&gt;.  

As of 2025, no formal accreditation or standardized training exists for [[w:Pastoral_counseling|pastoral counselors]] in Sri Lanka, hampering the legitimacy of Catholic counseling. Udeshini Perera remarks that mental health stigma, lack of standardized training, research regarding Catholic counseling effectiveness, and acceptance of the combination of religion and science in a professional setting present challenges for Catholic pastoral counseling in the country. Additionally, Catholic psychiatry in Sri Lanka appears to be under-researched, and evidence of its empirical effects on followers appears sparse. Further research is needed in assessing the empirical effects of Catholic counseling in Sri Lanka.  

=== Islam ===
The literature on the empirical effects of Islamic-based psychotherapy in Sri Lanka is limited. Research is limited to a 2012 case study of a 21-year-old Muslim woman experiencing episodic possession states. The patient ceased attending psychiatric services and opted for religious rituals. The patient reported, in a follow-up visit, that the possession states had been absent for 3 months since her switch to religious rituals. The woman and her family attributed the apparent improvement of her condition to religious rituals&lt;ref&gt;{{Cite journal|last=Hanwella|first=Raveen|last2=de Silva|first2=Varuni|last3=Yoosuf|first3=Alam|last4=Karunaratne|first4=Sanjeewani|last5=de Silva|first5=Pushpa|date=2012|title=Religious Beliefs, Possession States, and Spirits: Three Case Studies from Sri Lanka|url=http://www.hindawi.com/journals/crips/2012/232740/|journal=Case Reports in Psychiatry|language=en|volume=2012|pages=1–3|doi=10.1155/2012/232740|issn=2090-682X|pmc=3437272|pmid=22970398}}&lt;/ref&gt;.

Future recommendations would be to conduct research on the foundations of Islamic psychiatry in the country, and to observe the rituals implemented and their effects on patients. Studies have found that Islamic prayer can be an effective means of &quot;support and coping&quot;&lt;ref name=&quot;:15&quot; /&gt;. Seven world-wide case studies using Islamic-based psychotherapy on patients, consisting of religious rituals such as scriptural reading from the [[w:Quran|Quran]], teaching of fundamental Islamic concepts (such as ''[[w:Tawakkul|tawakkul]]''), and active implementation of contemplation (''[[w:Tadabbur|tadabbur]]''), have reported positive effects in decreasing cognitive and emotional symptoms associated with &quot;religious, obsessive-compulsive disorder, depression, agoraphobia, generalized anxiety disorder, grief, and substance use disorder.”&lt;ref&gt;{{Cite journal|last=Kurhade|first=Chhaya Shantaram|last2=Jagannathan|first2=Aarti|last3=Varambally|first3=Shivarama|last4=Shivanna|first4=Sushrutha|date=2022-01|title=Religion-based interventions for mental health disorders: A systematic review|url=https://journals.lww.com/10.4103/ijoyppp.ijoyppp_14_21|journal=Journal of Applied Consciousness Studies|language=en|volume=10|issue=1|pages=20–33|doi=10.4103/ijoyppp.ijoyppp_14_21|issn=2949-6993}}&lt;/ref&gt; Additionally, a community-based study of elderly patients in Bangalore, India receiving Islamic-based psychotherapy observed decreased exhibitions of sleep disorders, eating disorders, and emotional distress&lt;ref&gt;{{Cite journal|last=Hafeez|first=Nimin|last2=Sanjay|first2=Thittamaranahalli Varadappa|last3=Puthussery|first3=Yannick Poulose|last4=Madhusudan|first4=Muralidhar|last5=Kariyappa|first5=Poornima Muddaiah|last6=Kulkarni|first6=Sridevi|last7=Raj|first7=Lavanya|date=2023-12-31|title=Spiritual practices among elderly, prevalence, pattern and associated factors: a community-based study from rural Bengaluru, India|url=https://jccpsl.sljol.info/articles/10.4038/jccpsl.v29i4.8610|journal=Journal of the College of Community Physicians of Sri Lanka|language=en|volume=29|issue=4|doi=10.4038/jccpsl.v29i4.8610|issn=1391-3174}}&lt;/ref&gt;. 

=== Hinduism ===
Despite Hindus being 12.6% of the population of Sri Lanka, the research on Hinduism-based therapy in the country is limited. Ayurvedic medicine, a form of medicine originating from ancient India, predominated the Sri Lankan medical landscape for over 2,000 years and even had a symbiotic relationship with Sinhalese medicine, which also played a significant and influential role in the country's medical framework&lt;ref name=&quot;:0&quot; /&gt;&lt;ref&gt;{{Cite journal|last=Udayanga|first=Samitha|date=2021-06-30|title=Cultural understanding of ‘spiritual well-being’ and ‘psychological well-being’ among Sinhalese Buddhists in Sri Lanka|url=https://sljss.sljol.info/article/10.4038/sljss.v44i1.7990/|journal=Sri Lanka Journal of Social Sciences|volume=44|issue=1|pages=33|doi=10.4038/sljss.v44i1.7990|issn=2478-1169}}&lt;/ref&gt;. Despite its historical dominance, Ayurvedic medicine has been challenged against modern evidence-based medical standards&lt;ref&gt;{{Cite book|url=https://philarchive.org/rec/DOMAAT|title=Ayurveda: Ancient Tradition or Pseudoscientific Practice? A Philosophical Inquiry|last=Dominic|first=Shubham K.}}&lt;/ref&gt;.

=== Comparative synthesis ===
Taking an overarching review of the role of religion in Sri Lanka, methods to improve mental well-being are practiced by adherents of Buddhism, Hinduism, Islam, and Christianity. These practices are implemented in traditionally-oriented mental health care, which has been reportedly preferred over psychiatric care at times. These rituals practiced across these religions indicate a common theme of psychologically integrated aspects of well-being. Interpretation of trauma is a central use in religion, with religious principles, such as karma and ''tawakkul'', serve as psychologically analogous mechanisms during times of distress.  

In terms of methodological comparisons to the studies described, qualitative interviews have documented Buddhist practices and principles, like Bodhipuja and the belief in karma, in response to traumatic events, while case studies found religious practices by other religious groups, such as a Muslim patient reading Islamic scripture and observing prayer, to reduce emotional distress. Peer-reviewed sources have documented Catholic practices and principles, such as ''Lectio Divina'' and unconditional positive regard, in improving mindfulness and emotional regulation. The paper acknowledges limitations in the evaluation of certain findings, such as in Islam and Hinduism. These shortcomings, however, are a reflection of the existing literature and its deficiencies. Empirical findings indicate mental health practices are complex and are multifaceted in their effects. 

Evidently, religion serves a parallel role to psychiatric services in improving mental health. Despite its perceived benefits, the findings surrounding religions' role in mental health suffer from conflicting, and sometimes contradictory, results. Additionally, a disproportionate amount of empirical findings seem to be Buddhist-predominant, while other religions are underrepresented in the research. Regarding research barriers, the methodological approaches implemented to study the practices of religious followers vary, though much of the research was brought from qualitative or case-based studies, impeding generalizability. Another noteworthy issue is that many studies do not utilize standardized, psychiatric measures.

== Future Outlook ==
Despite significant changes to the mental health environment in Sri Lanka, the current legal framework shaping mental health in the country has not been updated since 1956. A Cambridge University Press article detailed many limitations of the Mental Disease Ordinance of 1956, including discrepancies between the legal provisions of involuntary admissions and modern practices, potential exposure to trauma through extra-legal detentions of the mentally ill, and an absence of legal guidelines addressing the restraint of violent patients&lt;ref name=&quot;:6&quot; /&gt;. Participants from Sri Lanka reported in a comparative legislative questionnaire that they felt the mental health laws were &quot;outdated&quot; and descriptions of clinical roles remained ambiguous&lt;ref name=&quot;:16&quot; /&gt;. A draft mental health legislation from 2007 included provisions for human rights, but due to &quot;bureaucratic processes&quot; and a &quot;lack of consensus&quot;, the draft has not been officially approved.

These limitations pose challenges to the standardization of mental healthcare admissions and may impact the rights of detained patients. Detained patients may have their human rights violated due to a lack of updated legal framework, thereby impeding the identification of such violations. Additionally, with the lack of clarity on clinical roles, clinical responsibilities may not be routinely recognized and observed, leading to role confusion and potential legal ramifications&lt;ref name=&quot;:16&quot;&gt;{{Cite journal|last=Dey|first=Sangeeta|last2=Mellsop|first2=Graham|last3=Diesfeld|first3=Kate|last4=Dharmawardene|first4=Vajira|last5=Mendis|first5=Susitha|last6=Chaudhuri|first6=Sreemanti|last7=Deb|first7=Aniruddha|last8=Huq|first8=Nafisa|last9=Ahmed|first9=Helal Uddin|date=2019-10-24|title=Comparing legislation for involuntary admission and treatment of mental illness in four South Asian countries|url=https://ijmhs.biomedcentral.com/articles/10.1186/s13033-019-0322-7|journal=International Journal of Mental Health Systems|volume=13|issue=1|pages=67|doi=10.1186/s13033-019-0322-7|issn=1752-4458|pmc=6813093|pmid=31666805}}&lt;/ref&gt;. Lastly, current efforts should ideally move beyond just addressing poverty-centered matters, and expand efforts to domestic violence victims and children with disabilities, as shelters and specialized services are limited&lt;ref name=&quot;:82&quot;&gt;{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}&lt;/ref&gt;.

Stagnation in policy development leaves Sri Lanka without a practical, up-to-date, and comprehensive mental health framework, which could put both clinicians and patients at risk. Future reforms should include clarification on the treatment and detention process of involuntary admissions of patients and a clear delineation of clinical roles and their responsibilities. Without the necessary reforms to advance Sri Lankan mental health legislation, clinicians and vulnerable patients may suffer from a lack of comprehensive oversight. 

==Acknowledgements==
No acknowledgments have been made.

==References==
{{reflist|35em}}

[[Category:Mental health]]
[[Category:Sri Lanka]]</text>
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      <comment>/* Use of AI */ Reply</comment>
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      <text bytes="2144" sha1="qlq4xwb7tbbgte4gwyj2o7j28sge2th" xml:space="preserve">== Use of AI ==

Could we not use generative AI to write the introduction please? It sounds terrible and I feel like it goes against the integrity of the project. [[User:Supersazon|Supersazon]] ([[User talk:Supersazon|discuss]] • [[Special:Contributions/Supersazon|contribs]]) 18:49, 21 October 2025 (UTC)

:I wholeheartedly agree [[User:Laquin|Laquin]] ([[User talk:Laquin|discuss]] • [[Special:Contributions/Laquin|contribs]]) 18:12, 7 December 2025 (UTC)
::{{ping|Laquin}} &amp; {{ping|Supersazon}} Welcome to Wikiversity! Feel free to edit the introduction if you'd like. Also, don't be afraid to pitch in your views on the usage of GenAI on Wikiversity at [[Wikiversity:Colloquium#General_ban_on_direct_use_of_GenAI_output_with_exceptions]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:59, 7 December 2025 (UTC)
:::{{ping|Atcovi}} Thank you! I will take a look. :) [[User:Laquin|Laquin]] ([[User talk:Laquin|discuss]] • [[Special:Contributions/Laquin|contribs]]) 20:04, 7 December 2025 (UTC)
::::@[[User:Atcovi|Atcovi]]: I suspect we have been snookered by two hit and run trolls.
::::1)     Both the Supersazon and the Laquin user id’s were deleted some time ago. The comments on this page are the only contributions either one ever made. 
::::2)     The comments they left are generic. There is no evidence either of them read the introduction or the course materials.
::::3)     The text of the introduction is brief and, in my opinion, well written. The claim that it “Sounds terrible” is unsubstantiated and false.
::::4)     The introductory text is properly attributed. The prompt used directed the LMM to use the course text as its data source. 
::::Therefore, I recommend we discount their comments and recognize that the claim the “resource includes substantial content generated by artificial intelligence”. Is unsubstantiated.
::::Please remove the AI banner because it makes an unsubstantiated claim.
::::Thanks [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 14:17, 16 July 2026 (UTC)</text>
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      <comment>Gavin R Putland moved page [[Talk:WikiJournal Preprints/Coordinates Last: Vector Analysis Done Fast]] to [[Talk:Coordinates Last: Vector Analysis Done Fast]]: Conversion from preprint to learning resource.</comment>
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      <text bytes="333" sha1="acjzf1l80v9r63cylg5hcukgmdmyb1c" xml:space="preserve">I am open to advice as to whether this article is too long for a WikiJournal and, if so, whether it is more suitable for a Wikiversity page or a Wikibook. &amp;mdash;&amp;nbsp;[[User:Gavin R Putland|Gavin R Putland]] ([[User talk:Gavin R Putland|discuss]] • [[Special:Contributions/Gavin R Putland|contribs]]) 12:58, 9 November 2025 (UTC).</text>
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      <comment>/* The 16-cell 4-orthoplex */</comment>
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      <text bytes="87577" sha1="k93zkmnkwe1nfrv4pes399dcf3pr002" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords  form an ''edge polygon'' of the 16-cell {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges, for example in the characteristic rotation of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

We can also rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 24-cell edges. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in any invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit, and its entire orbit traces an isocline circle in 4-space over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords. There are two chiral ways we can rotate the 24-cell isoclinically in invariant great hexagon planes containing its edges, called the ''great hexagon left rotation'' and the ''great hexagon right rotation,'' respectively.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]
We can rotate the 24-cell isoclinically in 12 Clifford parallel invariant planes containing two &lt;math&gt;r_{1}&lt;/math&gt; edges each, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon left rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; 2{12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. The orbit of each vertex traces an isocline circle in 4-space over 12 &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. 

[[File:Regular_star_figure_8(3,1).svg|thumb|left|150px|{24/8}=8{3}&lt;small&gt; &lt;/small&gt;shows 8 of 32&lt;small&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; triangles in the 24-cell]]
We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six &lt;math&gt;r_{2}&lt;/math&gt; edges each, over &lt;math&gt;r_{4}&lt;/math&gt; isocline chords. This is the  ''great hexagon right rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_4&lt;/math&gt; 8{3} star polygon which constructs &lt;math&gt;1/r_4&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_4&lt;/math&gt; chord makes four 30° turns.  Eight Clifford parallel triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;r_4&lt;/math&gt; chords form a circular fibration of 8 twisted parallel strands {24/8}=8{3} that intersects each 24-cell vertex once. In three successive 60° isoclinic displacements each vertex circles a triangle and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the four distinct triangles which intersect at the vertex. The isocline curves over a self-intersecting dodecagram of 12 &lt;math&gt;r_4&lt;/math&gt; chords.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;6&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chord
! colspan=&quot;3&quot; |Isocline chord
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: gainsboro;&quot; |
|1.414~
|1.414~
|- style=&quot;background: gainsboro;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we found three distinct isoclinic rotations. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these and also three other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The edge chords form the rotation's edge {24}-gon, and lie in invariant planes of the rotation. The isocline chords form the rotation's Clifford {24}-gon and lie in the invariant planes completely orthogonal to the edge planes. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°.  

We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges in 6 Clifford parallel invariant great square planes containing four &lt;math&gt;t_{6}&lt;/math&gt; edges each, over &lt;math&gt;t_{6}&lt;/math&gt; isocline chords. The &lt;math&gt;t_6&lt;/math&gt; chord is the 16-cell-&lt;math&gt;r_2&lt;/math&gt; chord. The edge polygon and the Clifford polygon are both {24/6}=6{4}. This is the ''characteristic right rotation of the 24-cell''.  

The rotational curve over each 90° &lt;math&gt;t_6&lt;/math&gt; chord makes six 15° turns.  Six Clifford parallel skew triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;t_6&lt;/math&gt; chords form a circular fibration of six twisted parallel strands that intersects each 24-cell vertex once. &lt;s&gt;In every 360° of isoclinic rotation each vertex circles a skew great square and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the three distinct skew squares which intersect at the vertex and the three distinct skew squares which intersect at its antipodal vertex. The isocline curves over a self-intersecting {24}-gon of &lt;math&gt;t_6&lt;/math&gt; chords.&lt;/s&gt;  

...  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <id>2818444</id>
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      <text bytes="87576" sha1="ljutdehafoan2gscgr14qcvomuoaf8k" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). Consequently 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which contains 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). Consequently the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== The 16-cell 4-orthoplex ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. Their procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it also moves a distance of &lt;math&gt;-1&lt;/math&gt;. Fontaine and Hurley also demonstrated the significance of &lt;math&gt;1/r_i&lt;/math&gt; in Steinbach's Diagonal Product Formula, which says that every chord length is the sum of certain smaller chord lengths. The smaller chords are certain diagonals of the same regular polygon of a smaller edge length, specifically edge length &lt;math&gt;1/r_i&lt;/math&gt; rather than &lt;math&gt;1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Each chord is a distinct 4-vector with a length and a direction. Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation, where vertices circle over the chords of an &lt;math&gt;r_i&lt;/math&gt; polygon. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns. The angle between two &lt;math&gt;r_i&lt;/math&gt; chords is &lt;math&gt;180^\circ - i \times 45^\circ&lt;/math&gt;.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon {8/1} which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of the 16-cell form an ''edge polygon'' {8/2}=2{4}. The two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt; (it is &lt;math&gt;6\pi&lt;/math&gt; in this case), and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes twelve 45° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic revolution each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

We shall refer to this isoclinic rotation as the ''great square rotation characteristic of the 16-cell'', and note once again that it is Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_3&lt;/math&gt; {8/3} star polygon, which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== The 8-cell tesseract ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedrons]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in the great square rotation characteristic of the 16-cell, with the same effect on both alternate-position 16-cells. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords. They form a circular double helix which intersects each vertex of the tesseract once. The double helix is an 8-rung ladder twisted around 3 times, and bent into a circle in the fourth dimension with its ends joined. Each rung is a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges of Clifford parallel great squares, also isocline chords in great square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in great hexagon rotations. The green {12/5} dodecagram is a Clifford polygon.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges, for example in the characteristic rotation of the 16-cell, with the same effect on all three 16-cells. In 720° each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The rotational curve over each 90° &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. The triple helix is an 8-step circular staircase that twists around 3 times, and is bent into a torus in the fourth dimension. Each staircase step is a great triangle of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords.

We can also rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 24-cell edges. A complete 24-cell great circle edge plane revolution requires 720° like a complete 16-cell great circle edge plane revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each.  An isoclinic rotation by 60° in any invariant central plane containing a 24-cell edge takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the invariant planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square. All 24 vertices move at once on Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° rotational displacement is a one-twelfth segment of its geodesic orbit, and its entire orbit traces an isocline circle in 4-space over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords. There are two chiral ways we can rotate the 24-cell isoclinically in invariant great hexagon planes containing its edges, called the ''great hexagon left rotation'' and the ''great hexagon right rotation,'' respectively.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} shows 2 dodecagram isoclines of &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; chords in the 24-cell]]
We can rotate the 24-cell isoclinically in 12 Clifford parallel invariant planes containing two &lt;math&gt;r_{1}&lt;/math&gt; edges each, over &lt;math&gt;r_{5}&lt;/math&gt; isocline chords. This is the ''great hexagon left rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_5&lt;/math&gt; 2{12/5} star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;r_5&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. The orbit of each vertex traces an isocline circle in 4-space over 12 &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 5 times in a moving invariant rotation plane. In the course of a 720° revolution each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. 

[[File:Regular_star_figure_8(3,1).svg|thumb|left|150px|{24/8}=8{3}&lt;small&gt; &lt;/small&gt;shows 8 of 32&lt;small&gt; &lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; triangles in the 24-cell]]
We can rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing six &lt;math&gt;r_{2}&lt;/math&gt; edges each, over &lt;math&gt;r_{4}&lt;/math&gt; isocline chords. This is the  ''great hexagon right rotation characteristic of the 24-cell'', also Fontaine and Hurley's counterclockwise rotation over the &lt;math&gt;r_4&lt;/math&gt; 8{3} star polygon which constructs &lt;math&gt;1/r_4&lt;/math&gt;. The rotational curve over each 120° &lt;math&gt;r_4&lt;/math&gt; chord makes four 30° turns.  Eight Clifford parallel triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;r_4&lt;/math&gt; chords form a circular fibration of 8 twisted parallel strands {24/8}=8{3} that intersects each 24-cell vertex once. In three successive 60° isoclinic displacements each vertex circles a triangle and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the four distinct triangles which intersect at the vertex. The isocline curves over a self-intersecting dodecagram of 12 &lt;math&gt;r_4&lt;/math&gt; chords.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;6&quot; |6 distinct 180° chord pairs make 6 distinct isoclinic rotations
|-
! colspan=&quot;3&quot; |Edge chord
! colspan=&quot;3&quot; |Isocline chord
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_1&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;{24/1}={24}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-11.svg|100px]]&lt;br&gt;{24/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{11}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|15°
|165°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_2&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,1).svg|100px]]&lt;br&gt;{24/2}=2{12}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(12,5).svg|100px]]&lt;br&gt;{24/10}=2{12/5}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|30°
|150°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_3&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,1).svg|100px]]&lt;br&gt;{24/3}=3{8}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(8,3).svg|100px]]&lt;br&gt;{24/9}=3{8/3}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{9}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|45°
|135°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;{24/4}=4{6}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_8(3,1).svg|100px]]&lt;br&gt;{24/8}=8{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{8}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-5.svg|100px]]&lt;br&gt;{24/5}
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_24-7.svg|100px]]&lt;br&gt;{24/7}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{7}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: gainsboro;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro;&quot; |
|75°
|105°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;t_6&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;{24/6}=6{4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;t_{6}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: gainsboro;&quot; |
|1.414~
|1.414~
|- style=&quot;background: gainsboro;&quot; |
|90°
|90°
|}

By examining the chords &lt;math&gt;r_i&lt;/math&gt; of the 24-cell's Petrie {12}-gon we found three distinct isoclinic rotations. If we examine the chords &lt;math&gt;t_i&lt;/math&gt; of the 24-cell's {24}-gon we find these and also three other distinct isoclinic rotations.  

Each row of the table is a distinct isoclinic rotation of the 24-cell characterized by a pair of chords that sum to 180°. The edge chords form the rotation's edge {24}-gon, and lie in invariant planes of the rotation. The isocline chords form the rotation's Clifford {24}-gon and lie in the invariant planes completely orthogonal to the edge planes. The rotational angle between successive edge chords and the rotational angle between successive isocline chords also sum to 180°.  

We can rotate the 24-cell isoclinically in Clifford parallel invariant planes containing 16-cell edges in 6 Clifford parallel invariant great square planes containing four &lt;math&gt;t_{6}&lt;/math&gt; edges each, over &lt;math&gt;t_{6}&lt;/math&gt; isocline chords. The &lt;math&gt;t_6&lt;/math&gt; chord is the 16-cell-&lt;math&gt;r_2&lt;/math&gt; chord. The edge polygon and the Clifford polygon are both {24/6}=6{4}. This is the ''characteristic right rotation of the 24-cell''.  

The rotational curve over each 90° &lt;math&gt;t_6&lt;/math&gt; chord makes six 15° turns.  Six Clifford parallel skew triangle geodesic isoclines of circumference &lt;math&gt;2\pi&lt;/math&gt; over &lt;math&gt;t_6&lt;/math&gt; chords form a circular fibration of six twisted parallel strands that intersects each 24-cell vertex once. &lt;s&gt;In every 360° of isoclinic rotation each vertex circles a skew great square and returns to its original position, but the 24-cell returns to its original orientation only after each vertex has completed circuits of the three distinct skew squares which intersect at the vertex and the three distinct skew squares which intersect at its antipodal vertex. The isocline curves over a self-intersecting {24}-gon of &lt;math&gt;t_6&lt;/math&gt; chords.&lt;/s&gt;  

...  

{{Clear}}

== The 600-cell ==
[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are:

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation.

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;7&quot; |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short edge chord
! Section
! colspan=&quot;3&quot; |Long isocline chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_0&lt;/math&gt;
|0°
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{15}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
|0°
|180°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_1&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}=2{15/7}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{14}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: palegreen;&quot; |
|0.618~
|1.902~
|- style=&quot;background: palegreen;&quot; |
|12°
|168°
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_2&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{13}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: gainsboro;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
|24°
|156°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_3&lt;/math&gt;
|36°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;4&quot; |[[File:V1 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{12}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: yellow;&quot; |
|36°
|144°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_4&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;4&quot; |
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{11}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|48°
|132°
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_5&lt;/math&gt;
|60°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;4&quot; |[[File:V2 dodecahedron.png|100px]]&lt;br&gt;Dodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{10}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: palegreen;&quot; |
|60°
|120°
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{6}&lt;/math&gt;
|72°
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;4&quot; |[[File:V3 icosahedron.png|100px]]&lt;br&gt;Icosahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{9}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: yellow;&quot; |
|72°
|108°
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;4&quot; |&lt;math&gt;r_{7}&lt;/math&gt;
|90°
| rowspan=&quot;4&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;4&quot; |[[File:V4 icosidodecahedron.png|100px]]&lt;br&gt;Icosidodecahedron
| rowspan=&quot;4&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|90°
| rowspan=&quot;4&quot; |&lt;math&gt;r_{8}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|- style=&quot;background: seashell;&quot; |
|84°
|96°
|}

The list of 600-cell chords &lt;math&gt;r_{i}&lt;/math&gt; can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30/n}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon.

Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;\phi^{-1}&lt;/math&gt; is a icosahedron vertex figure, and the largest section at radial distance &lt;math&gt;\sqrt{2}&lt;/math&gt; is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;\sqrt{2}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance &lt;math&gt;\sqrt{2+\phi}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section).

[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;]]
We can rotate the 600-cell isoclinically in the great square rotation characteristic of the 16-cell,  with the same effect on 15 disjoint 16-cells. Each 90° displacement takes 15 pairs of completely orthogonal invariant great square planes to each other. In the course of a 720° revolution each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The rotational curve over each 90° chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; form a circular fibration of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once.

The 600-cell has another distinct isoclinic rotation in invariant great square planes, which has period 30 and visits every vertex of a 600-cell Petrie polygon. This [''great square left rotation characteristic of the 600-cell]'' takes place over &lt;math&gt;r_7&lt;/math&gt; edge chords and &lt;math&gt;r_8&lt;/math&gt; isocline chords. The {30/7} edge polygon is a skew helix of circumference &lt;math&gt;14\pi&lt;/math&gt; with each &lt;math&gt;r_7&lt;/math&gt; edge belonging to a distinct great square. The four {30/7} polygrams contribute one edge each to 30 great squares. Each 90° displacement takes every 16-cell to another 16-cell. The vertices of the invariant great squares each make seven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 90° {30/7} edge makes seven 12° turns. Four Clifford parallel {30/7} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once. The {30/8}=2{15/4} Clifford polygon is a compound of two skew {15/4} pentadecagrams of circumference &lt;math&gt;16\pi&lt;/math&gt; with each &lt;math&gt;r_8&lt;/math&gt; isocline chord belonging to a distinct 16-cell. The four {30/8} polygrams contribute one edge each to 30 great squares. The rotational curve over each 90° {30/8} isocline chord makes eight 12° turns. Four Clifford parallel {30/8} geodesics of circumference &lt;math&gt;16\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}
[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} &lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt; ]]
We can rotate the 600-cell isoclinically in the great hexagon rotation characteristic of the 24-cell, over &lt;math&gt;\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;\sqrt{3}&lt;/math&gt; isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° revolution each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; form a circular fibration of ten twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. 

The 600-cell has another distinct isoclinic rotation in invariant great hexagon planes, over &lt;math&gt;r_{4}=\sqrt{1}&lt;/math&gt; edge chords and &lt;math&gt;r_{11}=\sqrt{3}&lt;/math&gt; isocline chords This [''invariant great hexagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. Its {30/11} Clifford polygon is a skew helix where each &lt;math&gt;r_{11}&lt;/math&gt; isocline chord is the &lt;math&gt;\sqrt{3}&lt;/math&gt; diagonal of a great hexagon of a distinct 24-cell. The vertices of the invariant great hexagons of this rotation each make eleven orbits on a great circle within the moving invariant plane in the course of one complete revolution. The rotational curve over each 120° &lt;math&gt;r_{11}&lt;/math&gt; isocline chord makes eleven 12° turns. Four Clifford parallel {30/11} geodesic isoclines of circumference &lt;math&gt;22\pi&lt;/math&gt; over &lt;math&gt;r_{11}&lt;/math&gt; chords form a circular quadruple helix that intersects each 600-cell vertex once. 

We can rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its 36° &lt;math&gt;r_{3}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{12}&lt;/math&gt; isocline chords. This ''invariant great decagon rotation characteristic of the 600-cell'' has period 5 and takes disjoint 24-cells to each other. The rotational curve over each &lt;math&gt;r_{12}&lt;/math&gt; chord of its {5/2} Clifford polygon makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference &lt;math&gt;4\pi&lt;/math&gt; over five &lt;math&gt;r_{12}&lt;/math&gt; chords form a circular fibration of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. 

The rotation of the 600-cell by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on four Clifford parallel geodesic isoclines, displaced 144° in different directions.

The 600-cell has another distinct isoclinic rotation in invariant great decagon planes containing its 36° &lt;math&gt;r_{2}&lt;/math&gt; edges, over 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chords. This [''great decagon left rotation characteristic of the 600-cell]'' has period 30 and visits every vertex of a 600-cell Petrie polygon. The rotational curve over each 144° &lt;math&gt;r_{13}&lt;/math&gt; isocline chord makes thirteen 12° turns. Four Clifford parallel {30/13} geodesic isoclines of circumference &lt;math&gt;26\pi&lt;/math&gt; form a circular quadruple helix that intersects each 600-cell vertex once.

{{Clear}}

== The 5-cell 4-simplex ==

{| class=&quot;wikitable floatright&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;9&quot; |30 chords (15 180° pairs) make 15 distinct section polyhedra
|-
! colspan=&quot;3&quot; |Short chord
! Section
! colspan=&quot;3&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px]]&lt;br&gt;{30/15}=15{2}
|180°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|{{radic|4}}
|- style=&quot;background: palegreen;&quot; |
|0
|2
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px]]&lt;br&gt;{30/1}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px]]&lt;br&gt;{30/14}
|164.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|{{radic|3.927~}}
|- style=&quot;background: palegreen;&quot; |
|0.270~
|1.982~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px]]&lt;br&gt;{30/2}=2{15}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px]]&lt;br&gt;{30/13}
|154.8~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|{{radic|3.809~}}
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|1.952~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px]]&lt;br&gt;{30/3}=3{10}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px]]&lt;br&gt;{30/12}=6{5/2}
|144°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|{{radic|3.618~}}
|- style=&quot;background: yellow;&quot; |
|0.618~
|1.902~
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|{{radic|3.5}}
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|1.871~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px]]&lt;br&gt;{30/4}=2{15/2}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px]]&lt;br&gt;{30/11}
|135.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|{{radic|3.427~}}
|- style=&quot;background: palegreen;&quot; |
|0.757~
|1.851~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|{{radic|3.309~}}
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|1.819~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|{{radic|3.118~}}
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|1.766~
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px]]&lt;br&gt;{30/5}=5{6}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px]]&lt;br&gt;{30/10}=10{3}
|120°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|{{radic|3}}
|- style=&quot;background: palegreen;&quot; |
|1
|1.732~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|{{radic|2.809~}}
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|1.676~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|{{radic|2.691~}}
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|1.640~
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px]]&lt;br&gt;{30/6}=6{5}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px]]&lt;br&gt;{30/9}=3{10/3}
|108°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|{{radic|2.618~}}
|- style=&quot;background: yellow;&quot; |
|1.176~
|1.618~
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px]]&lt;br&gt;{30/8}=2{15/4}
|104.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|{{radic|2.5}}
|- style=&quot;background: palegreen;&quot; |
|1.224~
|1.581~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|{{radic|2.309~}}
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|1.520~
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|{{radic|2.191~}}
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|1.480~
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px]]&lt;br&gt;{30/7}
|90°
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|{{radic|2}}
|- style=&quot;background: seashell;&quot; |
|1.414~
|1.414~
|}

The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|list of thirty 120-cell chords]] &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section at radial distance &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell. Because  [[w:3-sphere|&lt;math&gt;\mathbb{S}^3&lt;/math&gt;]] is spherical, at radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. 

Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

{{Clear}}
== Finally the 120-cell ==
The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron.

The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure.

... 

{{Clear}}

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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The [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=makebot&amp;user=&amp;page=&amp;year=&amp;month=-1 bot status log] contains older entries of granting bot status.   Newer entries will appear in the [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=rights&amp;user=&amp;page=&amp;year=&amp;month=-1 user rights log].</text>
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{{Reflist}}

The manually updated list above may not be current.  Check [http://en.wikiversity.org/w/index.php?title=Special%3AListusers&amp;group=bot&amp;username= bots group user list] which is complete and see contribs for each bot.

The [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=makebot&amp;user=&amp;page=&amp;year=&amp;month=-1 bot status log] contains older entries of granting bot status.   Newer entries will appear in the [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=rights&amp;user=&amp;page=&amp;year=&amp;month=-1 user rights log].</text>
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        <id>3705</id>
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{{Wikiversity:Bots/item|CommonsDelinker|Siebrand|prevention of broken image links|2026-06-23}}
{{Wikiversity:Bots/item|JackBot|JackPotte|[[Special:DoubleRedirects]] automatically and [[Special:UncategorizedPages]] semiautomatically|2026-07-05}}
{{Wikiversity:Bots/item|MaintenanceBot|Dave Braunschweig|Various maintenance tasks.|2024-01-01}}
{{Wikiversity:Bots/item|MediaWiki default|MediaWiki default|setup edits in MediaWiki: namespace; the script is part of MediaWiki.&lt;ref&gt;[[User:MediaWiki default]] is the username used by a system maintenance script. It is ''not'' a bot and is unaffected by blocks.&lt;/ref&gt;|????-??-??&lt;ref&gt;Due to the bug described at [[phabricator:T36873]] the activity of this script no longer appears in Special:Contributions. It is not known when it was last run, or how often it has been used.&lt;/ref&gt;}}
{{Wikiversity:Bots/item|MediaWiki message delivery|MediaWiki message delivery|[[meta:MassMessage]] delivery|2026-07-13}}
{{Wikiversity:Bots/item|MGA73bot|MGA73|Working on files. See [[Wikiversity_talk:File_Review]] for background. Usually tasks are done with replace.py.|2024-09-02}}
{{Wikiversity:Bots/item|Leaderbot|Leaderboard|[[meta:Global reminder bot]]|N/A - it checks this wiki roughly every day, but only responds if needed}}
{{Wikiversity:Bots/item|WorkmarketBot|Evolution and evolvability|syncronising a list of tasks between WikiJournal_User_Group/Technical_editors/tasks and the external software Workmarket.|2024-11-29}}
|}

Notes:
{{Reflist}}

The manually updated list above may not be current.  Check [http://en.wikiversity.org/w/index.php?title=Special%3AListusers&amp;group=bot&amp;username= bots group user list] which is complete and see contribs for each bot.

The [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=makebot&amp;user=&amp;page=&amp;year=&amp;month=-1 bot status log] contains older entries of granting bot status.   Newer entries will appear in the [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=rights&amp;user=&amp;page=&amp;year=&amp;month=-1 user rights log].</text>
      <sha1>mitgld9yjbfgm7r2tki7i4s1fa2t14i</sha1>
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        <username>Mu301</username>
        <id>3705</id>
      </contributor>
      <minor />
      <comment>tidy</comment>
      <origin>2818429</origin>
      <model>wikitext</model>
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      <text bytes="2204" sha1="my1ok0jo7hifbwf7hfcbtqihnyoorad" xml:space="preserve">{| border=&quot;0&quot; align=&quot;center&quot; rules=&quot;all&quot; cellpadding=&quot;3px&quot; class=&quot;wikitable sortable&quot;
!Name of the bot
!Contributions
!Operator
!Functions
!Last activity
{{Wikiversity:Bots/item|CommonsDelinker|Siebrand|prevention of broken image links|2026-06-23}}
{{Wikiversity:Bots/item|JackBot|JackPotte|[[Special:DoubleRedirects]] automatically and [[Special:UncategorizedPages]] semiautomatically|2026-07-05}}
{{Wikiversity:Bots/item|MaintenanceBot|Dave Braunschweig|Various maintenance tasks.|2024-01-01}}
{{Wikiversity:Bots/item|MediaWiki default|MediaWiki default|setup edits in MediaWiki: namespace; the script is part of MediaWiki.&lt;ref&gt;[[User:MediaWiki default]] is the username used by a system maintenance script. It is ''not'' a bot and is unaffected by blocks.&lt;/ref&gt;|????-??-??&lt;ref&gt;Due to the bug described at [[phabricator:T36873]] the activity of this script no longer appears in Special:Contributions. It is not known when it was last run, or how often it has been used.&lt;/ref&gt;}}
{{Wikiversity:Bots/item|MediaWiki message delivery|MediaWiki message delivery|[[meta:MassMessage]] delivery|2026-07-13}}
{{Wikiversity:Bots/item|MGA73bot|MGA73|Working on files. See [[Wikiversity_talk:File_Review]] for background. Usually tasks are done with replace.py.|2024-09-02}}
{{Wikiversity:Bots/item|Leaderbot|Leaderboard|[[meta:Global reminder bot]]|N/A - it checks this wiki roughly every day, but only responds if needed}}
{{Wikiversity:Bots/item|WorkmarketBot|Evolution and evolvability|Syncronising a list of tasks between [[WikiJournal_User_Group/Technical_editors/tasks]] and the external software [https://www.workmarket.com/ Workmarket].|2024-11-29}}
|}

Notes:
{{Reflist}}

The manually updated list above may not be current.  Check [http://en.wikiversity.org/w/index.php?title=Special%3AListusers&amp;group=bot&amp;username= bots group user list] which is complete and see contribs for each bot.

The [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=makebot&amp;user=&amp;page=&amp;year=&amp;month=-1 bot status log] contains older entries of granting bot status.   Newer entries will appear in the [http://en.wikiversity.org/w/index.php?title=Special%3ALog&amp;type=rights&amp;user=&amp;page=&amp;year=&amp;month=-1 user rights log].</text>
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  <page>
    <title>Wikiversity talk:Inactivity policy</title>
    <ns>5</ns>
    <id>330057</id>
    <revision>
      <id>2818423</id>
      <parentid>2818098</parentid>
      <timestamp>2026-07-16T17:18:12Z</timestamp>
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        <username>Mu301</username>
        <id>3705</id>
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      <comment>/* wording */ Reply</comment>
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      <text bytes="5096" sha1="6zvf3vqymo58b8ct02uejogzzzt24m5" xml:space="preserve">== Notice to colloquium ==

What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC)
: A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC)
::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC)
::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC)

== Inactive curator template ==

Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC)
: I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC)
::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''&quot;no edits or no logged actions for 2 years&quot;'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC)
::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC)

== Communication with the SSM and deadlines ==

A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of &quot;peace&quot;.

I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now.

Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC)
: I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC)
::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC)

== wording ==

This may sound pendantic... the prhase &quot;are considered inactive if they have made no edits and logged actions within one year.&quot; Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame, but not necessarily both? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:39, 10 July 2026 (UTC)
: This means an edit and a logged action, together. I am not sure what is the point of desysopping a custodian who makes edits but otherwise does not use their permission(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:45, 10 July 2026 (UTC)
::I updated the draft language to remove any ambiguity. [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:18, 16 July 2026 (UTC)</text>
      <sha1>6zvf3vqymo58b8ct02uejogzzzt24m5</sha1>
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      <text bytes="5797" sha1="m6yrciem1iftyuxpy9gbxenjpwpbm2r" xml:space="preserve">== Notice to colloquium ==

What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC)
: A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC)
::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC)
::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC)

== Inactive curator template ==

Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC)
: I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC)
::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''&quot;no edits or no logged actions for 2 years&quot;'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC)
::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC)

== Communication with the SSM and deadlines ==

A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of &quot;peace&quot;.

I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now.

Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC)
: I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC)
::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC)

== wording ==

This may sound pendantic... the prhase &quot;are considered inactive if they have made no edits and logged actions within one year.&quot; Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame, but not necessarily both? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:39, 10 July 2026 (UTC)
: This means an edit and a logged action, together. I am not sure what is the point of desysopping a custodian who makes edits but otherwise does not use their permission(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:45, 10 July 2026 (UTC)
::I updated the draft language to remove any ambiguity. [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:18, 16 July 2026 (UTC)

== 'crat actions ==

For reference, please see the en.wikipedia section at [[meta:Admin activity review/Local inactivity policies]].
&lt;blockquote&gt;Bureaucrats: same as the Inactive administrators rule above, plus if a bureaucrat does not participate in bureaucrat activity for over three years, their rights may also be removed.&lt;/blockquote&gt;
As someone who uses this tool, I can tell you that these rights are infrequently used. If more than 12 months passes with no need to flip a custodian bit (a not particularity rare occurrence) it would result in every 'crat here having the bit simultaneously removed. Comments? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:25, 16 July 2026 (UTC)</text>
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      <text bytes="6257" sha1="5r9r70s3lammd64uxhja5rgk1ehbizk" xml:space="preserve">== Notice to colloquium ==

What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC)
: A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC)
::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC)
::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC)

== Inactive curator template ==

Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC)
: I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC)
::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''&quot;no edits or no logged actions for 2 years&quot;'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC)
::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC)

== Communication with the SSM and deadlines ==

A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of &quot;peace&quot;.

I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now.

Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC)
: I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC)
::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC)

== wording ==

This may sound pendantic... the prhase &quot;are considered inactive if they have made no edits and logged actions within one year.&quot; Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame, but not necessarily both? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:39, 10 July 2026 (UTC)
: This means an edit and a logged action, together. I am not sure what is the point of desysopping a custodian who makes edits but otherwise does not use their permission(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:45, 10 July 2026 (UTC)
::I updated the draft language to remove any ambiguity. [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:18, 16 July 2026 (UTC)

== 'crat actions ==

For reference, please see the en.wikipedia section at [[meta:Admin activity review/Local inactivity policies]].
&lt;blockquote&gt;Bureaucrats: same as the Inactive administrators rule above, plus if a bureaucrat does not participate in bureaucrat activity for over three years, their rights may also be removed.&lt;/blockquote&gt;
As someone who uses this tool, I can tell you that these rights are infrequently used. If more than 12 months passes with no need to flip a custodian bit (a not particularity rare occurrence) it would result in every 'crat here having the bit simultaneously removed. Comments? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:25, 16 July 2026 (UTC)
: Should this be a part of this policy, a notice would be recommended explaining that some bureaucrat haven't used their bit involving bureaucrat actions, and a week should be given whether they would like to retain it or not, or otherwise resign at [[m:SRP]]. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:56, 16 July 2026 (UTC)</text>
      <sha1>5r9r70s3lammd64uxhja5rgk1ehbizk</sha1>
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      <format>text/x-wiki</format>
      <text bytes="6618" sha1="cybkxlmq6jaq9e97t5belethnia2yv4" xml:space="preserve">== Notice to colloquium ==

What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC)
: A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC)
::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC)
::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC)

== Inactive curator template ==

Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC)
: I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC)
::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''&quot;no edits or no logged actions for 2 years&quot;'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC)
::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC)

== Communication with the SSM and deadlines ==

A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of &quot;peace&quot;.

I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now.

Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC)
: I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC)
::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC)

== wording ==

This may sound pendantic... the prhase &quot;are considered inactive if they have made no edits and logged actions within one year.&quot; Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame, but not necessarily both? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:39, 10 July 2026 (UTC)
: This means an edit and a logged action, together. I am not sure what is the point of desysopping a custodian who makes edits but otherwise does not use their permission(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:45, 10 July 2026 (UTC)
::I updated the draft language to remove any ambiguity. [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:18, 16 July 2026 (UTC)

== 'crat actions ==

For reference, please see the en.wikipedia section at [[meta:Admin activity review/Local inactivity policies]].
&lt;blockquote&gt;Bureaucrats: same as the Inactive administrators rule above, plus if a bureaucrat does not participate in bureaucrat activity for over three years, their rights may also be removed.&lt;/blockquote&gt;
As someone who uses this tool, I can tell you that these rights are infrequently used. If more than 12 months passes with no need to flip a custodian bit (a not particularity rare occurrence) it would result in every 'crat here having the bit simultaneously removed. Comments? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:25, 16 July 2026 (UTC)
: Should this be a part of this policy, a notice would be recommended explaining that some bureaucrat haven't used their bit involving bureaucrat actions, and a week should be given whether they would like to retain it or not, or otherwise resign at [[m:SRP]]. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:56, 16 July 2026 (UTC)
: Pinging bureaucrats [[User:Atcovi|Atcovi]], [[User:Dave Braunschweig|Dave Braunschweig]], [[User:Jtneill|Jtneill]] and [[User:Koavf|Koavf]] to this discussion. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:57, 16 July 2026 (UTC)</text>
      <sha1>cybkxlmq6jaq9e97t5belethnia2yv4</sha1>
    </revision>
    <revision>
      <id>2818438</id>
      <parentid>2818435</parentid>
      <timestamp>2026-07-16T22:24:29Z</timestamp>
      <contributor>
        <username>Mu301</username>
        <id>3705</id>
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      <text bytes="7158" sha1="4vz5vu2ddfzizn8yhzkgkpziaztd099" xml:space="preserve">== Notice to colloquium ==

What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC)
: A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC)
::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC)
::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC)

== Inactive curator template ==

Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC)
: I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC)
::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''&quot;no edits or no logged actions for 2 years&quot;'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC)
::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC)

== Communication with the SSM and deadlines ==

A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of &quot;peace&quot;.

I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now.

Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC)
: I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC)
::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC)

== wording ==

This may sound pendantic... the prhase &quot;are considered inactive if they have made no edits and logged actions within one year.&quot; Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame, but not necessarily both? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:39, 10 July 2026 (UTC)
: This means an edit and a logged action, together. I am not sure what is the point of desysopping a custodian who makes edits but otherwise does not use their permission(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:45, 10 July 2026 (UTC)
::I updated the draft language to remove any ambiguity. [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:18, 16 July 2026 (UTC)

== 'crat actions ==

For reference, please see the en.wikipedia section at [[meta:Admin activity review/Local inactivity policies]].
&lt;blockquote&gt;Bureaucrats: same as the Inactive administrators rule above, plus if a bureaucrat does not participate in bureaucrat activity for over three years, their rights may also be removed.&lt;/blockquote&gt;
As someone who uses this tool, I can tell you that these rights are infrequently used. If more than 12 months passes with no need to flip a custodian bit (a not particularity rare occurrence) it would result in every 'crat here having the bit simultaneously removed. Comments? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:25, 16 July 2026 (UTC)
: Should this be a part of this policy, a notice would be recommended explaining that some bureaucrat haven't used their bit involving bureaucrat actions, and a week should be given whether they would like to retain it or not, or otherwise resign at [[m:SRP]]. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:56, 16 July 2026 (UTC)
: Pinging bureaucrats [[User:Atcovi|Atcovi]], [[User:Dave Braunschweig|Dave Braunschweig]], [[User:Jtneill|Jtneill]] and [[User:Koavf|Koavf]] to this discussion. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:57, 16 July 2026 (UTC)

== Process ==

*For item #2 of process I would recommend requiring both talk page notification and also &quot;email this user.&quot; If someone is inactive they might not notice the talk page edit.

*For #3 we should make clear that a steward request can only be made after a community discussion has concluded. The draft policy seems to skip over the step of review by the community, how long the review should remain open, and other important due process details.
[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:24, 16 July 2026 (UTC)</text>
      <sha1>4vz5vu2ddfzizn8yhzkgkpziaztd099</sha1>
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      <text bytes="7768" sha1="gk56kjwem63ynuwlvxus3b2vdf22jlr" xml:space="preserve">== Notice to colloquium ==

What is the sence of noticing community about that? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:10, 8 June 2026 (UTC)
: A notice would be posted at the inactive SSM's user talk page, and a separate notification at the village pump listing the inactive support staff member(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:46, 25 June 2026 (UTC)
::And the reasoning behind why whole community should know, there is inactive staff? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:08, 2 July 2026 (UTC)
::: This is standard practice as the stewards have done this similar procedure per [[m:Admin activity review]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:06, 2 July 2026 (UTC)

== Inactive curator template ==

Just a note if this policy is agreet the template should be fixed. No it counts with 2 years. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:12, 8 June 2026 (UTC)
: I'm not sure what you are trying to explain. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:47, 25 June 2026 (UTC)
::I am saying that if the inactive period is changed, the {{tl|Inactive curator}} template text ''&quot;no edits or no logged actions for 2 years&quot;'' should be changed to the appropriate one. This is just a notice, not to forget to do so. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:10, 2 July 2026 (UTC)
::: I see, thank you for explaining. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:05, 2 July 2026 (UTC)

== Communication with the SSM and deadlines ==

A notification on the user's user page is a decent way to communicate with support staff. If they don't respond, it's clear that there's no point in waiting any longer and their rights have been revoked. On the contrary, if they respond, they suspect that they should start working on Wikiversity, but it may happen that they won't, i.e. SSM will respond, but they will continue to be inactive, so they will have another year of &quot;peace&quot;.

I would probably reduce the inactivity time to '''8 months''' (i.e. 6 months + 2 months, which may take to creat a custodian), but I would leave the response time at a '''month or more'''. I assume that sometimes the reason for inactivity is health problems or personal problems, and in such situations a person is usually not very reactive - i.e. they don't manage to respond quickly to all the requests that come to them. Another reason may be the busy work schedule of university teachers, who, for example, are on the job for 4 months during exams. This means, yes, you have been inactive for a while for some reason and then someone invites you to return to activity, but you are sick, or you are writing a scientific article, grant report, etc. and you don't have much time right now.

Here, it would perhaps require standardized posts for all SSM roles, where a notice would be written that according to the policy, a SSM cannot be inactive for a given period. ''Then a question whether they will resume activity within 2 months.'' Yes - rights retained, no/no answer - rights removed within a month. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:29, 8 June 2026 (UTC)
: I still think we should leave the timeframe as one year to maintain consistency with some other projects. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:40, 25 June 2026 (UTC)
::Well, why not. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:25, 25 June 2026 (UTC)

== wording ==

This may sound pendantic... the prhase &quot;are considered inactive if they have made no edits and logged actions within one year.&quot; Does this mean that any edit [[w:Logical conjunction|AND]] a log action are required in one year? Does it mean that either one is required in that time frame, but not necessarily both? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 04:39, 10 July 2026 (UTC)
: This means an edit and a logged action, together. I am not sure what is the point of desysopping a custodian who makes edits but otherwise does not use their permission(s). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:45, 10 July 2026 (UTC)
::I updated the draft language to remove any ambiguity. [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:18, 16 July 2026 (UTC)

== 'crat actions ==

For reference, please see the en.wikipedia section at [[meta:Admin activity review/Local inactivity policies]].
&lt;blockquote&gt;Bureaucrats: same as the Inactive administrators rule above, plus if a bureaucrat does not participate in bureaucrat activity for over three years, their rights may also be removed.&lt;/blockquote&gt;
As someone who uses this tool, I can tell you that these rights are infrequently used. If more than 12 months passes with no need to flip a custodian bit (a not particularity rare occurrence) it would result in every 'crat here having the bit simultaneously removed. Comments? [[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 17:25, 16 July 2026 (UTC)
: Should this be a part of this policy, a notice would be recommended explaining that some bureaucrat haven't used their bit involving bureaucrat actions, and a week should be given whether they would like to retain it or not, or otherwise resign at [[m:SRP]]. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:56, 16 July 2026 (UTC)
: Pinging bureaucrats [[User:Atcovi|Atcovi]], [[User:Dave Braunschweig|Dave Braunschweig]], [[User:Jtneill|Jtneill]] and [[User:Koavf|Koavf]] to this discussion. [[User:Codename Noreste|&lt;span style=&quot;color: blue&quot;&gt;Codename Noreste&lt;/span&gt;]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:57, 16 July 2026 (UTC)
::I am fine with CN's proposal. I think a more sensible policy would be to collapse bureaucrat and sysop rights together, so if someone hasn't changed users rights (bureaucrat) &lt;em&gt;or&lt;/em&gt; blocked a user, deleted a page, deleted a rev, imported a page, etc. (sysop rights), then that person's advanced user rights would fall under the inactivity policy. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:26, 16 July 2026 (UTC)

== Process ==

*For item #2 of process I would recommend requiring both talk page notification and also &quot;email this user.&quot; If someone is inactive they might not notice the talk page edit.

*For #3 we should make clear that a steward request can only be made after a community discussion has concluded. The draft policy seems to skip over the step of review by the community, how long the review should remain open, and other important due process details.
[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 22:24, 16 July 2026 (UTC)</text>
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      <comment>Gavin R Putland moved page [[WikiJournal Preprints/Coordinates Last: Vector Analysis Done Fast]] to [[Coordinates Last: Vector Analysis Done Fast]]: Conversion from preprint to learning resource.</comment>
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      <comment>Gavin R Putland moved page [[Talk:WikiJournal Preprints/Coordinates Last: Vector Analysis Done Fast]] to [[Talk:Coordinates Last: Vector Analysis Done Fast]]: Conversion from preprint to learning resource.</comment>
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|Description=Carry Lookahead Adders 2A traditional (20260716 - 20260715)
|Source={{own|Young1lim}}
|Date=2026-07-16
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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{{Information
|Description=Carry Lookahead Adders 2A traditional (20260716 - 20260715)
|Source={{own|Young1lim}}
|Date=2026-07-16
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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== Server switch ==

&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;

[[:m:Special:MyLanguage/Tech/Server switch 2020|Read this message in another language]]  •  [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-Tech%2FServer+switch+2020&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]

The [[foundation:|Wikimedia Foundation]] tests the switch between its first and secondary data centers.  This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster. To make sure everything is working, the Wikimedia Technology department needs to do a planned test. This test will show if they can reliably switch from one data centre to the other. It requires many teams to prepare for the test and to be available to fix any unexpected problems. &lt;!--

 They will switch all traffic back to the primary data center on '''Tuesday, October 27 2020'''. --&gt;

Unfortunately, because of some limitations in [[mw:Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made.  We apologize for this disruption, and we are working to minimize it in the future. 

'''You will be able to read, but not edit, all wikis for a short period of time.'''

*You will not be able to edit for up to an hour on Tuesday, 29 June 2021. The test will start at [https://zonestamp.toolforge.org/1624975200 14:00 UTC] (07:00 PDT, 10:00 EDT, 15:00 WEST/BST, 16:00 CEST, 19:30 IST,  23:00 JST, and in New Zealand at 02:00 NZST on Wednesday 30 June).
*If you try to edit or save during these times, you will see an error message.  We hope that no edits will be lost during these minutes, but we can't guarantee it.  If you see the error message, then please wait until everything is back to normal.  Then you should be able to save your edit.  But, we recommend that you make a copy of your changes first, just in case.

''Other effects'':

*Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped.
*There will be code freezes for the week of June 28.  Non-essential code deployments will not happen.

This project may be postponed if necessary.  You can [[wikitech:Switch_Datacenter#Schedule_for_2021_switch|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule.  There will be more notifications about this. A banner will be displayed on all wikis 30 minutes before this operation happens. '''Please share this information with your community.'''&lt;/div&gt;&lt;/div&gt; [[user:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] 01:23, 27 June 2021 (UTC)
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== Server switch ==

&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;

[[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-Tech%2FServer+switch&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]

The [[foundation:|Wikimedia Foundation]] tests the switch between its first and secondary data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster. To make sure everything is working, the Wikimedia Technology department needs to do a planned test. This test will show if they can reliably switch from one data centre to the other. It requires many teams to prepare for the test and to be available to fix any unexpected problems. 

They will switch all traffic back to the primary data center on '''Tuesday, 14 September 2021'''.

Unfortunately, because of some limitations in [[mw:Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. 

'''You will be able to read, but not edit, all wikis for a short period of time.'''

*You will not be able to edit for up to an hour on Tuesday, 14 September 2021. The test will start at [https://zonestamp.toolforge.org/1631628049 14:00 UTC] (07:00 PDT, 10:00 EDT, 15:00 WEST/BST, 16:00 CEST, 19:30 IST, 23:00 JST, and in New Zealand at 02:00 NZST on Wednesday, 15 September).
*If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case.

''Other effects'':

*Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped.
* We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards.

This project may be postponed if necessary.  You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. There will be more notifications about this. A banner will be displayed on all wikis 30 minutes before this operation happens. '''Please share this information with your community.'''&lt;/div&gt;&lt;/div&gt; [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User:SGrabarczuk (WMF)|&lt;span class=&quot;signature-talk&quot;&gt;{{int:Talkpagelinktext}}&lt;/span&gt;]]) 01:10, 11 September 2021 (UTC)
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== Bots need to upgrade to Pywikibot 6.6.1 ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;
Dear bot operators, bots running [[mw:Pywikibot|Pywikibot]] must upgrade to [https://doc.wikimedia.org/pywikibot/stable/changelog.html version 6.6.1] otherwise they will break when [[mw:MediaWiki_1.37/Deprecation_of_legacy_API_token_parameters|deprecated API parameters]] are removed. If you have any questions or need help in upgrading, please reach out using one of the [[mw:Special:MyLanguage/Manual:Pywikibot/Communication|Pywikibot communication channels]].

Thanks, [[m:User:Legoktm|Legoktm]] ([[m:User talk:Legoktm|talk]]) 18:02, 22 September 2021 (UTC)
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== Your wiki will be in read only soon ==

&lt;section begin=&quot;server-switch&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;

[[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-Tech%2FServer+switch&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]

The [[foundation:|Wikimedia Foundation]] tests the switch between its first and secondary data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster. To make sure everything is working, the Wikimedia Technology department needs to do a planned test. This test will show if they can reliably switch from one data centre to the other. It requires many teams to prepare for the test and to be available to fix any unexpected problems. 

All traffic will switch on '''{{#time:j xg|2023-03-01|en}}'''. The test will start at '''[https://zonestamp.toolforge.org/{{#time:U|2023-03-01T14:00|en}} {{#time:H:i e|2023-03-01T14:00}}]'''.

Unfortunately, because of some limitations in [[mw:Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. 

'''You will be able to read, but not edit, all wikis for a short period of time.'''

*You will not be able to edit for up to an hour on {{#time:l j xg Y|2023-03-01|en}}.
*If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case.

''Other effects'':

*Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped.
* We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards.
* [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes.

This project may be postponed if necessary.  You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. There will be more notifications about this. A banner will be displayed on all wikis 30 minutes before this operation happens. '''Please share this information with your community.'''&lt;/div&gt;&lt;section end=&quot;server-switch&quot;/&gt;

&lt;span dir=ltr&gt;[[m:User:Trizek (WMF)|Trizek (WMF)]] ([[m:User talk:Trizek (WMF)|{{int:talk}}]])&lt;/span&gt; 21:24, 27 February 2023 (UTC)
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== Your wiki will be in read-only soon ==

&lt;section begin=&quot;server-switch&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;

[[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-Tech%2FServer+switch&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]

The [[foundation:|Wikimedia Foundation]] tests the switch between its first and secondary data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster. To make sure everything is working, the Wikimedia Technology department needs to do a planned test. This test will show if they can reliably switch from one data centre to the other. It requires many teams to prepare for the test and to be available to fix any unexpected problems. 

All traffic will switch on '''{{#time:j xg|2023-04-26|en}}'''. The test will start at '''[https://zonestamp.toolforge.org/{{#time:U|2023-04-26T14:00|en}} {{#time:H:i e|2023-04-26T14:00}}]'''.

Unfortunately, because of some limitations in [[mw:Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. 

'''You will be able to read, but not edit, all wikis for a short period of time.'''

*You will not be able to edit for up to an hour on {{#time:l j xg Y|2023-04-26|en}}.
*If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case.

''Other effects'':

*Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped.
* We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards.
* [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes.

This project may be postponed if necessary.  You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. There will be more notifications about this. A banner will be displayed on all wikis 30 minutes before this operation happens. '''Please share this information with your community.'''&lt;/div&gt;&lt;section end=&quot;server-switch&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:21, 21 April 2023 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Bots&amp;oldid=24903511 --&gt;

== Your wiki will be in read-only soon ==

&lt;section begin=&quot;server-switch&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;

[[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-Tech%2FServer+switch&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]

The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster. To make sure everything is working, the Wikimedia Technology department needs to do a planned test. This test will show if they can reliably switch from one data centre to the other. It requires many teams to prepare for the test and to be available to fix any unexpected problems. 

All traffic will switch on '''{{#time:j xg|2023-09-20|en}}'''. The test will start at '''[https://zonestamp.toolforge.org/{{#time:U|2023-09-20T14:00|en}} {{#time:H:i e|2023-09-20T14:00}}]'''.

Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. 

'''You will be able to read, but not edit, all wikis for a short period of time.'''

*You will not be able to edit for up to an hour on {{#time:l j xg Y|2023-09-20|en}}.
*If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case.

''Other effects'':

*Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped.
* We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards.
* [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes.

This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. There will be more notifications about this. A banner will be displayed on all wikis 30 minutes before this operation happens. '''Please share this information with your community.'''&lt;/div&gt;&lt;section end=&quot;server-switch&quot;/&gt;

[[User:Trizek (WMF)|Trizek_(WMF)]] ([[m:User talk:Trizek (WMF)|talk]]) 09:30, 15 September 2023 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Bots&amp;oldid=25086542 --&gt;</text>
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|Permission={{self|GFDL|cc-by-sa-3.0,2.5,2.0,1.0}}
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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|Source={{own|Young1lim}}
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