Talk:Divergence theorem/Archive 1

This is an archive of past discussions about Divergence theorem. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Archive 1

Remarks about links

(Moved from my talk page)

Thank you for your changes to divergence theorem. However, I have some remarks about links.

First, one should not link to plural, rather to singular, so it's got to be [[divergence]]s and not [[divergences]]. Second, you should wonder if links are actually needed, a link to sink does not make any sence in that article.

Third, one should check where the links point to. Instead of boundary one should link to [[boundary (topology)|boundary]]. Fourth, a link to "conservation law" should be instead conservation law, because, as you notice, those two words go together.

These are minor things, but it helps give value to the links. Thanks. Oleg Alexandrov (talk) 10:38, 29 October 2005 (UTC)

Yes thanks for advice!--Light current 19:32, 29 October 2005 (UTC)

Example

By symmetry,

\iiint \limits _{W}ydV=\iiint \limits _{W}zdV=0

Why? --Abdull 19:30, 2 June 2006 (UTC)

A sphere looks the same if you look at its z-axis as if you look at its y-axis. Rotational symmetry. -lethe ^{talk +} 20:05, 2 June 2006 (UTC)

Clarified the spherical symmetry case

I have added a bit of clarification to the section for spherical symmetry. I was prompted by a change provided by 83.30.188.66 (talk · contribs), which was aiming for clarity, but technically introduced an error. I hope the rewording I have now supplied makes it a bit clearer. Further suggestions very welcome. —Duae Quartunciae (talk · cont) 11:48, 5 August 2007 (UTC)

WikiProject class rating

This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 09:48, 10 November 2007 (UTC)

Generalization of the divergence theorem to tensors

Could somebody with the requisite mathematical expertise please fill this in?

I copied a sentence into the intro, specifying the article wherein the generalization of this theorem can be found. I don't think this page itself is the right place to go into it. --Steve (talk) 21:08, 1 April 2008 (UTC)

Moved content off of page

I made dedicated articles for Gauss's law for magnetism and Gauss's law for gravity, and accordingly moved some material from here to those pages. (I went through and updated the links for hopefully all of the other wikipedia articles citing this material.) By the way, anyone interested in contributing to those articles is encouraged to do so. :-) --Steve (talk) 21:08, 1 April 2008 (UTC)

Do N-dimensional integrals need to have N squiggly integral signs? (N=2 or 3)

On the plus side, it helps people who only know a little calculus, and aren't familiar with the idea of one squiggly-integral sign denoting anything other than a one-dimensional integral. On the minus side, it makes equations harder to read and generally more cluttered. On the plus side, anyone who can understand it with one integral sign can understand it with three, but not necessarily vice-versa. On the minus side, I don't think any professional mathematician or physicist would write it this way (except maybe when they're writing introductory textbooks), and at least the physics articles I've seen on Wikipedia don't write it that way either.

Has anyone thought about this issue, pro and/or con? I have mixed feelings, myself. --Steve (talk) 21:07, 8 April 2008 (UTC)

Another Gauss theorem I found

I once (far far time ago) learned about other (of the many, Wikipedia should have a disambiguation page or something about this, its a serious issue) algorithm called "Gauss theorem", used for findings roots equations (of any power) when [independent coefficient]/[primary coefficient] is an integer by "trying" with different roots gotten by finding divisors of that division. How is this theorem called in Wikipedia-Other parts of the world other than my classroom? --Me (talk) 23:34, 19 September 2008 (UTC)

I found this...is that what you're talking about? --Steve (talk) 01:00, 20 September 2008 (UTC)

Here's a nice page: List of topics named after Carl Friedrich Gauss. Perhaps Gauss theorem and related terms should redirect there instead? Or at least a disambiguating note at the top of this article? --Steve (talk) 01:03, 20 September 2008 (UTC)

Conditions for the Divergence theorem

The condition of the div theorem says that F must be C¹. How can the theorem be applied then for example, for $u\nabla G$ , where G is a Green's function? Typically, Green's functions have a singularity, so they are not C¹. Does this mean the divergence theorem can be extended? --Janzz2k 17:21, 19 April 2007 (UTC)

The condition of the div theorem states that F must be C¹. However, the derivative does not need to be continuous. This can be proven by decomposing the volumetric domain into subregions where the function is C¹. It is readily seen that if the function is continuous, the terms corresponding to interior boundaries will cancel. —Preceding unsigned comment added by 128.115.27.10 (talk) 17:22, 16 March 2011 (UTC)

We can change "continuously differentiable" to just "differentiable", but I hope we can find a textbook source first. --Steve (talk) 17:37, 16 March 2011 (UTC)

Gauss's law for acoustics

Anyone interested to add something about application of Gauss's law in acoustics published in http://www.arxiv.com/abs/1104.0893 — Preceding unsigned comment added by 93.139.121.178 (talk) 19:08, 30 May 2011 (UTC)

Green Theorem

In my calculus book I remember that Green's theorem could be obtaneid as a particular case of Gauss theorem. I think all u had to was to remove one coordinate. — Preceding unsigned comment added by 85.18.50.180 (talk) 11:50, 5 June 2011 (UTC)

Updated images with relevant plots

Because this article is specifically about the divergence theorem and not about general Vector fields, I felt that the picture given in the example (found here) had little to offer to a reader's understanding of the divergence theorem or the example. I've since rendered and inserted a replacement image with the example field plotted. Glosser.ca (talk) 01:09, 5 November 2011 (UTC)

Other numbers of dimensions besides 3

(Inspired by the Green Theorem comment above.)

What do people think about discussing the two-dimensional (or n-dimensional) case, instead of just 3D? I know we've decided in the past to leave all generalizations to the Stokes' theorem article, but looking at that article, it seems to me like Stokes' theorem is more abstract and more general and more complicated than just the straight generalization of the divergence theorem. In particular, non-mathematicians (e.g. engineers) may be interested in the 2D divergence theorem but would not be well served by the Stokes' theorem article. :-) --Steve (talk) 18:47, 5 June 2011 (UTC)

Yes, a 2D example (ruducing to area/contour?) would be great! --94.221.119.182 (talk) 19:57, 29 November 2011 (UTC)

Actually the article is already written giving the N-dimensional generalized version. The only example in the article is already a 2D example. --Steve (talk) 01:18, 30 November 2011 (UTC)

Update to {{oiint}}

Sorry about the table in the 1st Example to complicate the markup, but the new template is gradually becoming popularized on WP, so a table was required to block the template integral expression and the lines of calculation tidily:

$\oiint$

\scriptstyle S

\mathbf {F} \cdot \mathbf {n} \,dS

{\begin{aligned}&=\iiint \limits _{W}\left(\nabla \cdot \mathbf {F} \right)\,dV\\&=2\iiint \limits _{W}\left(1+y+z\right)\,dV\\&=2\iiint \limits _{W}\,dV+2\iiint \limits _{W}y\,dV+2\iiint \limits _{W}z\,dV.\end{aligned}}

which translates to the mark-up: =(

:{|
|- valign="top"
| {{oiint
| intsubscpt = <math>\scriptstyle S</math>
| integrand = <math>\mathbf{F}\cdot\mathbf{n} \, dS</math>
}} ||   || <math>
\begin{align}
 &= \iiint\limits_W\left(\nabla\cdot\mathbf{F}\right) \, dV\\
 &= 2\iiint\limits_W\left(1+y+z\right) \, dV\\
 &= 2\iiint\limits_W \,dV + 2\iiint\limits_W y \,dV + 2\iiint\limits_W z \,dV.
\end{align}
</math>
|}

F = q(E+v×B) ⇄ ∑_ic_i 20:28, 2 April 2012 (UTC)

Integrals

I've just been going over my notes from Maths, and I've noticed that the gauss integral doesn't specify a closed surface ( $\oint _{S}$ ). Can anyone confirm that it doesn't need the closed surface part? (sorry, can't figure out the double closed integral) —Preceding unsigned comment added by Jporteous (talk • contribs) 14:33, 8 September 2008 (UTC)

Also see the previous section on this page. There are four ways to write a two-dimensional integral over a closed surface: Choose either one squiggly integral sign or two squiggly integral signs, and choose either to put in a little circle or leave it out. Under the integral sign it says "∂V", the boundary of a 3D volume, which is automatically a closed two-dimensional surface. So it's unambiguous any way you write it. Right now there's a second squiggly integral sign, to make it slightly clearer that it's a two-dimensional integral, but no circle, which makes it slightly less clear that it's a closed surface. One could maybe have it both ways and put a big circle passing through both integral signs, but that's tricky in LaTeX... --Steve (talk) 15:49, 8 September 2008 (UTC)

There's this way:

\int \!\!\!\!\!\oint \limits _{\partial V}\mathbf {F} \cdot \mathbf {n} \,dS

Does that look good? --Steve (talk) 15:53, 8 September 2008 (UTC)

WikiTeX only has the AMS-LaTeX package for maths, and that doesn't include an \oiint command. The esint package is a good fix for this; perhaps they could incorporate it in the future. 173.75.21.154 (talk) 05:25, 31 May 2011 (UTC)

If it answers future viewers the answer is yes - you can use the template: {{oiint}}, which renders as $\oiint$ . It’s now becoming standardized on wikipedia. Please click the integral image/link for more examples and the template parameters. F = q(E+v×B) ⇄ ∑_ic_i 20:41, 2 April 2012 (UTC)

"Correctly called Ostrogradsky's theorem"?

Claiming it is corrected called something based on who gave the first rigorous proof is interjecting the author's opinion in how theorem's should be named. Fermat didn't prove Fermat's Last Theorem, Stokes didn't prove what we now call Stokes' Theorem in geometry, etc. If someone wants to include it being called something else that's fine, but saying one way is correct is an opinion which doesn't belong here. RyanEberhart (talk) 19:50, 20 September 2012 (UTC)

Flawed Example

Here is the text from the article: "Let's say I have a rigid container filled with some gas. If the gas starts to expand but the container does not expand, what has to happen? Since we assume that the container does not expand (it is rigid) but that the gas is expanding, then gas has to somehow leak out of the container. (Or I suppose the container could burst, but that counts as both gas leaking out of the container and the container expanding.)"

This is not true. If the gas expands, due to, perhaps, an increase in temperature, then the pressure in the container will rise, without the volume of the container necessarily changing. This is why things explode in the first place—the pressure builds up due to gas expansion (temperature increase) but gas cannot escape, leading to a pressure build-up and eventual explosion. I don't know enough math to suggest a better example, but this one is severely flawed.

24.60.252.143 (talk) 15:57, 13 August 2014 (UTC)

I've removed the example from the article. Ozob (talk) 01:31, 14 August 2014 (UTC)

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Proof?

I'm not an expert on the topic, but could the article be improved by adding a proof?50.39.171.4 (talk) 19:33, 1 November 2018 (UTC)

Define i, j, k in the examples

In both given examples, you are using $\mathbf {i} ,\mathbf {j}$ and $\mathbf {k}$ . Since we are not used to it: Is this a standard notation for the unit vectors? — Preceding unsigned comment added by 129.27.232.39 (talk) 10:28, 9 December 2018 (UTC)

Piecewise smooth case

I haven't been able to find a proof or reference for the divergence theorem for surfaces with piecewise smooth boundary. The reference provided does not mention boundary smoothness at all, nor does it provide any proof or references. Does anyone know of a proof in the piecewise smooth case, and if so can they add it? — Preceding unsigned comment added by 129.94.177.30 (talk) 05:34, 7 January 2020 (UTC)

A statement and proof of the piecewise-smooth case in three dimensions can be found in this book. ^[1] Does this require a proof for an arbitrary number of dimensions? Or would this reference suffice? Benjamin Schulz (talk) 23:12, 4 April 2020 (UTC)

I would think all proofs of the divergence theorem require the surface to be piecewise smooth, in fact the divergence theorem only holds for piecewise smooth surfaces, as stated in the article. Can you even define a surface integral over a non-piecewise-smooth surface? --Chetvorno^TALK 02:00, 5 April 2020 (UTC)

The book "Calculus on manifolds - Spivak - 1968", on page 137, says the theorem is true for piecewise-smooth case. — Preceding unsigned comment added by Lrazevedo (talk • contribs) 15:04, 24 June 2020 (UTC)

References

^ Wade, William (2000). An Introduction to Analysis (2 ed.). Upper Saddle RIver, NJ: Prentice-Hall. pp. 479–480. ISBN 0-13-014409-6.

Wikiversity sister link added

I added a pedagogical introduction at v:Divergence theorem. As per regs, the link is situated at the bottom under "External links". The editors of this article can choose to mention and/or link to v:Divergence theorem in the lede or elsewhere else. If there is anything in the Wikipedia article you would like to have sent to Wikiversity, leave a note at v:User talk:Guy vandegrift and I will help you. At Wikiversity we use lots of subpages, giving us more freedom to add material that might not interest all readers.--Guy vandegrift (talk) 03:43, 19 March 2021 (UTC)

[1] Wade, William (2000). An Introduction to Analysis (2 ed.). Upper Saddle RIver, NJ: Prentice-Hall. pp. 479–480. ISBN 0-13-014409-6.

[1]