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Untitled

This article is based on the GFDL article from PlanetMath at http://planetmath.org/encyclopedia/Divergence.html

That's my contribution for the day done. The Anome

Re: removal of the condition that the region S be compact - I doubt that one can do this unconditionally.

Charles Matthews 18:53, 6 Feb 2004 (UTC)

You are correct. Either the manifold must be compact, or the integrand must have compact support. sorry for the sloppiness. i think i will just change it back.

Re: Infinite plane of mass

The behavior is an approximative case only. It is the closest to "ideal" when you are very close to the black hole's event horizon. At least that is what I remember from a website which I'll need to look up. :-)

--24.84.203.193 28 June 2005 14:17 (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)[reply]

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)[reply]

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

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Generalization

5.95.187.184 (talk)I would like to recall your attention about this section, in particular about the first subsection: Multiple dimensions. In fact, I think that this one would actually benefit in being inserted as a sub-subsection of the generalization to the Riemaniann Manifold, for which the equation is: ${\int _{M}{\text{div}}(X)\ \nu _{g}=\int _{\partial M}<X,N>\nu _{\hat {g}}}$ , where $\nu$ is for the riemaniann volume form of the riemannian Manifolds, namely, $(M,g)$ and $(\partial M,{\hat {g}})$ (the last one considered as a submanifold of $M$ with the inclusion as embedding), $X$ is a vector field, and $N$ is the versor normal field defined on $\partial M$ . Considered this equation, the subsection Multiple dimensions is just a corollary, with $M$ considered as a submanifold of $\mathbb {R} ^{n}$

@@ Line 72: / Line 72: @@
 is: <math>{\int_{M}\text{div}(X)\ \nu_g=\int_{\partial M}<X,N>\nu_{\hat{g}}}</math>, where <math>\nu</math> is for the
 riemaniann volume form of the riemannian Manifolds, namely,
-<math>(M,g)</math> and <math>(\partial M, \hat{g})</math>  (the last one considered as a submanifold of <math>M</math> with the inclusion as embedding), <math>X</math> is a vector field, and <math>N</math> is the versor normal field defined on <math>\partial M</math>. Considered this equation, the subsection '''Multiple dimensions''' is just a corollary, with <math>M=\mathbb{R}^n</math>.
+<math>(M,g)</math> and <math>(\partial M, \hat{g})</math>  (the last one considered as a submanifold of <math>M</math> with the inclusion as embedding), <math>X</math> is a vector field, and <math>N</math> is the versor normal field defined on <math>\partial M</math>. Considered this equation, the subsection '''Multiple dimensions''' is just a corollary, with <math>M</math> considered as a submanifold of <math>\mathbb{R}^n</math>