Talk:Stokes' theorem

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I have a question, does anybody know exactly what form of Stokes' theorem was asked on Stokes' exams at Cambridge? Also what level was the course: undergraduate or graduate?

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—Preceding unsigned comment added by 136.152.180.29 (talk) 19:02, 28 April 2008 (UTC)

This article may be too technical for a general audience. Please help improve this article by providing more context and better explanations of technical details to make it more accessible, without removing technical details.

Hello. Stokes' theorem is a great topic & I'm glad to see there is a nice article here. It seems the theorem has had a long & distinguished history. I wonder if someone wants to add a section on its history -- going from special cases through more general formulations and finally ending up with the version stated for differential forms. I don't know enough to write that stuff myself, although maybe I'll read up just for the fun of it! Happy editing, Wile E. Heresiarch 20:00, 1 Mar 2004 (UTC)

The Soviet Encyclopedia states that the general form is down to Poincare (c.1899) - with differential forms in general probably only defined in the following few years by Cartan. I think it's probably somewhat naive to look for a complete proof then, though. There were subsequent improvements, allowing a 'small' bad set on the boundary, for example.

Charles Matthews 21:59, 1 Mar 2004 (UTC)

I have a book by Arnold that calls it the 'Newton-Leibnitz-Gauss-Green-Ostrogradski-Stokes-Poincare formula' Billlion 14:19, 13 Sep 2004 (UTC)

What book, exactly? —Preceding unsigned comment added by 136.152.180.29 (talk) 19:00, 28 April 2008 (UTC)

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The classical Stokes theorem doesn't seem to follow from the general one as given here, since in the former v is a three-dimensional vectorfield while the latter wants a two-dimensional one. We need the more general formulation of Stokes theorem which talks about k dimensional submanifolds of M and k-1 forms. AxelBoldt

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If M is only piecewise smooth, is it still possible to unambiguously define C¹ functions on M? AxelBoldt

You definitely raise an interesting issue. I think a proper treatment of the piecewise case requires at least some indication of what sort of integration and differentiation we are performing in the statement of Stokes' theorem.

If differentiation is defined weakly (in the sense of distributions), then the theorem almost certainly requires the use of currents.

On the other hand, it is possible to relax the C¹ condition slightly, and stay more or less in the realm of calculus on manifolds. Specifically, decompose M into its smooth components M_i, which are manifolds with boundary. A differential form is (for lack of imagination) almost continuous if α is continuous on each M_i, and the pullback of α to every intersection is almost continuous. A differential form α is almost C¹ if it is C¹ on each M_i and dα is almost continuous. Stokes' theorem should then hold for almost C¹ forms, at least when the boundaries of the M_i are sufficiently nice.

Anyway, it is probably best to restrict the statement of the theorem to manifolds with boundary to avoid technical issues like this. Silly rabbit 16:57, 15 June 2006 (UTC)

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This article assumes quite a lot of mathematical background -- is it possible to do a simpler more "physical" treatment first, then do a more formal definition afterwards? The Anome

I agree, although I really think people looking up 'Stokes Theorem' will, by and large, have a mathematical background. One section of the page which seems redundant to me is the section expressing the formula in terms of dxdydz, which is just as, if not more, confusing than the curl notation.

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my textbook has this:

which doesn't seem quite the same as what is in the article:

∫_Σ rot v · dΣ = ∫_{∂ Σ} v · dr

any thoughts? -- Tarquin

It is equivalent. - Patrick 19:00 Jan 9, 2003 (UTC)

Hardly anyone uses the notation 'rot' anymore. The curl notation is much more standard and I think it's more appropriate. JMO. - Revolver

Contents 1 Stokes's Theorem? 2 Proof Process is too complexed 3 Examples 4 Requested move 5 Give an example 6 Reorganize the article?

[edit] Stokes's Theorem?

As per this apostrophes article from Economist.com's style guide, shouldn't "Stokes' theorem" be changed to "Stokes's theorem"? ✈ James C. 03:17, 2004 Aug 22 (UTC)

Actually I think Stokes theorem is probably even better. Charles Matthews 09:00, 22 Aug 2004 (UTC)

I agree. should this page then to be moved to "Stokes theorem", and "Stokes' theorem" be kept as a redirect? ✈ James C. 19:57, 2004 Aug 23 (UTC)

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moved from Stokes' theorem as per discussion on that page.

The theorem is named after Stokes because of his habit of using it on the prize examinations. It aquired its name about 1845 after his students began publishing papers refering to it under this title. This is documented in some of the older mathematics books but does not seem presently to be locatable with Google or other search engines.

It is documented in my Multivariable Calculus book, Multivariable Calculus, Fourth Edition by James Stewart. It also recommends G.E. Hutchinson's The Enchanted Voyage as a source for information about Stokes. --APW 03:35, 20 January 2006 (UTC)

[edit] Proof Process is too complexed

For one of Stokes theorem

         
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I think we could image its physical chart to early-learn.

--HydrogenSu 05:02, 22 December 2005 (UTC)

[edit] Examples

We really need an example here. (NB. I need an example, so I can't offer one!) - Drrngrvy 02:13, 12 January 2006 (UTC)

[edit] Give an example

To quote the beginning of the article:

Let M be an oriented piecewise smooth manifold of dimension n and let ω be an n−1 form that is a compactly supported differential form on M of class C¹. If ∂M denotes the boundary of M with its induced orientation, then yaddayaddayadda.

It's great to have a precise definition in this article, but (and to relive already asked questions) it'll be great if there was an easier introduction first, maybe stemming from an everyday physical problem. Thank you, --Abdull 17:45, 31 May 2006 (UTC)

Does anyone have a picture of this theorem in action? I've heard it having to do with gradients (maybe I misheard or misunderstood), and I came to this article to see how it worked. A picture would work wonders here, if possible.RSido 03:56, 28 February 2007 (UTC)

[edit] Reorganize the article?

I think this article should be reorganized to reflect that it's in a general-reference encyclopedia and not a specialized one intended for mathematicians. The introductory section should explain what the theorem is to someone who doesn't know it, and anyone who knows what an oriented piecewise smooth manifold is, is certainly already familiar with the classic formulation of the theorem, so the current intro doesn't really do much good. So the presentation should start with the classical version, including the physics-textbook "proof" (e.g. Kleppner and Kolenkow, An Introduction to Mechanics) showing the usual diagram with little circulating arrows inside grid squares showing how the contributions from opposite sides of the squares almost cancel. This shows the physical significance of the curl. It could then explain that the modern treatment is a generalization of this old theorem from vector calculus. For the "modern" version, vol. 1 of Spivak's Comprehensive Introduction to Differential Geometry has a somewhat more abstract, but less terse, treatment than Calculus on Manifolds. Phr (talk) 00:54, 10 August 2006 (UTC)

Yes, I agree with Phr and Abdull, the article is much too technical. You dont need to know about mainfolds, compactly supported differential forms and homology groups to understand Stokes theorem. What is needed is a clear physical explanation with pictures. (And I'm a mathematician!) Paul Matthews 15:12, 17 August 2006 (UTC)

I third the suggestion. The people most likely to look up this topic are students taking Vector Calculus or E-M. It should be introduced at their level, and the theorem's significance from the viewpoint of differential geometry moved out of the intro and into its own section. - Arsian120 05:59, 17 September 2006 (UTC)

Here is another opinion regarding the level of sophistication and complexity of the articles in this encyclopedia.

This encyclopedia is a “Hyper-link” document. Any article should start with a very basic description of the article in a language, simple enough to be understood by the average reading public. However, each article should branch deeper and deeper into the subject. With this, it is obvious that the terminology and the mathematical apparatus will go more and more complicated. If developed correctly, at some level, even mathematicians should experience difficulties reading the text beyond certain level, if the subject is not in the area of their own expertise. This would be an excellent illustration for the usefulness of the encyclopedia for the reading public with vastly different areas of interest and level of expertise. After all, I would never look in any encyclopedia for something that I already know very well.

Regards, Boris Spasov

Would the article be more approachable if it were split into two articles? One article could describe the 2-dimensional case common in a vector calculus-level class (after all, we already have an article about Green's Theorem). The other would concentrate on the general form. Thanks for the fish! (talk) 21:59, 17 April 2008 (UTC)