# Talk:Stokes' theorem

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## Exam

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?

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

this is gash —Preceding unsigned comment added by 78.105.240.153 (talk) 15:59, 10 November 2008 (UTC)

The linked exam, in the first question misspells "show" as "shew"bruno615 (talk) 23:33, 14 December 2008 (UTC)

- Here is a dictionary entry. "Shew" is an archaic spelling of "show" which was often used prior to the 20th century as a synonym for "prove". It is rarely used these days. siℓℓy rabbit (talk) 03:15, 15 December 2008 (UTC)

## History

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)

- Who first proved the Generalized form of the Theorem? Sugarfoot1001 02:47, 8 October 2012 (UTC)

## General formulation

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

## Piecewise

If *M* is only piecewise smooth, is it still possible to unambiguously define C^{1} 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 C
^{1}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 C^{1}if it is C^{1}on each*M*_{i}and*d*α is almost continuous. Stokes' theorem should then hold for almost C^{1}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)

## Physical approach

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.
- My guess is that most of those looking up Stokes' Theorem are physics students taking an intermediate electromagnetism class. These students, of whom I was once one, have a mathematical background, but it very rarely goes beyond a semester of real analysis. I doubt if the discussion currently in the article would be of much help to them. Tpudlik (talk) 06:32, 7 February 2010 (UTC)

## Notation for curl

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

- I would agree, the textbooks, lectures, and other material I've read or worked on generally shy away from the 'rot' notation. I think using curl notation would be more widely recognizable and more easily understood by most readers of this article. Memeca16 (talk) —Preceding undated comment added 03:05, 13 September 2010 (UTC).

## Spelling

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)

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

## The Name

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)

## Proof Process is too complexed

For one of Stokes theorem

or

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

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

## 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)

*The following discussion is an archived debate of the proposal.***Please do not modify it.**Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the debate was **move**. —Nightstallion *(?)* 00:44, 29 January 2006 (UTC)

## Requested move

Stokes theorem → Stokes' theorem – The theorem is named after Stokes, and thus his name should be used in the possessive (as in Euler's theorem or Lagrange's theorem). It was moved to Stokes theorem after disagreement over whether the proper title should be *Stokes' theorem* or *Stokes's theorem*, but clearly *Stokes theorem* is incorrect. The most common usage is *Stokes' theorem*. —Bkell 13:10, 24 January 2006 (UTC)

- Note that this was only moved by copy+paste a few days ago - should probably be speedy moved back. — sjorford (talk) 20:40, 25 January 2006 (UTC)

Okay, help me out if I'm wrong here. I've been trying to glean what's happened to this page by browsing through the history.

- The content of this page existed at Stokes theorem from 2001-Dec-2 to 2002-May-8.
- Maveric149 then copy+pasted the contents to Stokes' theorem.
- Content was at Stokes' theorem from 2002-May-8 to 2005-Feb-20.
- Anon user 141.211.120.73 copy+pasted the contents back to Stokes theorem.
- Current content has been at Stokes theorem since then (barring a brief move and revert to Stokes Theorem on 2005-Dec-10).

The talk pages have also been copy+pasted. We currently have this page as well as Talk:Stokes' theorem. I'm not sure what the correct name is, but it looks like we need to merge the histories. -- Fropuff 08:19, 26 January 2006 (UTC)

**Support**move because of common names principle. If this theorem was being discovered today I would try to get it named Stokes's theorem but I am more than a century late to advocate that. But, there needs to be a**history merger**before we can do anything. Stefán Ingi 10:48, 26 January 2006 (UTC)

*The above discussion is preserved as an archive of the debate.***Please do not modify it.**Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

## Give an example

To quote the beginning of the article:

**Let M be an oriented piecewise smooth manifold of dimension n and let be an n−1 form that is a compactly supported differential form on M of class C^{1}. 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)

## 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)

- For once, I agree with the technical tag. But the solution is fairly simple. Just move the freshman/sophomore level vector calculus stuff to the top, redo the lede to reflect the change, and bingo, accessibility. --C S (talk) 06:20, 14 August 2008 (UTC)

## Flux cutting and flux linkage

In the main sectionon Stokes' theorem, I reached the equations described by

Maxwell-Faraday equation Faraday's law of induction: C and S stationary

I was able to derive that myself.

The restriction to a stationary contour C leaves out some interesting electrical phenomena. These involve how to calculate induced emf when the contour and the magnetic field both vary with time. Such situations can occur in electrical machinery. I think that time differentiation with respect to the contour corresponds to flux cutting emf while the time derivative of the surface integral corresponds to flux linkage emf. Unfortunately, typical electrical engineering programs do not treat this subject well.

I think that a derivative of the contour integral that includes both the derivatives arising arising from the time variation of the contour and the magnetic field will give the correct answer.

Since posting this, I have carried out a few simple integrations for varying contours. One case is that for a constant field through a rectangle with dimensions x=vt and y. Another case was a circular contour with r=vt. In both cases, I found that the emf from law of induction gave the expected result. Moreover, this result agrees with the flux cutting law.

This leads me to believe that the law of induction will remain correct even with moving contours. This leaves open the question of nonuniform magnetic field. I will try a multinomial field in x and y. The real question will be: What happens if B is a function of time as well.
* —Preceding unsigned comment added by 99.165.14.155 (talk) 05:57, 27 September 2008 (UTC) *
PEBill (talk) 21:32, 27 July 2008 (UTC)

## Integration on manifolds?

Why does integration on manifolds redirect here? Is there a more appropriate redirect target? I'm curious what articles if any there are on topics related to integration on manifolds, such as integration on chains which is discussed ever so briefly at chain (algebraic topology). It seems to me that this is a topic which is most definitely article-worthy, or that at least we should make some effort to draw together related articles into a list or a category or somesuch. siℓℓy rabbit (talk) 12:36, 15 October 2008 (UTC)

- In fact,
*integration on manifolds*should better redirect to the*integration*section in the article differential forms. The generalized Stokes theorem as described here is only the most prominent application, but there are other ones. Thus the redirection should be changed, in fact! If someone reading this is from the staff: please do so. - Thanks in advance, 87.160.93.125 (talk) 16:12, 17 September 2009 (UTC)

## Fixing this article

It strikes me that great deal could be done to improve this article. As others have pointed out, the very technical lead section obscures the intuitive, more "down to earth" interpretation of this theorem (or at least some of its special cases).

Speaking of special cases or consequences of the theorem, there are many that aren't mentioned here, such as:

- Cauchy's integral formula and the residue theorem
- Brouwer's fixed-point theorem
- The shoelace formula

and also some more easily accessible, "physical" consequences:

- The mechanism of a planimeter
- Archimedes' principle

I would like to work myself on this article, but since I don't know for sure when I will have time to, I thought I'd bring up these points and solicit feedback!

4dhayman (talk) 01:34, 24 June 2012 (UTC)

## Example?

So I can get a better sense of this theorem:

Suppose I had some paiper-mache machine that I could use to cover an object with a single layer of paiper-mache. What information would the machine need to record to calculate the volume of the object with stokes' theorem?

68.111.85.224 (talk) 04:51, 22 June 2013 (UTC)

Or am I reading it wrong? Could someone calculate the surface area of an object with a planimeter-like device and a pencil?

68.111.85.224 (talk) 21:21, 22 June 2013 (UTC)

Actually, now I get it. The net rotation of stuff going through a surface is equal to the rotation of stuff at the edge of the surface.

68.111.85.224 (talk) 07:30, 26 June 2013 (UTC)