18
$\begingroup$

In the Wikipedia article on Stokes' theorem the following claim is advanced without any references given:

The main challenge in a precise statement of Stokes' theorem is in defining the notion of a boundary. Surfaces such as the Koch snowflake, for example, are well-known not to exhibit a Riemann-integrable boundary, and the notion of surface measure in Lebesgue theory cannot be defined for a non-Lipschitz surface. One (advanced) technique is to pass to a weak formulation and then apply the machinery of geometric measure theory; for that approach see the coarea formula.

Can someone explain what is the weak formulation announced here and how exactly can we obtain the general Stokes' theorem in $\mathbb{R}^{3}$ for surfaces with non-smooth boundaries?

$\endgroup$
4
  • 5
    $\begingroup$ This question is similar to some which arose before. The book by Sauvigny, Partial Differential Equations, gives a useful and simple criterion for application of Stokes's theorem with mild boundary singularities. $\endgroup$
    – Ben McKay
    Jan 20 at 11:42
  • 7
    $\begingroup$ An elementary trick that works well for many simple examples with real analytic singularities is to write out an explicit resolution, and pull back the differential form. Consider a cone. It is obvious how to map a cylinder smoothly to it, squishing one end of the cylinder. But Stokes's theorem on the cylinder is easy: a manifold with corners. So Stokes's theorem holds on the cone by pulling back the relevant differential forms. $\endgroup$
    – Ben McKay
    Jan 20 at 13:17
  • $\begingroup$ Dear @BenMcKay, thank you for your answer. I was more precisely looking for a weak formula in dimension 3 that could be similar to the generalized Gauss-Green theorem for sets of finite perimeter, if that is what is proposed in the wikipedia article. $\endgroup$
    – JabbEd
    Jan 20 at 14:35
  • 1
    $\begingroup$ Related question: mathoverflow.net/questions/358606/… $\endgroup$
    – mlk
    Jan 21 at 12:09

4 Answers 4

20
$\begingroup$

When I wrote that Wikipedia paragraph, I think I had in mind Theorem 5.16 in Evans & Gariepy, Measure Theory and the Fine Properties of Functions, which essentially proves Stokes' theorem for level sets of functions with bounded variation. Unfortunately, the proof given in the book is just citations to Lemmas 5.2 and 5.5, Theorem 5.15, and "the foregoing theory", so I can't be confident (via a cursory review) that "weak formulation+coarea" is the correct summary.

$\endgroup$
1
  • 20
    $\begingroup$ Reading a question that says 'what does this wikipedia paragraph mean?' and then an answer that starts with 'When I wrote that wikipedia paragraph' just made my day. $\endgroup$
    – quarague
    Jan 21 at 7:56
13
$\begingroup$

Harrison, Jenny, Stokes' theorem for nonsmooth chains. Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 235–242. This research announcement reports progress in developing a viable theory of integration of $n$-forms over possibly nonrectifiable $n$-dimensional domains in $\mathbb{R}^m$. As an example, this theory permits integration of a $C^2$ $1$-form over at least some Jordan curves in $\mathbb{R}^3$ with Hausdorff dimension greater than two.

I think it is safe to say that there is no universally accepted final form of Stokes's theorem, since we can try to allow rough differential forms and rough submanifolds and rough boundaries, with a huge range of different notions of roughness, corresponding to different function spaces.

$\endgroup$
5
$\begingroup$

Another, not easy to read book, is still the book of H. Federer (1969), Geometric Measure Theory.

$\endgroup$
3
$\begingroup$

As said in the article, you find these results in book on geometric measure theory. Maybe "Geometric Integration Theory" by Krantz and Parks can be of help. Theorem 6.2.12 (not in the preview on Google books) is one statement of Stokes, but maybe this is not as general as you would like it to be…

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.