11
$\begingroup$

I was wondering if Stokes' theorem could be formulated in a setting which could be easily applied in situations where the traditional form cannot, such as on manifolds with corners like a rectangle or on a cone. I was thinking of something like:

If $M$ is a n-dimensional oriented Lipschitz-manifold with boundary and $\omega$ a compactly supported locally Lipschitz $n-1$-form on $M$, then $$\int_{M}d\omega=\int_{\partial M}\omega.$$

The notion of the (exterior) derivative of a form would of course have to include some notion of almost everywhereness on $M$, like applying Rademacher's theorem to the functions $\omega\circ\phi$ for a countable cover with charts $\phi$. I wonder if this has been done or can be done at all.

$\endgroup$
  • $\begingroup$ I don't know about the more general case, but can't Stokes' theorem be applied to manifolds with corners using the usual notions anyway? I think so. $\endgroup$ – Sam Apr 24 '11 at 1:02
  • 5
    $\begingroup$ Yes, as Sam mentions Stokes' theorem applies without any trouble to manifolds with cubical (or even worse) corners. For Lipschitz manifold it holds as well. See for example: math.neu.edu/sites/default/files/salvi.h/Stokes%20theorem.pdf a Google search will give you many more references. $\endgroup$ – Ryan Budney Apr 24 '11 at 1:20
2
$\begingroup$

There appear to be plenty of references on google (as Ryan pointed out). Here is one on Stokes theorem for Lipschitz forms on a smooth manifold:

http://arxiv.org/pdf/0805.4144.pdf

Does that help?

$\endgroup$

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.