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.

  • $\begingroup$ 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
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    $\begingroup$ 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

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:


Does that help?


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