# Generalized Stokes' theorem

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?

• 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. Jan 20 at 11:42
• 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. Jan 20 at 13:17
• 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. Jan 20 at 14:35
• Related question: mathoverflow.net/questions/358606/…
– mlk
Jan 21 at 12:09

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.