TITLE: Generalized Stokes' theorem
QUESTION [18 upvotes]: 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?

REPLY [20 votes]: 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.