I wrote that Wikipedia paragraph more than a year ago, so my memory's a little fuzzy on the inspiration. 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 cursory review doesn't guarantee that I accurately identified the theorem, or that "weak formulation+coarea" is the correct summary.

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.