Maybe this is an elementary question, but I'm unable to find the appropriate reference for it. The Stokes theorem tells us that, for a $n+1$-dimensional manifold $M$ with boundary $\partial M$ and any differentiable $n$-form $\omega$ on $M$, we have $\int_{\partial M} \omega = \int_M d\omega $.
But Stokes theorem is also true, say, for a cone $M = \{(x,y,z) \in \mathbb{R}^3 \ \vert\ \ x^2 + y^2 = z^2, 0\leq z \leq 1 \}$, or a square in the plane, $M =\{(x,y) \in \mathbb{R}^2 \ \vert\ 0 \leq x, y\leq 1 \}$ which are not manifolds. So my questions are:
- Are these cone and square examples of what I think are called "manifold with corners"?
- If this is so, where can I find a reference for a version of Stokes' theorem for manifolds with corners?
- If "manifold with corners" is not, which is the appropriate setting (and a reference) for a Stokes' theorem that includes those examples?
Any hints will be appreciated.
EDIT: Since thanking individually everyone would be too long, let me edit my question to acknowledge all of your answers. Thank you very much: I've found what I was looking for and more.
Best Answer
The most general form of Stokes' theorem I know of is proved in the book Partial Differential Equations 1. Foundations and Integral Representations by Friedrich Sauvigny.
The aim in the book is to provide a version of the divergence theorem which holds also in cases where the boundary has certain singularities (as you described: the singular boundary has to have zero capacity). As a precursor they also prove the Stokes' theorem (they credit the proof to E. Heinz!).
Note that this is much more general than manifolds with corners, it encompasses your cone as well!