Looking at the generalizations of Stokes theorem, I did find a version for manifold with corners, but I was surprised this generalization doesn't contain a simple example such as the cone. So my question is the following :


Is the following version of Stokes theorem correct?


Let $M$ be a smooth submanifold without boundary of $\mathbb{R}^N$ of dimension $n\leq N$, and suppose that the topological boundary $\partial M$ of $M$ is the finite disjoint union of (smooth) submanifolds of $\mathbb{R}^N$, each of them of dimension $\leq n-1$.
Then for all $n-1$ form on $\mathbb{R}^N$ with compact support, we have $\int\_M d\omega = \sum\_i \int\_{B\_i} \omega$, where $(B\_i)$ are all the submanifolds of dimension $n-1$ in the decomposition of $\partial M$.


I am aware of a generalization of Stokes theorem that contains the cone, that is in the book of
*Partial Differential Equations 1* of Sauvigny, where the notion of "singular set with capacity zero" is described, but I admit I don't understand what it means. I suppose (but I am not sure) that the previous statement is true if one can show that submanifolds of dimension $\leq n-2$ in the topological boundary of $M$ are with capacity zero with respect to $M$ in the sense of Sauvigny. Is that correct?



Stokes theorem for manifolds with boundary as disjoint union of submanifolds
**TL;DR** Yes, i think this is correct and should follow directly from a general version of Stokes's theorem in Geometric Measure Theory.


Recall that a rectifiable set of dimension $m$ in ${\bf R}^n$ is a set of finite $m$-dimensional Hausdorff measure almost contained (wrt this measure) in the union of images of countably many Lipschitz functions from ${\bf R}^m$ to ${\bf R}^n$. Manifolds with boundaries, cones, etc are of course rectifiable.


The Stokes's theorem is valid for any rectifiable set with compact support. Actually this is a tautology because Stokes' formula is used to define the boundary of such a set as a rectifiable current (i.e as a linear form on the sets of differential forms, or equivalently as a differential form with distribution coefficients).


So the question is, when does the boundary of a rectifiable set is associated to a rectifiable set itself, such as to give a geometric meaning to the boundary? The Closure Theorem from Geometric Measure Theory implies that the boundary is a rectifiable set if and only if it has finite mass.


A general reference is the book *"Geometric Measure Theory"* by H. Federer.
If you are not interested in the lengthy and technical proofs, you can instead read the shorter book *"Geometric Measure Theory, a beginner's guide"* by Frank Morgan (chapter 4) which surveys the whole theory.


The first step in the theory is to define a notion of a tangent cone at a given point of a measurable set and to show that if the set is m-rectifiable, then this cone is actually an m-dimensional plane almost everywhere.


