It is very common in physics and engineering to apply the divergence theorem to compact spaces whose boundary is not smooth. For example, in the wikipedia link I just gave, the picture illustrating this very classic theorem shows a boundary of a 3D domain that is stratified (vertices are dimension 0, edges are of dimension 1, faces are dimension 2).
So my question: is the following true ? :
Let $U$ be a bounded open subset of a $n$-manifold (we can suppose to work in $\mathbb{R}^n$ if it helps), whose boundary is a disjoint union $S_0 \cup \cdots \cup S_{n-1}$ where
(i) $S_k$ is a $k$-submanifold for all $0\leq k \leq n-1$ (non necessarily connected),
(ii) for all $k$, $\overline S_k$ is the union of all $S_j$ for $j\leq k$.
Then, the classic formula for the divergence theorem holds for $U$ (an integral of a divergence is reduced to an integral on $S_{n-1}$).
Second question if the previous is true : is it still true if we remove hypothesis (ii) ?
I know there are results concerning manifold with corners and Stokes theorem, but I am not familiar enough with them to prove that the setting I am working with here gives a manifold with corner.
