I've found the following version of Stokes' theorem in G. Stolzenberg's lecture notes 19:

**Notation:**

for $1 \le n \le m$

$\Lambda(m, n) = \{ \lambda: \{1,...,n\} \rightarrow \{1,...,m\} \ | \ \lambda(1) < ... \lambda(n) \}$

$p_{\lambda}: \mathbb{R}^m \ni (x_1, ..., x_m) \rightarrow (x_{\lambda(1)}, ..., x_{\lambda(n)}) \in \mathbb{R}^n$

$\mathcal{L}^n$ is $n$-dimensional Lebesgue measure and $\mathcal{H}^n$ is Hausdorff measure

Now, **the theorem** itself:

Assume that

$(1)$ $N$ is an $n$-dimensional, oriented class $C^1$ submanifold of $\mathbb{R}^m$, $2 \le n \le m$

$(2) \ M \subset N$ is open, $int _N (\overline{M}^N) = M$ and $\partial _N M$ is either empty or is an $n-1$-dimensional class $C^1$ submanifold in $\mathbb{R}^{m}$

$(3) \ \overline{M}$ is compact, $\mathcal{H}^n(M) < \infty$, $\mathcal{H}^{n-1}(\partial_NM) < \infty$

$(4)$ Let's define $\delta: = \overline{M} \setminus N$ and assume that $(*) \ \ \forall \lambda \in \Lambda(m, n-1) : \ \mathcal{L}^{n-1}(p_{\lambda}(\delta)) = 0$

$(5) \ \omega \in \Omega_{n-1}^{(1)} (\mathbb{R}^m)$

$(6)$ We induce the orientation on $M$ and on the boundary: $\partial _NM$ from $N$

**Then** we have that:

$d \omega $ absolutely integrable on $M$, $\omega$ on $\partial_NM$ if it is not empty

and $\int_M d \omega = \int_{\partial_NM} \omega$

While proving this theorem, we assume, wlog, that $p_{\lambda} : \mathbb{R}^m \ni (x_1, ..., x_m) \rightarrow (x_1, ..., x_{n-1}) \in \mathbb{R}^{n-1}$

My **question** is:

Apparently, we can replace $(*)$ with $\mathcal{H}^{n-1}(\delta)=0$ which implies $(*)$.

I was wondering whether you could explain to me why this is true.

Also, could you recommend a book or a paper in which I can find something more about Stokes' theorem with corners (singular points)? I've already read

- Stokes theorem for manifolds with corners? and consequently:
Brian Conrad's notes on differential geometry: math.stanford.edu/~conrad/diffgeomPage/handouts.html (but the problem is that I need a source which has been published)

a chapter dedicated to Stokes' theorem in Sauvigny's "Partial

Differential Equations" (here, we simlarly consider the set of

singular boundary points which has capacity zero, although at the

moment I'm not able to decide which assumption is more general )I've also read this article but it's not connected to my main

theorem (presented above)

Could you recommend some books, papers in which I can find something about Stokes theorem which will "agree" with the theorem I wrote down here?

I would be extremely grateful for all your insight.

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