# Stokes theorem with corners

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.

• looks like you're heading for geometric measure theory. – Fan Zheng Mar 18 '15 at 1:02
• Federer's Geometric Measure Theory? There is in chapter four: Homological Integration Theory a subsection 4.5 called Normal currents of dimension $n$ in $\mathbb{R}^n$ in which Gauss-Green theorem is discussed. Is that what you mean? – Jacobb Mar 18 '15 at 6:17
• @FanZheng Could you please recommend a few books, papers concerning geometric measure theory which might be helpful in my case? – Jacobb Mar 18 '15 at 18:01
• One of the main basic results in Geometric Measure Theory is a version of the Stokes theorem for manifolds with a locally Lipschitz boundary, due to de Giorgi and Federer. It seems to me that such boundaries are general enough to include corners in the $\mathscr{C}^1$ sense. A proof of this theorem can be found in Federer's book and (I think) in Whitney's book "Geometric Integration Theory" quoted in Zurab Silagadze's answer below. – Pedro Lauridsen Ribeiro Mar 25 '15 at 22:57
• This theorem is the one called the "Gauss-Green theorem" in Federer's book which you mentioned above. Have a look as well at the paper by M. Taylor, M. Mitrea and A. Vasy, "Lipschitz domains, domains with corners, and the Hodge Laplacian". Commun. PDE 30 (2005) 1445-1462, arXiv:math/0408438. – Pedro Lauridsen Ribeiro Mar 25 '15 at 23:07

• MR0760450 de Rham, Georges Differentiable manifolds. Forms, currents, harmonic forms. Translated from the French by F. R. Smith. With an introduction by S. S. Chern. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 266. Springer-Verlag, Berlin, 1984. x+167 pp. ISBN: 3-540-13463-8 58A14 (55N99 58-01). See page 23.

A direct proof is not difficult: Remove the corners $C$ (codim $\ge 2$ in $M$) from the boundary. Then Stokes theorem holds for differential forms with compact support in $M\setminus C$. But you can approximate differential forms on $M$ with compact support by differential forms on $M\setminus C$ with compact support and the integrals over the manifold and over the boundary converge.

# EDIT:

Namely, step on a differential form on $M$ with a smooth bump function which vanishes on an $\epsilon$-neighborhood of $C$ and which is 1 off the $2\epsilon$-neighborhood of $C$. Since $C$ is a union of codim $\ge 2$ strata, the measure of these neighborhoods goes to 0 with $\epsilon\to0$

• Thank you. Could you explain to me how we can approximate forms on the manifold $M$ with compact support by forms on the manfifold with singularities removed: $M \setminus C$? (I suppose this is the key observation when it comes to that remark about Hausdorff measure zero). – Jacobb Mar 20 '15 at 15:07
• See also Serge Lang's book on Fundamentals of differentiable geometry (Chapter XVII, §3). – ACL Mar 20 '15 at 15:49
• @Jacobb Just multiply the form with the bump function. It becomes 0 near $C$. – Peter Michor Mar 20 '15 at 16:15
• @Jacobb You need odd forms only if you want to work on non-orientable manifolds. Another version of the same topic is: look at forms on the orientable double cover and look the the eigenspaces of the action of the idempotent deck transformation on differential forms. Those with eigenvalue -1 correspond to the odd forms, those with eigenvalue correspond to the even forms. See chapter III of mat.univie.ac.at/~michor/dgbook.pdf – Peter Michor Mar 21 '15 at 12:29
• @jacobb: 1. chains are formal combinations of manifolds with corners. 2. My bump function bumps from constant 1 down to zero. – Peter Michor Mar 25 '15 at 19:00

This book http://www.math.wustl.edu/~sk/books/root.pdf (Geometric Integration Theory, by S.G. Krantz and H.R. Parks) is a self-contained introduction to geometric measure theory. See also Hassler Whitney's classic "Geometric Integration Theory".

http://www.ams.org/journals/bull/1993-29-02/S0273-0979-1993-00429-4/ (Stokes' theorem for nonsmooth chains, by J. Harrison) provides a generalization of the Stokes' theorem.