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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:

  1. Are these cone and square examples of what I think are called "manifold with corners"?
  2. If this is so, where can I find a reference for a version of Stokes' theorem for manifolds with corners?
  3. 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.

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You may find usefull the article of Joyce "On manifolds with corners" arxiv.org/abs/0910.3518. Square is considered as a manifold with corners, but the cone usually not, it seems. –  Dmitri Jan 25 '10 at 10:13
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A cone is not a manifold with corners. A cone on the other hand is a stratified space, and the proof of Stokes' that Orbicular mentions works for them, and even more general objects. –  Ryan Budney Jan 25 '10 at 14:04
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You may want to take a look at Brian Conrad's notes on differential geometry: math.stanford.edu/~conrad/diffgeomPage/handouts.html –  Johnson Jia Jan 25 '10 at 22:39

5 Answers 5

up vote 12 down vote accepted

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!

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John Lee's excellent book "Introduction to smooth manifolds" has a chapter on manifolds with corners, in which he proves Stokes' theorem for them.

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Triangulate your manifold $M$ so that $\partial M$ is triangulated as well. Then prove Stokes' theorem for diffeomorphic images of a standard simplex, as in de Rham's "Variétées différentiables".

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If you are looking for an online reference, you can check out Brian Conrads course notes on differential geometry. Near the bottom of that page, you can find the handout with Stokes theorem for manifolds with corners.

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You could take a look at Ch. XXIII paragraph 6 in Lang's Real and Functional Analysis entitled "Stokes' Theorem with Singularities". This version works for the cone too, I think. I have not read it myself, though.

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