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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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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 form $n$-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.
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Stokes theorem for manifolds with corners?

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 manifold $M$ with boundary $\partial M$ and any differentiable 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.