# Questions tagged [stokes-theorem]

Stokes' theorem is a result about integration of differential forms.

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### Prove Poincaré Lemma for $1$-form

Let $U\subseteq\mathbb{R}^n$ be an open set that contains $0$, and for all $t\in[0,1]$ and $x\in U$, $tx\in\mathbb{R}^n$. Show that every closed differentiable 1-form $w$, (i.e. $dw=0$) is an exact ...
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### Can the Gauss-Bonnet theorem be proven from Stokes's theorem?

In a comment to this question, John Ma claims that the Gauss-Bonnet theorem can be proven from Stokes's theorem, but does not explain how. For two dimensions, Stokes's theorem says that for any ...
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### Question on the proof of Stokes' Theorem in Spivak

The following is a quick outline of the proof of Stokes' Theorem as found in a Comprehensive Introduction to Differential Geometry Vol. 1 by Spivak. Theorem (Local Stokes' Theorem). Let $M$ be a ...
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### Why is it that the surface integral of the flux of a vector field is the same as the surface integral of the vector field itself?

In other words, this: http://www.math.ucla.edu/~archristian/teaching/32b-w17/week-7.pdf Is this just a definition because what we really care about is how much the vectors are "pushing" through the ...
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### integral of a k-form over an oriented compact manifold

I see in my course the following theorem: If $\omega$ is an exact k-form over an oriented compact manifold M of dimension $k$, then $\int_M \omega=0$. I don't have a proof of this theorem and I ...
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