Differentiable Manifolds/Integration of forms

Theorem (Stokes' theorem):

Let $M$ be an oriented, $n$-dimensional smooth manifold with boundary and let $\omega \in \Omega _{c}^{n-1}(M)$. Then

$\int _{M}d\omega =\int _{\partial M}\omega$,

where $\partial M$ bears the orientation inherited from $M$.

Proof: We first prove the special cases where $M=\mathbb {R} ^{n}$ or $M=\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}\geq 0\}$. In the first case, we restrict attention to a form of the form $\omega =fdx_{1}\wedge \cdots \wedge dx_{n-1}$, so that $d\omega ={\frac {\partial f}{\partial x_{n}}}(-1)^{n-1}dx_{1}\wedge \cdots \wedge dx_{n}$. Now $\mathbb {R} ^{n}$ has no boundary, so that

$\int _{\partial \mathbb {R} ^{n}}\omega =\int _{\emptyset }\omega =0$

by the definition of the integral of forms. Indeed, we have the empty atlas for $\emptyset$, which is oriented. Also, we have, using Fubini's theorem,

${\begin{aligned}\int _{\mathbb {R} ^{n}}d\omega &=(-1)^{n-1}\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }{\frac {\partial f}{\partial x_{n}}}(x_{1},\ldots ,x_{n})dx_{1}\cdots dx_{n}\\&=0,\end{aligned}}$

since $f$ and hence ${\frac {\partial f}{\partial x_{n}}}$ have compact support, proving the statement for $M=\mathbb {R} ^{n}$. We proceed to the half-space, ie. we set $M=\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}\geq 0\}$. A general ($n-1$)-form may be written as

$\omega =\sum _{k=1}^{n}f_{k}dx_{1}\wedge \cdots \wedge {\widehat {dx_{k}}}\wedge \cdots \wedge dx_{n}$,

so that we have

$d\omega =\left(\sum _{k=1}^{n}(-1)^{k-1}{\frac {\partial f_{k}}{\partial x_{k}}}\right)dx_{1}\wedge \cdots \wedge dx_{n}$.

Hence,

${\begin{aligned}\int _{M}d\omega &=\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }\int _{0}^{\infty }\sum _{k=1}^{n}(-1)^{k-1}{\frac {\partial f_{k}}{\partial x_{k}}}dx_{1}\cdots dx_{n}\\&=\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }f_{n}(0,x_{2},\ldots ,x_{n})dx_{2}\cdots dx_{n}+\overbrace {\sum _{k=2}^{n}(-1)^{k-1}\int _{0}^{\infty }\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }{\frac {\partial f_{k}}{\partial x_{k}}}dx_{k}dx_{2}\cdots {\widehat {dx_{k}}}\cdots dx_{n}dx_{1}} ^{=0}\end{aligned}}$

and

$\int _{\partial M}\omega =\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }f_{n}(0,x_{2},\ldots ,x_{n})dx_{2}\cdots dx_{n}$

and the two integrals coincide. Having dealt with the two special cases, we may proceed to the general case. Hence, suppose we have an oriented manifold $M$ with oriented atlas $(U_{\alpha },\varphi _{\alpha })_{\alpha \in A}$ such that each $V_{\alpha }:=\varphi _{\alpha }(U_{\alpha })$ is equal to either $\{(x_{1},\ldots ,x_{n}|x_{1}\geq 0\}$ or $\mathbb {R} ^{n}$, and let $\omega \in \Omega _{c}^{n-1}(M)$, where $n=\dim M$. Then by definition of the integral of a top form over an oriented manifold, whenever $(\rho _{\alpha })_{\alpha \in A}$ is a partition of unity subordinate to $(U_{\alpha })_{\alpha \in A}$, we have

${\begin{aligned}\int _{M}d\omega =\sum _{\alpha \in A}\int _{U_{\alpha }}\rho _{\alpha }d\omega =\sum _{\alpha \in A}\int _{V_{\alpha }}\rho _{\alpha }\varphi _{\alpha }^{*}(d\omega )=\sum _{\alpha \in A}\rho _{\alpha }\int _{V_{\alpha }}d\varphi _{\alpha }^{*}(\omega )=\sum _{\alpha \in A}\rho _{\alpha }\int _{\partial V_{\alpha }}\varphi _{\alpha }^{*}(\omega )=\int _{\partial M}\omega .\end{aligned}}$ $\Box$