# Differentiable Manifolds/Integration of forms

Theorem (Stokes' theorem):

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

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

where ${\displaystyle \partial M}$ bears the orientation inherited from ${\displaystyle M}$.

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

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

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

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

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

so that we have

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

Hence,

{\displaystyle {\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

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

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

{\displaystyle {\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}}} ${\displaystyle \Box }$