Proof: We first prove the special cases where
or
. In the first case, we restrict attention to a form of the form
, so that
. Now
has no boundary, so that
![{\displaystyle \int _{\partial \mathbb {R} ^{n}}\omega =\int _{\emptyset }\omega =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a433d9fe676b667b8a3605c09baeca9a3bd48be)
by the definition of the integral of forms. Indeed, we have the empty atlas for
, 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9df33f59ef80793094713e6baf58c10131efd951)
since
and hence
have compact support, proving the statement for
. We proceed to the half-space, ie. we set
. A general (
)-form may be written as
,
so that we have
.
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5bb96e7527edfd777f9799a8c79f571be90807)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ac87e589e4fe7fec79c488ae1d9d73fe27d012)
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
with oriented atlas
such that each
is equal to either
or
, and let
, where
. Then by definition of the integral of a top form over an oriented manifold, whenever
is a partition of unity subordinate to
, we have
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)