Theorem 6.1 (Stoke's theorem)
Let
![$ M$](img54.gif)
be an oriented manifold with boundary of dimension
![$ n$](img164.gif)
and
let
![$ \omega$](img474.gif)
be a differential form of degree
![$ n-1$](img679.gif)
with compact
support then
Proof.
We cover
![$ M$](img54.gif)
with a covering by co-ordinate charts
![$ (U_\alpha, \psi_\alpha)$](img148.gif)
and choose a partition of unity
![$ \phi_\alpha $](img635.gif)
subordinate to this
cover. Notice that because
![$ \sum_\alpha \phi_\alpha =1$](img696.gif)
we have
![$ \sum_\alpha d\phi_\alpha = 0$](img697.gif)
and hence
and
So it suffices prove that
or equivalently Stoke's theorem for differential forms with compact
support
on
![$ \mathbb{R}^n_+$](img681.gif)
. Let us assume then that
![$ \omega$](img474.gif)
is a differential form on
![$ U$](img2.gif)
, where
![$ U$](img2.gif)
is
of the form
![$ U =
\mathbb{R}^n_+ \cap V$](img709.gif)
for
![$ V $](img34.gif)
open in
![$ \mathbb{R}^n$](img12.gif)
.
As as
![$ \omega$](img474.gif)
is has compact support it has bounded support
so there is an
![$ R > 0$](img710.gif)
such that if
![$ \vert x^i\vert > R$](img711.gif)
for all
![$ i = 1, \dots, n$](img407.gif)
then
![$ \omega(x) = 0$](img712.gif)
.
Write
![$ x = (t, y)$](img713.gif)
for
![$ y \in
\mathbb{R}^{n-1}$](img714.gif)
and
![$ \omega = f dt\wedge dy^1 \dots dy^{n-1}$](img715.gif)
so
we have