In Tu's An Introduction to Manifolds, he states Stokes's theorem as:
For any smooth $(n-1)$-form $\omega$ with compact support on the oriented $n$-dimensional manifold $M$, $$\int_M d\omega = \int_{\partial M}\omega.$$
In his proof he chooses an atlas $\{(U_\alpha, \phi_\alpha)\}$ for $M$ in which each $U_\alpha$ is diffeomorphic to either $\mathbb{R}^n$ or $\mathcal{H}^n$ via an orientation-preserving diffeomorphism. He then says
Suppose Stokes's theorem holds for $\mathbb{R}^n$ and for $\mathcal{H}^n$. Then it holds for all the charts $U_\alpha$ in our atlas, which are diffeomorphic to $\mathbb{R}^n$ or $\mathcal{H}^n$.
I am having trouble seeing how Stokes's theorem applying to $\mathbb{R}^n$ and $\mathcal{H}^n$ implies it also holds for all the charts $U_\alpha$. Is he using the diffeomorphism and a pull back to make some sort of change of variables that preserves the integral?
Best Answer
Let $F: \mathbb R^n \to U$ be a diffeomorphism. If you know that
$$\tag{1} \int_{\mathbb R^n} d\omega =0$$
for all compactly support smooth differential $n-1$ form $\omega$ on $\mathbb R^n$, show that
$$\tag{2} \int_U d\beta = 0$$ for all compactly support smooth differential $n-1$ form $\beta$ on $U$.
Sketch of proof: For any such $\beta$, $\omega = F^*\beta$ is a compactly support differential $(n-1)$ forms on $\mathbb R^n$. Use (1) on this $\omega$ to obtain (2).
Similar for $\mathbb H^n$.