State Stokes' Theorem.

Prove that, if $M^{m}$ is a compact connected manifold and $\Phi: U \rightarrow \mathbb{R}^{m}$ is a surjective chart on $M$, then for any $\omega \in \Omega^{m}(M)$ there is $\eta \in \Omega^{m-1}(M)$ such that $\operatorname{supp}(\omega+d \eta) \subseteq \Phi^{-1}\left(\mathbf{B}^{m}\right)$, where $\mathbf{B}^{m}$ is the unit ball in $\mathbb{R}^{m}$.

[You may assume that, if $\omega \in \Omega^{m}\left(\mathbb{R}^{m}\right)$ with $\operatorname{supp}(\omega) \subseteq \mathbf{B}^{m}$ and $\int_{\mathbb{R}^{m}} \omega=0$, then $\exists \eta \in \Omega^{m-1}\left(\mathbb{R}^{m}\right)$ with $\operatorname{supp}(\eta) \subseteq \mathbf{B}^{m}$ such that $\left.d \eta=\omega .\right]$

By considering the $m$-form

$$\omega=x_{1} d x_{2} \wedge \ldots \wedge d x_{m+1}+\cdots+x_{m+1} d x_{1} \wedge \ldots \wedge d x_{m}$$

on $\mathbb{R}^{m+1}$, or otherwise, deduce that $H^{m}\left(S^{m}\right) \cong \mathbb{R}$.