Let $M$ be a compact differentiable manifold of dimension $n$ without boundary. Show that $M$ is orientable if and only if there exists a diffential $n$-form $\omega$ defined on $M$ and which is everywhere nonzero.
To show the "only if" part, I'm trying to use the fact that there exists a differentiable partition of unity on $M$, $\{\varphi_i\}_{i=1}^m$, where the support of $\varphi_i$ is contained in some $V_{\alpha_i}$ such that $\{V_{\alpha_i}\}_{i=1}^m$ is a covering of $M$. *($M$ is compact)
Hence, I'm trying to define $\omega$ a $n$-differential form which representation on $U_{\alpha_i}$ (such that $f_{\alpha_i}:U_{\alpha_i}\to V_{\alpha_i}$ is a parametrization of $M$) is given by $$\omega_{\alpha_i}=\varphi_i\,dx_1\wedge\dots\wedge dx_n$$
I don't know if it's correct until this point. But, if it is, I can't show that $\omega$ is well defined, and I didn't use the fact that $M$ is orientable. Also, I have no idea how to show the "if" part.