Let $M$ be a manifold and let $U_{\alpha}$ and $U_{\beta}$ be coordinate charts with coordinates $x^{\alpha}$ and $x^{\beta}$, respectively.
How to show that $f_{\alpha} : U_{\alpha}\times\mathbb{R}^n\to TM_{|U_{\alpha}}$, $(p,a^{\alpha})\to\sum_{i} a^\alpha_i\frac{\partial}{\partial x^\alpha_i}|_{p}$ is a bundle isomorphism?