TITLE: Existence of a smooth frame for a vector bundle
QUESTION [5 upvotes]: Let $p:E\rightarrow M$ be a smooth n dimensional bundle and $M$ a smooth manifold. My question is does there always exist a local frame for this vector bundle. A local frame is a collection of smooth sections $r_1,...,r_n:U\rightarrow E$ such that $r_1(m),...,r_n(m) $ is a basis for $p^{-1}(m)$. If such a frame exist how can we find it? Thank you!

REPLY [2 votes]: Let $\phi:p^{-1}(U)\to U\times \mathbb{R}^k$ be a local trivialization and $\{e_1,...,e_k\}$ the canonical basis of $\mathbb{R}^k$. Define $r_i:U\to E$ by $r_i(p):=\phi^{-1}(p,e_i)$ for all $p\in U$. Then $\{r_1,...,r_k\}$ is the desired frame.