Let $r \in \mathbb{N} \cup \{0\}$, let $\pi: E \to M$ be a [smooth vector bundle](/page/Smooth%20Vector%20Bundle) of rank $r$ over a [smooth manifold](/page/Smooth%20Manifold) $M$, and let $U \subset M$ be open. There is a natural bijection between local trivialisations $\tau: E|_U \to U \times \mathbb{R}^r$ of $E$ over $U$ and ordered smooth local frames $(s_1,\dots,s_r)$ of $E$ over $U$, where $E|_U := \pi^{-1}(U)$ and each $s_i: U \to E|_U$ is a smooth section of $\pi|_{E|_U}:E|_U\to U$ such that $(s_1(x),\dots,s_r(x))$ is a basis of the fiber $E_x:=\pi^{-1}(\{x\})$ for every $x \in U$. When $r=0$, the frame is the empty ordered tuple and $\mathbb{R}^0$ is the zero [vector space](/page/Vector%20Space).