Let $M$ be a smooth manifold, let $\pi: E \to M$ be a smooth real vector bundle of rank $k$, and let $\mathcal{C}^\infty_M$ denote the sheaf of smooth real-valued functions on $M$. Define the sheaf of smooth sections $\mathcal{E}$ by assigning to each [open set](/page/Open%20Set) $V \subset M$ the $\mathcal{C}^\infty_M(V)$-module

\begin{align*} \mathcal{E}(V) := \Gamma^\infty(V,E) = \{s: V \to E \mid s \text{ is smooth and } \pi \circ s = \operatorname{id}_V\}. \end{align*}

Then $\mathcal{E}$ is a locally free sheaf of $\mathcal{C}^\infty_M$-modules of rank $k$. Equivalently, for every point $p \in M$, there exists an open neighbourhood $U \subset M$ of $p$ and an isomorphism of sheaves of $\mathcal{C}^\infty_M|_U$-modules

\begin{align*} \mathcal{E}|_U \cong (\mathcal{C}^\infty_M|_U)^k. \end{align*}