Module of vector bundles

Question

If E→M and F→M are two vector bundles over manifold M, then $C^{\infty}$(E,F) is a $C^{\infty}$(M) module. I am confused about this claim for the operation of module. To f$\in$$C^{\infty}$(E,F) and g$\in$$C^{\infty}$(M), how could g operate on f? f is from E to F while g is from M to R.

Answer 1

Let $p:E\to M$ and $q:F\to M$ be the projections, so $f$ is a smooth map $E\to F$ such that $qf=p$ and $f$ is linear on each fiber. Define $g\cdot f$ by $g\cdot f(x)=g(p(x))f(x)$ for $x\in E$ (here $g(p(x))\in\mathbb{R}$ and we are multiplying $f(x)$ by it using the vector space structure of $F_{p(x)}$). It is then straightforward to verify that $g\cdot f\in C^\infty(E,F)$, and this makes $C^\infty(E,F)$ a module over $C^\infty(M)$.