How can I show that every derivation of $C^\infty(M)$ on a smooth manifold can be represented by a vector field?

Question

How can I show that every derivation of $C^\infty(M)$ on a smooth manifold can be represented by a vector field? I want to show that the space of vector fields is isomorphic to the space of derivations of $C^\infty(M)$. I know the proof when $M = \mathbb{R^n}$, but would like to do it for a general smooth manifold.

Answer 1

If you know the proof on $\mathbb{R}^n$, it is the same in every chart, of $(U_i)_{i\in I}$ and if $X_i$ is the vector field defined on $U_i$, the restriction of $X_i$ and $X_j$ on $U_i\cap U_j$ are equal.