This is problem 19.11 from Loring Tu's An Introduction to Manifolds [1 ed.].
Vector fields as derivations of $C^\infty$ functions.
In Subsection 14.1 we showed that a $C^\infty$ vector field $X$ on a manifold $M$ gives rise to a derivation of $C^\infty(M)$.
We will now show that every derivation of $C^\infty (M)$ arises from one and only one vector field. To distinguish the vector filed from the derivation, we will temporarily denote the derivation arising from $X$ by $\phi(X)$. Thus, for any $f \in C^\infty(M)$,
$$(\phi(X)f)(p)=X_p f \; \text{for all } p \in M.$$
(a) Let $\mathscr{F} = C^\infty (M).$ Prove that $\phi: \mathfrak{X}(M) \to Der(C^\infty(M))$ is an $\mathscr{F}-$linear map.
(b) Show that $\phi$ is injective.
(c) If $D$ is a derivation of $C^\infty(M)$ and $p \in M$, define $D_p : C_p^\infty(M) \to C_p^\infty(M)$ by
$$D_p[f]=[D\tilde{f}]\in C_p^\infty(M),$$ where $[f]$ is the germ of $f$ at $p$ and $\tilde{f}$ is a global extension of $f$. Show that $D_p[f]$ is well-defined.
(d) Show that $D_p$ is a derivation of $C_p^\infty(M)$.
(e) Prove that $\phi : \mathfrak{X} \to Der(C^\infty(M))$ is an isomorphism of $\mathscr{F}-$modules.
I proved (a)-(d), however, I can't figure out how to show the isomorphism in (e). How can we prove surjectivity of the linear map $\phi$ using (c) and (d)?
Best Answer
Let $D\in Der(C^\infty(M))$. For any $p\in M$ and $f\in C_pM$ define $X_p(f)=D_p(f)(p)$.
Then check:
$\bullet$ $X_p:C_pM\to \mathbb R$ , $f\mapsto X_p(f)$ is a tangent vector at $p$
$\bullet$ $X:M\to TM$, $p\mapsto X_p$ is a smooth vectorfield (Hint: Proposition 14.3)
$\bullet$ $\phi(X)=D$