Differentiating Composition of Smooth Functions

[Nusc]

Homework Statement

Let f: M \rightarrow N, g:N \rightarrow K, and h = g \circ f : M \rightarrow K. Show that h_{*} = g_{*} \circ f_{*}.

Proof:

Let M,N and K be manifolds and f and g be C^\infinity functions.

Let p \in M. For any u \in F^{\infinity}(g(f((p))) and any derivation D at p.

[g \circ f)_* D](u) = D(u \circ g \circ f) = (f_{*}D)(u \circ g) = (g_{*}(f_{*}D))(u)

Homework Equations

The Attempt at a Solution

 

[Nusc]

Should be C^\infty and F^\infty(g(f((p)))