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)))