- #1

- 753

- 2

## Homework Statement

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

Proof:

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

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

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