# Differential of a smooth map

## Definition

Given:

• Two smooth manifolds [ilmath](M,\mathcal{A})[/ilmath] and [ilmath](N,\mathcal{B})[/ilmath] (which may have different dimensions) and are with or without boundary
• A smooth map [ilmath]F:M\rightarrow N[/ilmath]

For each [ilmath]p\in M[/ilmath] we define a map

• $dF_p:T_p(M)\rightarrow T_{F(p)}N$ called the differential of [ilmath]F[/ilmath] at [ilmath]p[/ilmath][1] as
• (really hard to write - I want a $dF_p:v\mapsto(\text{something})$)

Given:

• $v\in T_p(M)$ that is to say $v:C^\infty(M)\rightarrow\mathbb{R}$
• $f\in C^\infty(N)$

The differential acts on [ilmath]f[/ilmath] as follows:

• $dF_p(v)(f) = v(f\circ F)$