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

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

Given:

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

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

• [math]dF_p(v)(f) = v(f\circ F)[/math]

See also

• Smooth map
• Smooth manifold
• Smooth function