Differential of a smooth map

Definition

Given:

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

For each $p\in M$ we define a map

• $$dF_p:T_p(M)\rightarrow T_{F(p)}N$$ called the differential of $F$ at $p$ 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 $f$ as follows:

• $$dF_p(v)(f) = v(f\circ F)$$

See also

• Smooth map
• Smooth manifold
• Smooth function