Differential of a smooth map

Let $f:M\to N$ be a smooth map between smooth manifolds, and let $p\in M$ . The differential (or pushforward) of $f$ at $p$ is the linear map

\mathrm{d}f_p:T_pM\longrightarrow T_{f(p)}N

between tangent spaces characterized as follows.

Choose smooth charts $(U,\varphi)$ around $p$ and $(V,\psi)$ around $f(p)$ with $f(U)\subset V$ . Writing $\psi\circ f\circ\varphi^{-1}:\varphi(U)\to\psi(V)$ as a smooth map between open subsets of Euclidean space, $\mathrm{d}f_p$ is the unique linear map whose matrix in these coordinates is the Jacobian of $\psi\circ f\circ\varphi^{-1}$ at $\varphi(p)$ . This definition is independent of the chosen charts.

Functoriality

The differential is functorial: if $g:N\to P$ is smooth, then

\mathrm{d}(g\circ f)_p=\mathrm{d}g_{f(p)}\circ \mathrm{d}f_p, \qquad \mathrm{d}(\mathrm{id}_M)_p=\mathrm{id}_{T_pM}.

Examples

A coordinate computation. For $f:\mathbb{R}^2\to\mathbb{R}$ , $f(x,y)=x^2y$ , the differential at $(x,y)$ is the linear functional $\mathrm{d}f_{(x,y)}(u,v)=(2xy)u+(x^2)v.$
Projection. For $\pi:M\times F\to M$ , the differential at $(m,f)$ is the projection $\mathrm{d}\pi_{(m,f)}:T_mM\oplus T_fF\to T_mM$ onto the first factor.
Left translation on a Lie group. If $G$ is a Lie group and $L_g:G\to G$ is left translation by $g$ , then $\mathrm{d}(L_g)_h:T_hG\to T_{gh}G$ is a linear isomorphism for every $h\in G$ (in fact $L_g$ is a diffeomorphism).