Differential (pushforward) of a smooth map

Let $M,N$ be smooth manifolds and let $F:M\to N$ be a smooth map .

Definition (differential at a point). For each $p\in M$, the differential (or pushforward) of $F$ at $p$ is the linear map

between tangent spaces defined as follows (using the derivation model of tangent vectors): if $v\in T_pM$ is a derivation at $p$, then $dF_p(v)\in T_{F(p)}N$ is the derivation given by

for every smooth function $g$ defined near $F(p)$.

Equivalently, if $v$ is represented by the velocity of a smooth curve $\gamma:(-\epsilon,\epsilon)\to M$ with $\gamma(0)=p$, then $dF_p(v)$ is represented by the velocity of $F\circ\gamma$ at $0$.

Bundle map form. The assignments $p\mapsto dF_p$ assemble into a smooth map between total spaces of tangent bundles,

covering $F$ (meaning $\pi_N\circ dF = F\circ \pi_M$, where $\pi_M:TM\to M$ and $\pi_N:TN\to N$ are the bundle projections). This viewpoint uses the tangent bundle functorially.

Chain rule. If $G:N\to P$ is another smooth map, then for every $p\in M$,

The differential detects local rank properties: $F$ is a smooth immersion iff $dF_p$ is injective for all $p$, and a smooth submersion iff $dF_p$ is surjective for all $p$. In particular, the notion of regular value is expressed in terms of surjectivity of $dF_p$ along a fiber .

Examples

1. A map $\mathbb{R}^2\to\mathbb{R}^2$. Let $F(x,y)=(x^2,y^2)$. Then in standard coordinates,

   $$dF_{(x,y)}(u,v) = (2x\,u,\;2y\,v).$$

   At $(1,1)$ the map has rank $2$, while at $(0,1)$ it has rank $1$, and at $(0,0)$ it has rank $0$.

2. Projection is a submersion. Let $\pi:\mathbb{R}^2\to\mathbb{R}$ be $\pi(x,y)=x$. Then

   $$d\pi_{(x,y)}(u,v)=u,$$

   which is surjective for every $(x,y)$. Thus $\pi$ is a smooth submersion , and its fibers are vertical lines.

3. Inclusion is an immersion. Let $i:S^1\hookrightarrow \mathbb{R}^2$ be the standard inclusion. For each $p\in S^1$, the map $di_p:T_pS^1\to T_p\mathbb{R}^2$ is injective, so $i$ is a smooth immersion (in fact a smooth embedding ).