differential

The differential is a way to map between tangent spaces. It works for manifolds, algebraic varieties, etc… but the definition I’ll write down is for manifolds (though nothing changes really.)

It is a coordinate-free analogue to the Jacobian.

Definition

For a map between manifolds $F:M\to N$ and a point $p\in M$, define

$$\begin{array}{rl}d{F}_p:T_{p}M& \to T_{F(p)}N\\ v& \mapsto \left(\begin{array}{rl}d{F}_p(v):{\mathcal{O}}_{N,F(p)}& \to \mathbb{R}\\ f& \mapsto v(F^{\ast }f)\end{array}\right)\end{array}$$

Written somewhat nicer,

$$d{F}_p(v)(f)=v(f\circ F).$$

Note this is linear, and because $v$ is a derivation, $d{F}_p(v)$ is a derivation (this is glossing over some of the calculation though).

Chain rule

Given manifolds (or affine algebraic varieties) $X,Y,Z$ with maps

$$\phi :X\to Y\psi :Y\to X$$

for a point $x\in X$, we know we can find the differentials

$$d{\phi }_x:T_{x}X\to T_{\varphi (x)}Yd{\psi }_{\varphi (x)}:T_{\varphi (x)}Y\to T_{\psi (\varphi (x))}Z$$

Then

$$d(\psi \circ \phi {)}_x=d{\psi }_{\phi (x)}\circ d{\phi }_x$$

Rank

We can define the rank of a smooth map at a point $x\in M$ (can be generalized I guess, but I don’t need that now…) as

$${\text{rank}}_x(F)=\text{rank}(d{F}_x)$$