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 and a point , define

Written somewhat nicer,

Note this is linear, and because is a derivation, is a derivation (this is glossing over some of the calculation though).

Chain rule

Given manifolds (or affine algebraic varieties) with maps

for a point , we know we can find the differentials

Then

Rank

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