aka derivative

Definition of the differential of a map

Special case: differential of a map from a surface

–> For the differential of a function from $R^n$ to $R$, the linear map can be represented as an $n$-dimensional vector, which corresponds to the Gradient of the function.

Differential of a map between manifolds

The differential at $a \in M$ of a Smooth function $F: M \to N$ between Differentiable manifolds is the Homomorphism of the Tangent spaces

$DF_a: T_a M \to T_{F(a)} N$

defined by

$DF_a(X_a)(f) = X_a (f \circ F )$

It is a Functor between Tangent vectors (seen as maps from $C^\infty (M) \to \mathbb{R}$.

In terms of coordinates, we see that the basis vectors are mapped as

$DF_a (\frac{\partial}{\partial x_i})_a (f) = \sum_j \frac{\partial F_j}{\partial x_i} (a) (\frac{\partial}{\partial y_j})_{F(a)} (f)$

This shows that tangent vectors are Contravariant

This functor is Covariantly functorial