Differential of a map

This section explains the differential (or derivative) of a map between some pair of spaces $R^n$ and $R^m$ . Here we use $\mathbb{R}^2$ and $\mathbb{R}^3$ , but everything is clearly generalizable.

Given a map $F$ from some connected subset $U$ of $\mathbb{R}^2$ to a subset $V$ of $\mathbb{R}^3$ , the differential $dF_p$ at a point $p \in U$ maps vectors in $U$ to vectors in $V$ . In particular, $v = dF_p(w)$ is the mapping of the vector $w$ into $\mathbb{R}^3$ . Since $F$ is defined as a mapping of points and has no notion of vectors, the differential is defined by examining a curve $\alpha(t)$ passing through $p$ and having velocity at $p$ equal to $w$ , that is, $\alpha(0) = p$ and $\alpha'(0) = w$ .

The curve $\alpha(t)$ gets mapped to the curve $\beta(t) = F \circ \alpha(t)$ and $v$ is defined as $\beta'(0)$ . In the canonical basis, it can be shown that $\beta'(0) = J_p \alpha'(0)$ or $v = J_p w$ , where $J_p$ is the Jacobian matrix of partial derivatives of $F$ at $p$ . Hence the differential map is linear and does not depend on the actual curve $\alpha(t)$ .

The Jacobian representation is convienient, since many standard results from calculus applied to maps result in simple matrix manipulations. The differential of a composition of two maps $F$ and $G$ ends up being the product of their respective Jacobians. Similarly, if the Jacobian of a map is invertible then the inverse function theorem applies and one can then talk of the inverse of a map $F^{-1}$ .