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}$ .

Copyright © 2005 Adrian Secord. All rights reserved.