This section explains the differential (or derivative) of a map between some pair of spaces and . Here we use and , but everything is clearly generalizable.
Given a map from some connected subset of to a subset of , the differential at a point maps vectors in to vectors in . In particular, is the mapping of the vector into . Since is defined as a mapping of points and has no notion of vectors, the differential is defined by examining a curve passing through and having velocity at equal to , that is, and .
The curve gets mapped to the curve and is defined as . In the canonical basis, it can be shown that or , where is the Jacobian matrix of partial derivatives of at . Hence the differential map is linear and does not depend on the actual curve .
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 and 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 .
Copyright © 2005 Adrian Secord. All rights reserved.