Pushforward (differential)

High Quality Content by WIKIPEDIA articles! Suppose that f : M ¿ N is a smooth map between smooth manifolds; then the differential of f at a point x is, in some sense, the best linear approximation of f near x. It can be viewed as generalization of the total derivative of ordinary calculus. Explicitly, it is a linear map from the tangent space of M at x to the tangent space of N at f(x). Hence it can be used to push forward tangent vectors on M to tangent vectors on N. The differential of a map f is also called, by various authors, the derivative or total derivative of f, and is sometimes itself called the pushforward.