Pushforward (differential)

Pushforward (differential)

Suppose that φ : M → N is a smooth map between smooth manifolds; then the differential of φ at a point x is, in some sense, the best linear approximation of φ near x. It can be viewed as a 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 φ(x). Hence it can be used to push tangent vectors on M forward to tangent vectors on N.

The differential of a map φ is also called, by various authors, the derivative or total derivative of φ, and is sometimes itself called the pushforward.