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

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