Pushforward (differential): Total Derivative, Linear Map, Tangent Space, Smooth Function, Open Set, Jacobian Matrix and Determinant

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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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Pagina's: 88, Paperback, Betascript Publishers