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Concept
Pushforward (differential)
Summary
In differential geometry, pushforward is a linear approximation of smooth maps on tangent spaces. Suppose that \varphi:M\to N is a smooth map between smooth manifolds; then the differential of \varphi at a point x, denoted d\varphi_x, is, in some sense, the best linear approximation of \varphi near x. It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, the differential is a linear map from the tangent space of M at x to the tangent space of N at \varphi(x), d\varphi_x: T_xM \to T_{\varphi(x)}N. Hence it can be used to push tangent vectors on M forward to tangent vectors on N. The differential of a map \varphi is also called, by various authors, the derivative or total derivative of \varphi.
Motivation
Let \varphi: U \to V be
Official source
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