Pushforward (differential)

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In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that φ : M → N {\displaystyle \varphi \colon M\to N} is a smooth map between smooth manifolds; then the differential of φ {\displaystyle \varphi } at a point x {\displaystyle x} , denoted d φ x {\displaystyle \mathrm {d} \varphi _{x}} , is, in some sense, the best linear approximation of φ {\displaystyle \varphi } near x {\displaystyle 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 {\displaystyle M} at x {\displaystyle x} to the tangent space of N {\displaystyle N} at φ ( x ) {\displaystyle \varphi (x)} , d φ x : T x M → T φ ( x ) N {\displaystyle \mathrm {d} \varphi _{x}\colon T_{x}M\to T_{\varphi (x)}N} . Hence it can be used to push tangent vectors on M {\displaystyle M} forward to tangent vectors on N {\displaystyle N} . The differential of a map φ {\displaystyle \varphi } is also called, by various authors, the derivative or total derivative of φ {\displaystyle \varphi } .