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 } .
Motivation Let φ : U → V {\displaystyle \varphi :U\to V} be a smooth map from an open subset U {\displaystyle U} of R m {\displaystyle \mathbb {R} ^{m}} to an open subset V {\displaystyle V} of R n {\displaystyle \mathbb {R} ^{n}} . For any point x {\displaystyle x} in U {\displaystyle U} , the Jacobian of φ {\displaystyle \varphi } at x {\displaystyle x} (with respect to the standard coordinates) is the matrix representation of the total derivative of φ {\displaystyle \varphi } at x {\displaystyle x} , which is a linear map
d φ x : T x R m → T φ ( x ) R n {\displaystyle d\varphi _{x}:T_{x}\mathbb {R} ^{m}\to T_{\varphi (x)}\mathbb {R} ^{n}}
between their tangent spaces. Note the tangent spaces T x R m , T φ ( x ) R n {\displaystyle T_{x}\mathbb {R} ^{m},T_{\varphi (x)}\mathbb {R} ^{n}} are isomorphic to R m {\displaystyle \mathbb {R} ^{m}} and R n {\displaystyle \mathbb {R} ^{n}} , respectively. The pushforward generalizes this construction to the case that φ {\displaystyle \varphi } is a smooth function between any smooth manifolds M {\displaystyle M} and N {\displaystyle N} .
The differential of a smooth map Let φ : M → N {\displaystyle \varphi \colon M\to N} be a smooth map of smooth manifolds. Given x ∈ M , {\displaystyle x\in M,} the differential of φ {\displaystyle \varphi } at x {\displaystyle x} is a linear map
d φ x : T x M → T φ ( x ) N {\displaystyle d\varphi _{x}\colon \ T_{x}M\to T_{\varphi (x)}N\,}
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