Editing Pushforward (differential) (section)

== Motivation ==
Let <math>\varphi: U \to V</math> be a [[Smooth function#Smooth functions on and between manifolds|smooth map]] from an [[Open subset#Euclidean space|open subset]] <math>U</math> of <math>\R^m</math> to an open subset <math>V</math> of <math>\R^n</math>. For any point <math>x</math> in <math>U</math>, the [[Jacobian matrix and determinant|Jacobian]] of <math>\varphi</math> at <math>x</math> (with respect to the standard coordinates) is the [[matrix (mathematics)|matrix]] representation of the [[total derivative]] of <math>\varphi</math> at <math>x</math>, which is a [[linear map]]

:<math>d\varphi_x:T_x\R^m\to T_{\varphi(x)}\R^n</math>

between their tangent spaces. Note the tangent spaces <math>T_x\R^m,T_{\varphi(x)}\R^n</math> are isomorphic to <math>\mathbb{R}^m</math> and <math>\mathbb{R}^n</math>, respectively. The pushforward generalizes this construction to the case that <math>\varphi</math> is a smooth function between ''any'' [[Manifold#Differentiable manifolds|smooth manifolds]] <math>M</math> and <math>N</math>.