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Suppose that φ : ''M'' → ''N'' is a smooth map between smooth manifolds; then the differential of φ at a point ''x'' is, in some sense, the best linear approximation of φ near ''x''. It can be viewed as a 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 φ(''x''). Hence it can be used to ''push'' tangent vectors on ''M'' ''forward'' to tangent vectors on ''N''. The differential of a map φ is also called, by various authors, the derivative or total derivative of φ, and is sometimes itself called the pushforward. == Motivation == Let φ : ''U'' → ''V'' be a smooth map from an open subset ''U'' of R''m'' to an open subset ''V'' of R''n''. For any point ''x'' in ''U'', the Jacobian of φ at ''x'' (with respect to the standard coordinates) is the matrix representation of the total derivative of φ at ''x'', which is a linear map : We wish to generalize this to the case that φ is a smooth function between ''any'' smooth manifolds ''M'' and ''N''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Pushforward (differential)」の詳細全文を読む スポンサード リンク
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