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| {{About|pushforward operations in [[differential geometry]], which are associated with the '''differential''' of a [[smooth map]] between [[smooth manifold]]s|other uses of this term in [[mathematics]]|Pushforward (disambiguation){{!}}Pushforward}}
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| [[File:pushforward.svg|thumb|upright=1.5|alt="If a map, φ, carries every point on manifold M to manifold N then the pushforward of φ carries vectors in the tangent space at every point in M to a tangent space at every point in N"|If a map, φ, carries every point on manifold M to manifold N then the pushforward of φ carries vectors in the tangent space at every point in M to a tangent space at every point in N]]
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| Suppose that φ : ''M'' → ''N'' is a [[smooth map]] between [[smooth manifold]]s; 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''.
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| 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'''.
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| == Motivation ==
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| Let φ : ''U'' → ''V'' be a [[Smooth function#Smooth functions between manifolds|smooth map]] from an [[Open subset#Euclidean space|open subset]] ''U'' of '''R'''<sup>''m''</sup> to an open subset ''V'' of '''R'''<sup>''n''</sup>. For any point ''x'' in ''U'', the [[Jacobian_matrix_and_determinant|Jacobian]] of φ at ''x'' (with respect to the standard coordinates) is the [[matrix (mathematics)|matrix]] representation of the [[total derivative]] of φ at ''x'', which is a [[linear map]]
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| :<math>\mathrm d \varphi_x:\mathbf R^m\to\mathbf R^n.</math>
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| We wish to generalize this to the case that φ is a smooth function between ''any'' [[Manifold#Differentiable manifolds|smooth manifolds]] ''M'' and ''N''.
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| == The differential of a smooth map ==
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| Let φ : ''M'' → ''N'' be a smooth map of smooth manifolds. Given some ''x'' ∈ ''M'', the '''differential''' of φ at ''x'' is a linear map
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| :<math>\mathrm d \varphi_x:T_xM\to T_{\varphi(x)}N\,</math>
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| from the [[tangent space]] of ''M'' at ''x'' to the tangent space of ''N'' at φ(''x''). The application of dφ<sub>''x''</sub> to a tangent vector ''X'' is sometimes called the '''pushforward''' of ''X'' by φ. The exact definition of this pushforward depends on the definition one uses for tangent vectors (for the various definitions see [[tangent space]]).
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| If one defines tangent vectors as equivalence classes of curves through ''x'' then the differential is given by
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| :<math>\mathrm d \varphi_x(\gamma'(0)) = (\varphi \circ \gamma)'(0).</math>
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| Here γ is a curve in ''M'' with γ(0) = ''x''. In other words, the pushforward of the tangent vector to the curve γ at 0 is just the tangent vector to the curve φ∘γ at 0.
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| Alternatively, if tangent vectors are defined as [[derivation (abstract algebra)|derivations]] acting on smooth real-valued functions, then the differential is given by
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| :<math>\mathrm d\varphi_x(X)(f) = X(f \circ \varphi).</math>
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| Here ''X'' ∈ ''T<sub>x</sub>M'', therefore ''X'' is a derivation defined on ''M'' and ''f'' is a smooth real-valued function on ''N''. By definition, the pushforward of ''X'' at a given ''x'' in ''M'' is in ''T''<sub>φ(''x'')</sub>''N'' and therefore itself is a derivation.
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| After choosing [[manifold_(mathematics)|charts]] around ''x'' and φ(''x''), ''F'' is locally determined by a smooth map
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| :<math>\widehat{\varphi} : U \to V</math>
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| between open sets of '''R'''<sup>''m''</sup> and '''R'''<sup>''n''</sup>, and dφ<sub>''x''</sub> has representation (at ''x'')
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| :<math>\mathrm d \varphi_x\left(\frac{ \partial }{\partial u^a}\right) = \frac{\partial \widehat{\varphi}^b}{\partial u^a} \frac{ \partial }{\partial v^b},</math>
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| in the [[Einstein summation notation]], where the partial derivatives are evaluated at the point in ''U'' corresponding to ''x'' in the given chart.
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| Extending by linearity gives the following matrix
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| :<math>(\mathrm d\varphi_x)_a^{\;b}= \frac{\partial \widehat{\varphi}^b}{\partial u^a}.</math>
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| Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map φ at each point. Therefore, in some chosen local coordinates, it is represented by the [[Jacobian matrix]] of the corresponding smooth map from '''R'''<sup>''m''</sup> to '''R'''<sup>''n''</sup>. In general the differential need not be invertible. If φ is a [[local diffeomorphism]], then the pushforward at ''x'' is invertible and its inverse gives the [[pullback (differential geometry)|pullback]] of ''T''<sub>φ(''x'')</sub>''N''.
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| The differential is frequently expressed using a variety of other notations such as
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| :<math>D\varphi_x,\; (\varphi_*)_x, \;\varphi'(x).</math>
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| It follows from the definition that the differential of a [[function composition|composite]] is the composite of the differentials (i.e., [[functor]]ial behaviour). This is the ''chain rule'' for smooth maps.
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| Also, the differential of a [[local diffeomorphism]] is a [[linear isomorphism]] of tangent spaces.
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| ==The differential on the tangent bundle ==
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| The differential of a smooth map φ induces, in an obvious manner, a [[bundle map]] (in fact a [[vector bundle homomorphism]]) from the [[tangent bundle]] of ''M'' to the tangent bundle of ''N'', denoted by dφ or φ<sub>∗</sub>, which fits into the following [[commutative diagram]]:
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| [[Image:SmoothPushforward-01.png|center]]
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| where π<sub>''M''</sub> and π<sub>''N''</sub> denote the bundle projections of the tangent bundles of ''M'' and ''N'' respectively.
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| Equivalently (see [[bundle map]]), φ<sub>∗</sub> = dφ is a bundle map from ''TM'' to the [[pullback bundle]] φ*''TN'' over ''M'', which may in turn be viewed as a [[section_(fiber bundle)|section]] of the [[vector bundle]] Hom(''TM'', φ*''TN'') over ''M''. The bundle map dφ is also denoted by Tφ and called the '''tangent map'''. In this way, ''T'' is a [[functor]].
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| == Pushforward of vector fields ==
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| Given a smooth map φ : ''M'' → ''N'' and a [[vector field]] ''X'' on ''M'', it is not usually possible to define a pushforward of ''X'' by φ as a vector field on ''N''. For example, if the map φ is not surjective, there is no natural way to define such a pushforward outside of the image of φ. Also, if φ is not injective there may be more than one choice of pushforward at a given point. Nevertheless, one can make this difficulty precise, using the notion of a vector field along a map.
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| A [[vector bundle|section]] of φ*''TN'' over ''M'' is called a '''vector field along φ'''. For example, if ''M'' is a submanifold of ''N'' and φ is the inclusion, then a vector field along φ is just a section of the tangent bundle of ''N'' along ''M''; in particular, a vector field on ''M'' defines such a section via the inclusion of ''TM'' inside ''TN''. This idea generalizes to arbitrary smooth maps.
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| Suppose that ''X'' is a vector field on ''M'', i.e., a section of ''TM''. Then, applying the differential pointwise to ''X'' yields the '''pushforward''' φ<sub>∗</sub>''X'', which is a vector field along φ, i.e., a section of φ*''TN'' over ''M''.
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| Any vector field ''Y'' on ''N'' defines a [[pullback bundle|pullback section]] φ*''Y'' of φ*''TN'' with (φ*''Y'')<sub>''x''</sub> = ''Y''<sub>φ(''x'')</sub>. A vector field ''X'' on ''M'' and a vector field ''Y'' on ''N'' are said to be '''φ-related''' if φ<sub>∗</sub>''X'' = φ*''Y'' as vector fields along φ. In other words, for all ''x'' in ''M'', dφ<sub>''x''</sub>(''X'')=''Y''<sub>φ(''x'')</sub>.
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| In some situations, given a ''X'' vector field on ''M'', there is a unique vector field ''Y'' on ''M'' which is φ-related to ''X''. This is true in particular when φ is a [[diffeomorphism]]. In this case, the pushforward defines a vector field ''Y'' on ''N'', given by
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| :<math>Y_x=\varphi_*(X_{\varphi^{-1}(x)}).</math>
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| A more general situation arises when φ is surjective (for example the [[fiber bundle|bundle projection]] of a fiber bundle). Then a vector field ''X'' on ''M'' is said to be '''projectable''' if for all ''y'' in ''N'', dφ<sub>''x''</sub>(''X<sub>x</sub>'') is independent of the choice of ''x'' in φ<sup>−1</sup>({''y''}). This is precisely the condition that guarantees that a pushforward of ''X'', as a vector field on ''N'', is well defined.
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| ==See also==
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| *[[Pullback (differential geometry)|Pullback]]
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| ==References==
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| * John M. Lee, ''Introduction to Smooth Manifolds'', (2003) Springer Graduate Texts in Mathemγatics 218.
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| * Jürgen Jost, ''Riemannian Geometry and Geometric Analysis'', (2002) Springer-Verlag, Berlin ISBN 3-540-42627-2 ''See section 1.6''.
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| * [[Ralph Abraham]] and [[Jerrold E. Marsden]], ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X ''See section 1.7 and 2.3''.
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| [[Category:Generalizations of the derivative]]
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| [[Category:Differential geometry]]
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| [[Category:Smooth functions]]
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