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Smoothness
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===Smooth functions on and between manifolds=== Given a [[Differentiable manifold|smooth manifold]] <math>M</math>, of dimension <math>m,</math> and an [[Atlas (topology)|atlas]] <math>\mathfrak{U} = \{(U_\alpha,\phi_\alpha)\}_\alpha,</math> then a map <math>f:M\to \R</math> is '''smooth''' on <math>M</math> if for all <math>p \in M</math> there exists a chart <math>(U, \phi) \in \mathfrak{U},</math> such that <math>p \in U,</math> and <math>f \circ \phi^{-1} : \phi(U) \to \R</math> is a smooth function from a neighborhood of <math>\phi(p)</math> in <math>\R^m</math> to <math>\R</math> (all partial derivatives up to a given order are continuous). Smoothness can be checked with respect to any [[Chart (topology)|chart]] of the atlas that contains <math>p,</math> since the smoothness requirements on the transition functions between charts ensure that if <math>f</math> is smooth near <math>p</math> in one chart it will be smooth near <math>p</math> in any other chart. If <math>F : M \to N</math> is a map from <math>M</math> to an <math>n</math>-dimensional manifold <math>N</math>, then <math>F</math> is smooth if, for every <math>p \in M,</math> there is a chart <math>(U,\phi)</math> containing <math>p,</math> and a chart <math>(V, \psi)</math> containing <math>F(p)</math> such that <math>F(U) \subset V,</math> and <math>\psi \circ F \circ \phi^{-1} : \phi(U) \to \psi(V)</math> is a smooth function from <math>\R^n.</math> Smooth maps between manifolds induce linear maps between [[tangent space]]s: for <math>F : M \to N</math>, at each point the [[Pushforward (differential)|pushforward]] (or differential) maps tangent vectors at <math>p</math> to tangent vectors at <math>F(p)</math>: <math>F_{*,p} : T_p M \to T_{F(p)}N,</math> and on the level of the [[tangent bundle]], the pushforward is a [[vector bundle homomorphism]]: <math>F_* : TM \to TN.</math> The dual to the pushforward is the [[Pullback (differential geometry)|pullback]], which "pulls" covectors on <math>N</math> back to covectors on <math>M,</math> and <math>k</math>-forms to <math>k</math>-forms: <math>F^* : \Omega^k(N) \to \Omega^k(M).</math> In this way smooth functions between manifolds can transport [[Sheaf (mathematics)|local data]], like [[vector field]]s and [[differential form]]s, from one manifold to another, or down to Euclidean space where computations like [[Integration on manifolds|integration]] are well understood. Preimages and pushforwards along smooth functions are, in general, not manifolds without additional assumptions. Preimages of regular points (that is, if the differential does not vanish on the preimage) are manifolds; this is the [[preimage theorem]]. Similarly, pushforwards along embeddings are manifolds.<ref>{{cite book |last1=Guillemin |first1=Victor |last2=Pollack |first2=Alan |title=Differential Topology |location=Englewood Cliffs |publisher=Prentice-Hall |year=1974 |isbn=0-13-212605-2 }}</ref>
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