Difference between revisions of "Smooth map"

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(Created page with "==Definition== A map {{M|f:M\rightarrow N}} between two smooth manifolds {{M|(M,\mathcal{A})}} and {{M|(N,\mathcal{B})}} (of not necessarily the same dimen...")
 
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'''Note: ''' not to be confused with [[Smooth function|smooth function]]
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==Definition==
 
==Definition==
 
A map {{M|f:M\rightarrow N}} between two [[Smooth manifold|smooth manifolds]] {{M|(M,\mathcal{A})}} and {{M|(N,\mathcal{B})}} (of not necessarily the same dimension) is said to be smooth<ref>Introduction to smooth manifolds - John M Lee - Second Edition</ref> if:
 
A map {{M|f:M\rightarrow N}} between two [[Smooth manifold|smooth manifolds]] {{M|(M,\mathcal{A})}} and {{M|(N,\mathcal{B})}} (of not necessarily the same dimension) is said to be smooth<ref>Introduction to smooth manifolds - John M Lee - Second Edition</ref> if:
* <math>\forall p\in M\exists\ (U,\varphi)\in\mathcal{A},\ p\in U\text{ and }(V,\psi)\in\mathcal{B}</math> such that <math>F(U)\subseteq V\wedge \psi\circ F\circ\varphi^{-1}</math> is [[Smooth|smooth]]
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* <math>\forall p\in M\exists\ (U,\varphi)\in\mathcal{A},\ p\in U\text{ and }(V,\psi)\in\mathcal{B}</math> such that <math>F(U)\subseteq V\wedge[\psi\circ F\circ\varphi^{-1}:\varphi(U)\rightarrow\psi(V)]</math> is [[Smooth|smooth]]
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==See also==
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* [[Smooth function]]
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* [[Smooth manifold]]
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==References==
 
==References==
 
<references/>
 
<references/>
  
 
{{Definition|Manifolds}}
 
{{Definition|Manifolds}}

Revision as of 20:44, 13 April 2015

Note: not to be confused with smooth function

Definition

A map f:MN between two smooth manifolds (M,A) and (N,B) (of not necessarily the same dimension) is said to be smooth[1] if:

  • pM (U,φ)A, pU and (V,ψ)B
    such that F(U)V[ψFφ1:φ(U)ψ(V)]
    is smooth

See also

References

  1. Jump up Introduction to smooth manifolds - John M Lee - Second Edition