Difference between revisions of "Smooth map"

From Maths
Jump to: navigation, search
m
m
Line 5: Line 5:
 
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}:\varphi(U)\rightarrow\psi(V)]</math> is [[Smooth|smooth]]
 
* <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]]
 +
<!--
  
 
===Via commutative diagrams===
 
===Via commutative diagrams===
Line 21: Line 22:
 
* {{M|(U,\varphi)\in\mathcal{A} }}
 
* {{M|(U,\varphi)\in\mathcal{A} }}
 
* {{M|(V,\psi)\in\mathcal{B} }}
 
* {{M|(V,\psi)\in\mathcal{B} }}
 
+
-->
  
 
==See also==
 
==See also==

Revision as of 21:36, 14 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