Difference between revisions of "Smooth map"
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'''Note: ''' not to be confused with [[Smooth function|smooth function]] | '''Note: ''' not to be confused with [[Smooth function|smooth function]] | ||
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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=== | ||
| + | A map is smooth if the following diagram commutes: | ||
| + | |||
| + | [math]\begin{CD} | ||
| + | M {{CD Right Arrow|F}} N\\ | ||
| + | {{CD Down Arrow|\varphi}} {{CD Down Arrow||\psi}}\\ | ||
| + | \varphi(U) {{CD Right Arrow|G|2==\psi\circ F\circ\varphi^{-1} }} \psi(V) | ||
| + | \end{CD}[/math] | ||
| + | |||
| + | Where: | ||
| + | * {{M|G}} is [[Smooth|smooth]] | ||
| + | ** (given by {{M|1=G=\psi\circ\varphi^{-1}:\varphi(U)\rightarrow\psi(V) }}) | ||
| + | * {{M|M,N}} are [[Smooth manifold|smooth manifolds]] (with [[Smooth structure|smooth structures]]) {{M|\mathcal{A},\mathcal{B} }} respectively | ||
| + | * {{M|(U,\varphi)\in\mathcal{A} }} | ||
| + | * {{M|(V,\psi)\in\mathcal{B} }} | ||
| + | |||
==See also== | ==See also== | ||
Revision as of 14:27, 14 April 2015
[math]\newcommand{\bigudot}{ \mathchoice{\mathop{\bigcup\mkern-15mu\cdot\mkern8mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}}{\mathop{\bigcup\mkern-13mu\cdot\mkern5mu}} }[/math][math]\newcommand{\udot}{\cup\mkern-12.5mu\cdot\mkern6.25mu\!}[/math][math]\require{AMScd}\newcommand{\d}[1][]{\mathrm{d}^{#1} }[/math] Note: not to be confused with smooth function
Definition
A map [ilmath]f:M\rightarrow N[/ilmath] between two smooth manifolds [ilmath](M,\mathcal{A})[/ilmath] and [ilmath](N,\mathcal{B})[/ilmath] (of not necessarily the same dimension) is said to be smooth[1] 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
Via commutative diagrams
A map is smooth if the following diagram commutes:
[math]\begin{CD} M @> F > > N\\ @V \varphi V V @V V\psi V\\ \varphi(U) @> G >=\psi\circ F\circ\varphi^{-1} > \psi(V) \end{CD}[/math]
Where:
- [ilmath]G[/ilmath] is smooth
- (given by [ilmath]G=\psi\circ\varphi^{-1}:\varphi(U)\rightarrow\psi(V)[/ilmath])
- [ilmath]M,N[/ilmath] are smooth manifolds (with smooth structures) [ilmath]\mathcal{A},\mathcal{B} [/ilmath] respectively
- [ilmath](U,\varphi)\in\mathcal{A} [/ilmath]
- [ilmath](V,\psi)\in\mathcal{B} [/ilmath]
See also
References
- ↑ Introduction to smooth manifolds - John M Lee - Second Edition
