Difference between revisions of "Smooth map"
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| + | {{Extra Maths}} | ||
'''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]] | ||
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| + | <!--See p35 intro to smooth manifolds - there are equiv definitions, but there's more to them--> | ||
| + | <!-- | ||
| + | ===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== | ||
Latest revision as of 21:37, 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
See also
References
- ↑ Introduction to smooth manifolds - John M Lee - Second Edition
