Difference between revisions of "Chart"
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'''Note:''' Sometimes called a coordinate chart | '''Note:''' Sometimes called a coordinate chart | ||
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| + | '''Note:''' see [[Transition map]] for moving between charts, and [[Smoothly compatible charts]] for the smooth form. | ||
==Definition== | ==Definition== | ||
| − | A coordinate chart - or just chart on a | + | A coordinate chart - or just chart on a [[Topological manifold|topological manifold]] of dimension {{M|n}} is a pair {{M|(U,\varphi)}}<ref>John M Lee - Introduction to smooth manifolds - Second Edition</ref> where: |
* {{M|U\subseteq M}} that is open | * {{M|U\subseteq M}} that is open | ||
* {{M|\varphi:U\rightarrow\hat{U} }} is a [[Homeomorphism|homeomorphism]] from {{M|U}} to an [[Open set|open]] subset {{M|1=\hat{U}=\varphi(U)\subseteq\mathbb{R}^n}} | * {{M|\varphi:U\rightarrow\hat{U} }} is a [[Homeomorphism|homeomorphism]] from {{M|U}} to an [[Open set|open]] subset {{M|1=\hat{U}=\varphi(U)\subseteq\mathbb{R}^n}} | ||
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* [[Atlas]] | * [[Atlas]] | ||
* [[Manifold]] | * [[Manifold]] | ||
| + | * [[Transition map]] | ||
| + | * [[Smoothly compatible charts]] | ||
==References== | ==References== | ||
Latest revision as of 06:32, 7 April 2015
Note: Sometimes called a coordinate chart
Note: see Transition map for moving between charts, and Smoothly compatible charts for the smooth form.
Definition
A coordinate chart - or just chart on a topological manifold of dimension [ilmath]n[/ilmath] is a pair [ilmath](U,\varphi)[/ilmath][1] where:
- [ilmath]U\subseteq M[/ilmath] that is open
- [ilmath]\varphi:U\rightarrow\hat{U} [/ilmath] is a homeomorphism from [ilmath]U[/ilmath] to an open subset [ilmath]\hat{U}=\varphi(U)\subseteq\mathbb{R}^n[/ilmath]
Names
- [ilmath]U[/ilmath] is called the coordinate domain or coordinate neighbourhood of each of its points
- If [ilmath]\varphi(U)[/ilmath] is an open ball then [ilmath]U[/ilmath] may be called a coordinate ball, or cube or whatever is applicable.
- [ilmath]\varphi[/ilmath] is called a local coordinate map or just coordinate map
- The component functions [math](x^1,\cdots,x^n)=\varphi[/math] are defined by [math]\varphi(p)=(x^1(p),\cdots,x^n(p))[/math] and are called local coordinates on U
Shorthands
- To emphasise coordinate functions over coordinate map, we may denote the chart by [math](U,(x^1,\cdots,x^n))[/math] or [math](U,(x^i))[/math]
- [ilmath](U,\varphi)[/ilmath] is a chart containing [ilmath]p[/ilmath] is shorthand for "[ilmath](U,\varphi)[/ilmath] is a chart whose domain, [ilmath]U[/ilmath], contains [ilmath]p[/ilmath]"
Comments
- By definition each point of the manifold is contained in some chart
- If [ilmath]\varphi(p)=0[/ilmath] the chart is said to be centred at [ilmath]p[/ilmath] (see below)
Centred chart
If [ilmath]\varphi(p)=0[/ilmath] then the chart [ilmath](U,\varphi)[/ilmath] is said to be centred at [ilmath]p[/ilmath]
- Given any chart whose domain contains [ilmath]p[/ilmath] it is easy to obtain a chart centred at [ilmath]p[/ilmath] simply by subtracting the constant vector [ilmath]\varphi(p)[/ilmath]
See also
References
- ↑ John M Lee - Introduction to smooth manifolds - Second Edition
