# Chart

From Maths

**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