# Definition:Local Coordinates

## Definition

Let $X$ be an $n$-dimensional manifold.

Let $p \in X$, and let $U \subseteq X$ be a neighbourhood of $p$.

Then a set of mappings $x_i: U \to \R$, $1 \le i \le n$, satisfying:

$a = b \iff \forall i: \map {x_i} a = \map {x_i} b$

is called a set of local coordinates.

When the neighbourhood $U$ is to be stressed, one may also say local coordinates for $U$.

Similarly, when the element $p$ is to be stressed, one may also say local coordinates around $p$.

## Also known as

Local coordinates $\tuple {x_1, x_2, \ldots x_n}$ are also know as the component functions of the local coordinate map $\phi$ defined by:

$\map \phi p = \tuple {\map {x_1} p, \map {x_2} p, \ldots \map {x_n} p}$