\begin{definition}[Definition:Local Coordinates]
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 $p$'''.
\end{definition}