Definition:Local Coordinates

Definition
For neighborhood $U$ of a point $p$, $U \subset X^n$ in an $n$-dimensional manifold $X^n$, local coordinates are a set of functions $\left\{{x_i}\right\}_{i=1}^n$ defined $x_i:X^n \to \R$ defined so that if $a,b \in U$ and $\forall i, x_i(a)=x_i(b)$ then $ p=q$.