Definition:Local Coordinates

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

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

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


 * $a = b \iff \forall i: x_i \left({a}\right) = x_i \left({b}\right)$

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 point $p$ is to be stressed, one may also say local coordinates around $p$.