Definition:Local Coordinates
From ProofWiki
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 $p$.
