Definition:Co-ordinate

Definition

Let $M$ be a Locally Euclidean space of dimension $d$.

Let $U \subseteq M$ be an open connected set.

Let $I \subseteq \R^d$ be an open subset of Euclidean space.

Co-ordinate Map

If $\phi : U \to I$ is a homeomorphism, then $\phi$ is called a co-ordinate map.

Co-ordinate System

The pair (U,\phi) is called a co-ordinate system.

If $m \in M$ such that $\phi(m) = 0$, then the co-ordinate system is said to be centred at $m$.

If $\phi(U)$ is an open cube about the origin, then $(U,\phi)$ is a cubic co-ordinate system.

Co-ordinate Functions

If $r_i$, $i=1,\ldots,d$ are defined by

$r_i : \R^d \owns (y_1,\ldots, y_d) \mapsto y_i \in \R$

then the functions $x_i = r_i \circ \phi$ are called the co-ordinate functions of $\phi$.