====== Definition ======

===== coordinate system =====

a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space.(Wikipedia)

===== coordinate change =====

A coordinate change is a set of functions $f^r(x^1, ..., x^N)$ $(r = 1, ..., N)$.

One generally supposes that $J' = \left|\frac{\partial x'^{r}}{\partial x^{s}}\right|$ does not vanish.

The new coordinates of a point $(x^1, ..., x^N)$ are $(x'^1, ..., x'^N)$ where $x'^r = f^r(x^1, ..., x^N)$ $(r = 1, ..., N)$.