Coordinate Functions aka Charts

A coordinate function $\chi$ maps a point in a coordinate patch of a manifold to a coordinate tuple of real numbers:

where $x$ represents a convenient tuple structure. $x$ is usually represented as an “up vector” indexed by superscripts. The number of components of $x$ is equal to the number of dimensions of the manifold.

Coordinate transformations

Assume we have two coordinate functions $\chi$ and $\chi’$. The coordinate transformation from $\chi’$ coordinates to $\chi’$ coordinates is just the composition $\chi \circ \chi’^{-1}$, where $\chi’^{-1} $ is the functional inverse of $\chi’$