A coordinate chart is a way of expressing the points of a small neighborhood, usually on a manifold 
, as coordinates in Euclidean
space. An example from geography is the coordinate chart given by the functions
of latitude and longitude.
This coordinate chart is not valid on the whole globe, since it doesn't give unique
coordinates at the north or south
pole (which way is east from the north pole?).
Technically, a coordinate chart is a map
 |
where 
is an open set in 
,
is an open set in 
and 
is the dimension of the manifold. Often, through notational
abuse, the open set 
is equated with 
,
and calculations on the manifold are done in the coordinate chart. This technique
has the drawback that it must be checked whether a change of coordinates affects
the result of a calculation.
The map 
must be one-to-one, and in fact must be a Homeomorphism.
On a smooth manifold, it must be a diffeomorphism,
although if the chart defines the smooth structure then this is a tautology. Similarly,
on a complex manifold, the map 
is holomorphic.
If there are two neighborhoods 
and 
with coordinate charts 
and 
, the transition function 
is well-defined
since coordinate charts are one-to-one.
