In mathematics and its applications, a coordinate system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. This concept is part of the theory of manifolds. "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other commutative ring. For complicated spaces, it is often not possible to provide one consistent coordinate system for the entire space. In this case, a collection of coordinate systems, called graphs, are put together to form an atlas covering the whole space. A simple example (which motivates the terminology) is the surface of the earth.
Although a specific coordinate system is useful for numerical calculations in a given space, the space itself is considered to exist independently of any particular choice of coordinates. From this point of view, a coordinate on a space is simply a function from the space (or a subset of the space) to the scalars. When the space has additional structure, one restricts attention to the functions which are compatible with this structure. Examples include:
The coordinates on a space transform naturally (by pullback) under the group of automorphisms of the space, and the set of all coordinates is a commutative ring called the coordinate ring of the space.
