Coordinate Chart

A coordinate chart is a way of expressing the points of a small neighborhood, usually on a manifold M, 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 U is an open set in M, V is an open set in R^n and n is the dimension of the manifold. Often, through notational abuse, the open set U is equated with V, 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 phi 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 phi is holomorphic.

If there are two neighborhoods U_1 and U_2 with coordinate charts phi_1 and phi_2, the transition function phi_2 degreesphi_1^(-1) is well-defined since coordinate charts are one-to-one.

See also

Atlas, Complex Manifold, Euclidean Space, Manifold, Smooth Manifold, Transition Function

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Coordinate Chart." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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