linguistic:translation, structural:reformatting, linguistic:voice_change, stylistic:tone
Charts

A coordinate map, a coordinate chart, or simply a chart, of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure. For a topological manifold, the simple space is a subset of some Euclidean space R n {\displaystyle \mathbb {R} ^{n}} and interest focuses on the topological structure. This structure is preserved by homeomorphisms, invertible maps that are continuous in both directions.

In the case of a differentiable manifold, a set of charts called an atlas, whose transition functions (see below) are all differentiable, allows us to do calculus on it. Polar coordinates, for example, form a chart for the plane R 2 {\displaystyle \mathbb {R} ^{2}} minus the positive x -axis and the origin. Another example of a chart is the map χ mentioned above, a chart for the circle.