coordinate chart

coordinate chart (plural coordinate charts)

1. (differential geometry) A homeomorphism from a local neighborhood around a point on a topological n-manifold in a Banach space to an n-dimensional Banach space (typically a subspace of the space the manifold is embedded in).

   Synonyms: chart, coordinate map, surface patch

   • 2016 September 30, Kristopher Tapp, Differential Geometry of Curves and Surfaces (Undergraduate Texts in Mathematics)‎, Springer, →ISBN, →OCLC, page 125:

     A set ${\displaystyle S\subset \mathbb {R} ^{3}}$ is called a regular surface if each of its points has a neighborhood in ${\displaystyle S}$ that is diffeomorphic to an open set in ${\displaystyle \mathbb {R} ^{2}}$. That is, for every ${\displaystyle p\in S}$, there exists a neighborhood, ${\displaystyle V}$, of ${\displaystyle p}$ in ${\displaystyle S}$, an open set ${\displaystyle U\subset \mathbb {R} ^{2}}$, and a diffeomorphism ${\displaystyle \sigma :U\rightarrow V}$. Such a diffeomorphism ${\displaystyle \sigma }$ is called a surface patch or a coordinate chart. A collection of surface patches that together cover all of the points of ${\displaystyle S}$ is called an atlas for ${\displaystyle S}$.

• Some authors refer to the pair of the neighborhood the mapping is defined on and the mapping together (i.e. ${\displaystyle (U,\varphi )}$ instead of ${\displaystyle \varphi }$, where ${\displaystyle U=\operatorname {dom} (\varphi )}$) as the coordinate chart while referring to the mapping alone as the coordinate map.

• transition function
• atlas