coordinate chart

English

Noun

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)‎^[1], Springer, →ISBN, →OCLC, page 125:

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

Usage notes

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

Translations

Translations

• Finnish: kartta (fi)
• German: Karte (de) f, Flachmacher m

See also

• transition function
• atlas