coordinate chart

English

coordinate chart (plural coordinate charts)

(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 1005779369, 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.