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)‎^[1], Springer, →ISBN, →OCLC, page 125:
  A set $S \subset ℝ^{3}$ is called a regular surface if each of its points has a neighborhood in $S$ that is diffeomorphic to an open set in $ℝ^{2}$ . That is, for every $p \in S$ , there exists a neighborhood, $V$ , of $p$ in $S$ , an open set $U \subset ℝ^{2}$ , and a diffeomorphism $σ : U \to V$ . Such a diffeomorphism $σ$ 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 $S$ .

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