coordinate system in nLab

Contents

Contents

Idea

A topological manifold XX of dimension n∈ℕn \in \mathbb{N} is by definition a topological space that is locally homeomorphic to a Cartesian space ℝ n\mathbb{R}^n. A choice of such morphism

ϕ:ℝ n→≃X| im(ϕ)↪X \phi : \mathbb{R}^n \stackrel{\simeq}{\to} X|_{im(\phi)} \hookrightarrow X

is a coordinate system or coordinate chart or just chart on the image of ϕ\phi.

This generalises to other sorts of manifolds.

An atlas is the collection of coordinate charts defining a manifold structure.

graphics grabbed from Frankel

• coordinate transformation, diffeomorphism

• Riemann normal coordinates, Walker coordinates

• polar coordinates