Graph Coordinates

Let \(U\subseteq \mathbb{R}^k\) be open, \(f\colon U\to \mathbb{R}^n\) is a continuous function, and \(\Gamma(f)=\{(x,f(x))\mid x\in U\}\subseteq \mathbb{R}^{n+k}\) its graph . A global chart on \(\Gamma(f)\) is given by \((\Gamma(f), \varphi)\), where \(\varphi\colon \Gamma(f)\to U\) with

\[ \varphi(x,y)=y, \quad (x,y)\in \Gamma(f). \]

This chart is called graph coordinates.