Let \(M\) be a \(n\)-dimensional manifold . A pair \((U, \phi)\) is called local chart on \(M\) if \(U\subset M\) is open and \(\phi\) is a homeomorphism of \(U\) onto an open subset of \(\mathbb{R}^n\). The map \(\varphi\) is called (local) coordinate map, and \(U\) is called coordinate domain. If, in addition, \(\varphi(U)\) is an open ball in \(\mathbb{R}^n\), then \(U\) is called coordinate ball .