Let \(M\) be a \(n\)-manifold and \((U, \varphi)\) a local chart on \(M\). Then for \(i \in \{1,\ldots,n\}\) the coordinate map \(x^i_{M}\colon U \to \mathbb{R}\) corresponding to \((U, \varphi)\) is defined by

\begin{equation*} x_{M}^i(p)=(\varphi(p)_i=x_{\mathbb{R}^n}^i\circ \varphi(p) \end{equation*}

for \(p\in U\). Usually we omit \(M\) in the superscript and just write \(x^i\).

The tuple \((x^1,\ldots,x^n)\) is called local coordinates corresponding to \((U, \varphi)\).

See also Link to heading

• partial derivatives as tangent vectors