Let \((U,\varphi)\) be a local chart on a smooth manifold \(M\). Then

\begin{equation*} p\mapsto \frac{\partial }{\partial x^i}\bigg|_p \end{equation*}

is a vector field on \(M\). It is called \(i\)-th coordinate vector field, and we abbreviate it by \(\partial_i\).

Links Link to heading

Results Link to heading

Remark

At every point the dual basis of the coordinate vector field is the coordinate covector field .