Let \((U,\varphi)\) be a local chart on a manifold \(M\). Then
\begin{equation*} p\mapsto dx^i{|}_p \end{equation*}is a covector field on \(M\), where \(dx^i\) denotes the differential of the coordinate map . It is called \(i\)-th coordinate covector field.
Remark
- the tuple \((dx^1, \ldots, dx^n)\) is a local coframe .
- at every point the dual basis of the coordinate vector field is the coordinate covector field