\[ \newcommand{\d}{\mathrm{d}} \newcommand{\e}{\mathrm{e}} \newcommand{\i}{\mathrm{i}} \]

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

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

is a vector field on \(M\). It is called \(i\)-th coordinate vector field.

If \(M\) is smooth then the coordinate vector field is also smooth [@lee2013smooth_manifolds, Proposition 8.1].

Remark
  • The tuple \((\frac{\partial }{\partial x^1}, \ldots, \frac{\partial }{\partial x^n})\) is a local frame .
  • At every point the dual basis of the coordinate vector field is the coordinate covector field .
  • We abbreviate the \(i\)-th coordinate vector field by \(\partial_i\).