Consider coordinate vector fields on a smooth manifold. Then the tuple \((\frac{\partial }{\partial x^1}, \ldots, \frac{\partial }{\partial x^n})\) is a smooth local frame [@lee2013smooth_manifolds].
\[
\newcommand{\d}{\mathrm{d}}
\newcommand{\e}{\mathrm{e}}
\newcommand{\i}{\mathrm{i}}
\]