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

Let \(X\) be a vector field and \((\frac{\partial }{\partial x^i})\) be the coordinate vector fields on some open subset \(U\subset M\). Then

\begin{equation*} X=X^i\frac{\partial }{\partial x^i}, \end{equation*}

where \(X^i\colon U\to \mathbb{R}\) denote the components of \(X\) in the given chart.

Remarks
  • Using the compact notation of coordinate vector fields, we get \begin{equation*} X=X^i\partial_i. \end{equation*}