Let \(\gamma\colon I\to M\) be a curve and \((x^i)\) local coordinates. We write the components functions of \(\gamma\) as
\begin{equation*} \gamma(t)=(x^1(t),\ldots,x^n(t)). \end{equation*}Using the local representation of the covariant derivative \(D_t\) it follows that \(\gamma\) is geodesic if and only if
\begin{equation*} \ddot{x}^k(t)+\dot{x}^i(t)\dot{x}^j(t)\Gamma^k_{ij}(x(t))=0. \end{equation*}