Covariant Derivative Along Curves in Local Coordinates

The covariant derivative along a curve \(\gamma\colon I\to M\) in local coordinates is given by

\begin{equation*} D_tV(t)=\bigl(\dot{V}^k(t)+\dot{\gamma}^i(t)V^j(t)\Gamma^k_{ij}(\gamma(t))\bigr), \end{equation*}

where \(\Gamma^k_{ij}\) denotes the connection coefficients of the connection \(\nabla\) determining \(D_t\).