Let \(X=X^iE_i\) and \(Y=Y^iE_i\), where \((E_i)\) is a local frame . The connection \(\nabla_XY\) has the local expression
\begin{equation*} \nabla_X Y = (X(Y^k)+X^iY^j\Gamma_{ij}^k)E_k, \end{equation*}where \(\Gamma_{ij}^k\) are the connection coefficients of \(\nabla\).