Let \(X=X^i\partial_i\) and \(Y=Y^i\partial_i\), where \((\partial_i)\) are the coordinate vector field. Then \(\nabla_X Y\) in local coordinates is given by
\begin{equation*} \nabla_X Y = (X(Y^k)+X^iY^j\Gamma_{ij}^k)E_k, \end{equation*}where \(\Gamma_{ij}^k\) are the Christoffel symbols of \(\nabla\).