In the book on Riemannian Geometry by John Lee ("Riemannian manifolds: an Introduction to Curvature") the author gives an exercise on page 54 involving connections:
Let $\triangledown$ be a linear connection. If $\omega$ us a 1-form and $X$ is a vector field, show that the coordinate expression for $\triangledown_X \,\omega$ is $ \triangledown_X \, \omega = \left(X^i \partial_i \omega_k - X^i w_j \Gamma^j_{ik} \right) \, dx^k $ where $\{\, \Gamma^k_{ij} \,\}$ are the Christoffel symbols of the given connection $\triangledown$ on TM.
I am a little stuck here, I appologize as I sense this should be really easy to solve. My guess is that I could use the property of the natural pairing $ \langle \partial_j , \ dx^k \, \rangle = \delta_k^j $ together with the property $ \begin{eqnarray} &0 = \triangledown_{\partial_i} \delta_k^j = \triangledown_{\partial_i} \langle dx^k, \delta_j \, \rangle = \langle \triangledown_{\partial_i} \, dx^k, \partial_j \, \rangle + \langle dx^k, \triangledown_{\partial_i} \partial_j \, \rangle \\ &= \langle \triangledown_{\partial_i} \, dx^k, \partial_j \, \rangle + \langle dx^k, \Gamma^k_{ij} \partial_k \, \rangle = \langle \triangledown_{\partial_i} \, dx^k, \partial_j \, \rangle + \Gamma^k_{ij} \end{eqnarray} \quad \text{(is this correct ?)} $ but I am not sure how to proceed or where I am having errors in my argument so far.. any help would be great ! sorry again if this is obvious .. thanks !