# Transforming covariant and contravariant derivatives

The covariant derivative of a contravector field is given by: $$\begin{equation} D_{k} A^{i} \equiv A^{i}_{\parallel k} = A^{i}_{\mid k} + \Gamma^{i}_{kp} A^{p} \end{equation}$$ With $$A^{i}_{\mid k} = \partial_{k} A^{i}$$. While the covariant derivative of a covector field is: $$\begin{equation} A_{i\parallel k} = A_{i\mid k} - \Gamma^{p}_{ik} A_{p} \end{equation}$$ This of course makes perfect sense to me because the derivatives transform exactly like a tensor of rank (1,1) and (0,2). However, since they are tensors, they should transform to each other, once the metric tensor $$g$$ is contracted with them. $$\begin{equation} A_{i \parallel k} = g_{ip} A^{p}_{\parallel k} \qquad A^{i}_{\parallel k} = g^{ip} A_{p\parallel k} \end{equation}$$ This is where I struggle. I have tried it from bouth directions and failed but here is what I came up with. $$\begin{equation} A^{i}_{\parallel k} = g^{ip} A_{p\parallel k} = g^{ip} A_{p \mid k} - g^{ip} \Gamma^{b}_{pk} A_{b}= A^{i}_{\mid k} - g^{ip} \frac{g^{bd}}{2} \left(\frac{\partial g_{pd}}{\partial x^k} + \frac{\partial g_{kd}}{\partial x^p} - \frac{\partial g_{pk}}{\partial x^d} \right) A_b \end{equation}$$ $$\begin{equation} =A^{i}_{\mid k} + \frac{g^{ip}}{2} \left(\frac{\partial g_{pk}}{\partial x^d} + \frac{\partial g_{pd}}{\partial x^k} - \frac{\partial g_{kd}}{\partial x^p} \right) A^d - g^{ip} \frac{\partial g_{pd}}{\partial x^k} A^d = A^i_{\mid k} + \Gamma^i_{kd} A^d - g^{ip} \frac{\partial g_{pd}}{\partial x^k} A^d \end{equation}$$ So it is the last term that is troubling me. I would expect it to be zero but I just can not see why this should be the case.

You made your mistake in the first line. Between the second and the third equal signs you raised the index through a partial derivative, but $$g^{ip} A_{p|k} \neq A^{i}_{|k} .$$ (It would be true for the covariant derivative.)
$$\begin{equation} (g^{ip} A_p)_{\mid k}=A^i_{\mid k} \end{equation}$$ $$\begin{equation} g^{ip}_{\mid k} g_{pd} A^d + g^{ip} A_{p \mid k} \end{equation}$$ Since $$(g^{ip} g_{pd})_{\mid k} = 0$$ we can conclude: $$\begin{equation} A^i_{\mid k} = -g_{pd\mid k} g^{ip} A^d + g^{ip} A_{p \mid k} \end{equation}$$ Which solves my problem.