Covariant derivative

The partial derivative of a vector $\partial_\alpha V^\beta$ does not transform like a tensor. The covariant derivative $\nabla_\alpha V^\beta$ does, and is the generalization of the partial derivative to curved spacetime. It is defined in terms of the Christoffel connection. Conceptually, it takes into account the variation of the basis vectors over the manifold.

\begin{align*} \nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma^\nu_{\mu\lambda} V^\lambda. \end{align*}

It is simple to show that the covariant derivative transforms like a tensor. Let $L^\mu{}_\nu = \frac{\partial x^\mu}{\partial x^\nu}$ . Then for $\nabla_\mu V^\nu$ to be a tensor, we demand

\begin{align*} \nabla_{\mu'}V^{\nu'} &= \partial_{\mu'} V^{\nu'} + \Gamma^{\nu'}_{\mu'\lambda'} V^{\lambda'} = L^\mu{}_{\mu'} L^{\nu'}{}_\nu \nabla_\mu V^\nu \\ &= L^\mu{}_{\mu'} L^{\nu'}{}_\nu \partial_\mu V^\nu + L^\mu{}_{\mu'} V^\nu \partial_\mu L^{\nu'}{}_\nu + \Gamma^{\nu'}_{\mu'\lambda'} L^{\lambda'}{}_\lambda V^\lambda \\ &= L^\mu{}_{\mu'} L^{\nu'}{}_\nu \partial_\mu V^\nu + L^\mu{}_{\mu'} L^{\nu'}{}_\nu \Gamma^\nu_{\mu\lambda} V^\lambda \\ L^\mu{}_{\mu'} L^{\nu'}{}_\nu \Gamma^\nu_{\mu\lambda} V^\lambda &= \Gamma^{\nu'}_{\mu'\lambda'} L^{\lambda'}{}_{\lambda} V^\lambda + L^\mu{}_{\mu'} V^\lambda \partial_\mu L^{\nu'}_\nu \\ \Gamma^{\nu'}_{\mu'\lambda'} &= L^\mu{}_{\mu'} L^\lambda{}_{\lambda'} L^{\nu'}{}_{\nu} \Gamma^\nu_{\mu\lambda} + L^{\mu}{}_{\mu'} L^\lambda{}_{\lambda'} \partial_\mu L^{\nu'}{}_{\lambda}. \end{align*}

The Christoffel symbol $\Gamma$ is defined such that its transformation perfectly cancels out the aditional terms introduced by the partial derivative.