How to find covariant derivative of a contravariant tensor? Let I have a contravariant tensor $A^\alpha$, I want to find covariant derivative of the contravariant tensor,
From the transformation of the contravariant tensor ($A^\alpha=\partial_\gamma x^\alpha A^\gamma$):
$$\partial_\mu A^\alpha=\partial_\mu(\partial_\gamma x^\alpha A^\gamma)$$
$$=A^\gamma \partial_\mu\partial_\gamma x^\alpha +\partial_\gamma x^\alpha \partial_\mu A^\gamma$$
Taking, $\partial_\mu\partial_\gamma x^\alpha=\Gamma_{\mu\gamma}^\alpha$
$$\partial_\mu A^\alpha=A^\gamma \Gamma_{\mu\gamma}^\alpha +\partial_\gamma x^\alpha \partial_\mu A^\gamma$$
Looking at internet sources, I can say that $$D_\mu A^\alpha=\partial_\mu A^\alpha+ \Gamma^{\alpha}_{\alpha\mu}A^\alpha$$
But how it's true? I can't find it anyhow. In YT, most of people were writing it from their head (or rather conceptually?)
 A: First of all, the equation $\partial_\mu A^\alpha=\partial_\mu(\partial_\beta x^\gamma A_\gamma)$ cannot be true as the contravariant index $\alpha$ disappears and the covariant index $\beta$ appears. Furthermore the definition of the Christoffel symbols isn't $\Gamma_{\mu\beta}^\gamma=\partial_\mu\partial_\beta x^\gamma$, but:
\begin{equation}
\Gamma_{\mu\beta}^\gamma
:=\frac{\partial x^\gamma}{\partial\xi^\nu}\frac{\partial^2\xi^\nu}{\partial x^\mu\partial x^\beta},
\end{equation}
where $\xi$ is a free-falling coordinate system without gravity according to the equivalence principle.
The construction of the covariant derivative arises from the problem that $\partial_\mu A^\sigma$ does not transform like a tensor. Let $\alpha$ and $\overline{\alpha}$ denote the transformation matrices, then:
\begin{align*}
\partial_\kappa'{A'}^\rho
&=\frac{\partial x^\mu}{\partial{x'}^\kappa}\frac{\partial}{\partial x^\mu}
\left(\frac{\partial{x'}^\rho}{\partial x^\sigma}A^\sigma\right)
=\frac{\partial x^\mu}{\partial{x'}^\kappa}\frac{\partial{x'}^\rho}{\partial x^\sigma}\frac{\partial A^\sigma}{\partial x^\mu}
+\frac{\partial x^\mu}{\partial{x'}^\kappa}\frac{\partial^2{x'}^\rho}{\partial x^\mu\partial x^\sigma}A^\sigma \\
&=\overline{\alpha}_\kappa^\mu\alpha_\sigma^\rho\partial_\mu A^\sigma
-\frac{\partial^2x^\xi}{\partial{x'}^\kappa\partial{x'}^\lambda}\alpha_\sigma^\lambda\alpha_\xi^\rho A^\sigma.
\end{align*}
The first term is that of a tensor transformation, but there is a additional second term. It can be eliminated using the Christoffel symbols, that also do not transform like a tensor:
\begin{align*}
{\Gamma'}_{\kappa\lambda}^\rho
&=\frac{\partial{x'}^\rho}{\partial \xi^\alpha}\frac{\partial^2 \xi^\alpha}{\partial {x'}^\kappa{x'}^\lambda}
=\frac{\partial{x'}^\rho}{\partial x^\sigma}\frac{\partial x^\sigma}{\partial \xi^\alpha}\frac{\partial}{\partial {x'}^\kappa}\left(\frac{\partial\xi^\alpha}{\partial x^\nu}\frac{\partial x^\nu}{\partial {x'}^\lambda}\right) \\
&=\frac{\partial{x'}^\rho}{\partial x^\sigma}\frac{\partial x^\sigma}{\partial \xi^\alpha}\left(\frac{\partial\xi^\alpha}{\partial x^\mu\partial x^\nu}\frac{\partial x^\mu}{\partial {x'}^\kappa}\frac{\partial x^\nu}{\partial {x'}^\lambda}+\frac{\partial\xi^\alpha}{\partial x^\nu}\frac{\partial^2 x^\nu}{\partial {x'}^\kappa\partial {x'}^\lambda}\right) \\
&=\frac{\partial {x'}^\rho}{\partial x^\sigma}\frac{\partial x^\mu}{\partial {x'}^\kappa}\frac{\partial x^\nu}{\partial {x'}^\lambda}\Gamma_{\mu\nu}^\sigma
+\frac{\partial {x'}^\rho}{\partial x^\tau}\frac{\partial^2 x^\tau}{\partial {x'}^\kappa\partial {x'}^\lambda}
=\alpha_\sigma^\rho\overline{\alpha}_\kappa^\mu\overline{\alpha}_\lambda^\nu\Gamma_{\mu\nu}^\sigma
+\alpha_\tau^\rho\frac{\partial^2 x^\tau}{\partial {x'}^\kappa\partial {x'}^\lambda}.
\end{align*}
A suitable combination of both equations now yields the transformation of a tensor:
\begin{align*}
\partial_\kappa'{A'}^\rho+{\Gamma'}_{\kappa\lambda}^\rho{A'}^\lambda
&=\overline{\alpha}_\kappa^\mu\alpha_\sigma^\rho\partial_\mu A^\sigma
-\frac{\partial^2x^\xi}{\partial{x'}^\kappa\partial{x'}^\lambda}\alpha_\sigma^\lambda\alpha_\xi^\rho A^\sigma
+\left(\overline{\alpha}_\kappa^\mu\overline{\alpha}_\lambda^\nu\alpha_\sigma^\rho\Gamma_{\mu\nu}^\sigma
+\alpha_\tau^\rho\frac{\partial^2 x^\tau}{\partial {x'}^\kappa\partial {x'}^\lambda}\right)\alpha_\nu^\lambda A^\nu \\
&=\overline{\alpha}_\kappa^\mu\alpha_\sigma^\rho\left(\partial_\mu A^\sigma+\Gamma_{\mu\nu}^\sigma A^\nu\right).
\end{align*}
This tensor is the covariant derivative $\nabla_\mu A^\sigma:=\partial_\mu A^\sigma+\Gamma_{\mu\nu}^\sigma A^\nu$.
