= Solution
{c}
For a contravariant <vector field> $A^\lambda$, the <covariant derivative> is
$$
\boxed{\nabla_\nu A^\lambda=A^\lambda{}_{;\nu}
=\partial_\nu A^\lambda+\Gamma^\lambda_{\mu\nu}A^\mu.}
$$
The <Levi-Civita connection> is the unique <torsion-free connection> satisfying <metric compatibility>. Its <Christoffel symbols> in a <coordinate basis> are
$$
\boxed{\Gamma^\lambda_{\mu\nu}
=\frac12g^{\lambda\sigma}
\left(\partial_\mu g_{\sigma\nu}
+\partial_\nu g_{\sigma\mu}-\partial_\sigma g_{\mu\nu}\right),
\qquad\Gamma^\lambda_{\mu\nu}=\Gamma^\lambda_{\nu\mu}.}
$$
There is generally no symmetry interchanging the upper index with a lower one. The <Christoffel symbols> themselves are not the components of a <tensor>.
An ordinary <partial derivative> of the components of a <vector field> includes the apparent change caused by the varying <coordinate basis>. The connection term corrects for that change, so $\nabla_\nu A^\lambda$ obeys the <tensor component transformation law>. It gives coordinate-independent ways to express <parallel transport>, <geodesics> and <stress-energy conservation> in curved <spacetime>.
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