= Solution
The <covariant derivative> differentiates a <vector field> while accounting for how its <coordinate basis> changes between neighboring points. For the <Levi-Civita connection>,
$$
\boxed{
v^\beta{}_{;\gamma}
=\nabla_\gamma v^\beta
=\partial_\gamma v^\beta+\Gamma^\beta{}_{\gamma\delta}v^\delta.
}
$$
The <Christoffel symbol> term makes this a <tensor> with one contravariant and one covariant index; ordinary <partial derivatives> of vector components do not generally have that transformation law. Equivalently, one compares the neighboring vector with the original after <parallel transport>.
For a curve $x^\gamma(\lambda)$ with tangent $U^\gamma=dx^\gamma/d\lambda$, contraction gives the <covariant derivative along a curve>
$$
\boxed{
\frac{Dv^\beta}{d\lambda}
=U^\gamma\nabla_\gamma v^\beta
=\frac{dv^\beta}{d\lambda}
+\Gamma^\beta{}_{\gamma\delta}U^\gamma v^\delta.
}
$$
It is the rate of change measured relative to a <parallel transport> frame along the curve. It vanishes precisely when $v$ undergoes <parallel transport> along that curve. If $v=U$, it is the covariant acceleration, and its vanishing is the <geodesic equation> with an <affine parameter>.
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