= Solution
A <vector field along a map> $\gamma$ assigns $V(t)\in T_{\gamma(t)}M$ smoothly to each parameter value. Equivalently, it is a <section of a vector bundle> $\gamma^*TM$. The <pullback connection> defines its <covariant derivative along a curve>, $D_tV$. To make the definition explicit, use <connection coefficients> with the convention
$$
\nabla_{\partial_i}\partial_j=\Gamma^k{}_{ij}\partial_k.
$$
For $V=V^k(t)\partial_k|_{\gamma(t)}$ the derivative is
$$
D_tV=\left(\frac{dV^k}{dt}
+\Gamma^k{}_{ij}(\gamma(t))\dot\gamma^i(t)V^j(t)\right)\partial_k.
$$
The <change of frame of a vector-bundle connection> ensures this expression is independent of the <coordinate chart>. It remains meaningful when $\dot\gamma=0$ and does not require extending $V$ to one ambient <vector field>.
\b[The field is parallel exactly when $\boxed{D_tV=0}$]. A <geodesic> for $\nabla$, with its specified affine parameter, is a <smooth curve> whose own tangent is parallel:
$$
\boxed{D_t\dot\gamma=0,
\qquad \ddot\gamma^k+\Gamma^k{}_{ij}(\gamma)\dot\gamma^i\dot\gamma^j=0.}
$$
This definition applies to any <affine connection>, whether or not it is a <metric connection>. A non-affine change of parameter generally gives a <pregeodesic> instead of this parametrized <geodesic>.
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