= Covariant derivative along a curve
{title2=$D_tV$}
A <connection on a vector bundle> pulls back along a smooth curve $\gamma$ to differentiate its time-dependent sections. For the <tangent bundle>, a section $V(t)$ of the <pullback tangent bundle> has
$$
(D_tV)^i=\dot V^i+\Gamma^i{}_{jk}(\gamma(t))\dot\gamma^j V^k.
$$
The <Leibniz rule> under a change of frame proves independence of the coordinates. The derivative includes the time dependence of the coefficients, so it is meaningful even at a zero tangent velocity. A <vector field along a map> need not extend to one ambient field if the curve self-intersects. Parallel fields satisfy $D_tV=0$, a <linear ordinary differential equation> whose unique solutions define <parallel transport>.
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