= Vector field along a curve
{title2=$V\in\Gamma(\gamma^*TM)$}
= Vector fields along a curve
{synonym}
A vector field along a curve $\gamma:I\to M$ is a smooth <section of a vector bundle> with $V(t)\in T_{\gamma(t)}M$. Its values may differ at distinct times mapping to the same base point. The <covariant derivative along a curve> differentiates its time-dependent coefficients using the pulled-back <connection on a vector bundle>; it does not require extending it to one ambient vector field.
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