Covariant derivative along a curve

A connection differentiates a vector field $V(t)$ along a curve $\gamma (t)$ by $D_{t}V={\nabla }_{\dot{\gamma }}V$. In coordinates, $(D_{t}V{)}^a={\dot{V}}^a+{\Gamma }^a{}_{bc}{\dot{\gamma }}^b{V}^c$.