Covariant derivative of a section in a vector field direction

For a connection as in Connection on a smooth vector bundle, a smooth vector field $X\in \mathrm{\Gamma }(TM)$ and a section $s\in \mathrm{\Gamma }(E)$, define the covariant derivative in direction $X$ by ${\mathrm{\nabla }}_{X}s=(\mathrm{\nabla }s)(X),({\mathrm{\nabla }}_{X}s)(p)=(\mathrm{\nabla }s{)}_p(X(p))\mathrm{.}$ It is a smooth section: in any local coordinates and bundle frame, if the matrix of $\mathrm{\nabla }s$ has entries $b_i^a$ and $X$ has components $X^i$, its components are the finite sums ${\sum }_i{b}_i^a{X}^i$. For a single tangent vector $v\in T_{p}M$, the notation ${\mathrm{\nabla }}_{v}s$ means $(\mathrm{\nabla }s{)}_p(v)$. In particular the direction is used only at $p$, while the section can be differentiated there.

Zero direction gives zero; zero section gives zero by real linearity of $\mathrm{\nabla }$. For a zero-dimensional base the sum is empty. Empty base and rank-zero bundle also give the unique zero section. A direction is not required to be nonzero or to extend along any prescribed curve. This is evaluation of supplied data and needs no choice.