Taken from Gravitation: Foundations and Frontiers by T. Padmanabhan Pg $197$
Consider a bunch of particles moving along geodesics in a spacetime. Let $x_i = x_i (\tau,v)$ denote the family of geodesics, where $\tau$ represents an affine parameter along the geodesics and $v$ labels a particular geodesic. The vector $$n_i = (\partial x_i / \partial v) \delta v \equiv v_i$$ $\delta v$ denotes the deviation between two neighboring geodesics parametrized by the values $v$ and $v +dv$. From the definition of the covariant derivative and the relation $(\partial u_i / \partial v) = \partial v_i / \partial τ$ (where $u_i$ is the tangent vector to the geodesic), it follows that:
$$ v^k \nabla_k u^i = u^k \nabla_k v^i$$
I'm still confused about this passage. Can someone provide an explicit example of family of curves dependent of $v$? And secondly, how does one go from $$(\partial u_i / \partial v) = \partial v_i / \partial τ$$ to $$ v^k \nabla_k u^i = u^k \nabla_k v^i$$?