I try to understand the difference between the classical definition of covariant derivative :
$$\nabla_{i}V^{j}=\partial_{i}V^{j}+V^{k}\Gamma_{ik}^{j}\quad\quad(1)$$
(where index $i$ represents curvilinear cooordinates $x^{i}$ and $V^{j}$ the $j-\text{th}$ component of vector $V$)
and the definition of called " covariant derivative of a vector field $V$ along a vector field $Z$ " and noted :
$$\nabla_{Z}V\quad\quad(2)$$
What's the expression of (2) and how to make the link with equation (1) ?
I saw that equation (2) was used like this in the demonstration of Ricci theorem :
$$Z \langle X,Y \rangle = \langle \nabla_Z X, Y\rangle + \langle X, \nabla_Z Y\rangle$$
Sorry if this question seems to be evident but I am just starting to learn basics of geometry differential.
Thanks for your help