Does covariant derivative include magnitude change of a vector as well as direction change of the same vector? Does covariant derivative include magnitude change of a vector as well as direction change of the vector? In some explanations I followed I have not noticed mentioning of magnitude change along with direction change of a vector.
 A: The short answer is yes, it does. A covariant derivative describes the change of a vector field under parallel transport along a given vector. It does not single out changes in direction from changes in magnitude, but rather returns a vector that describes the change of both. To extract one or the other, you can take the dot product and orthogonal projection with the given vector.
To get intuition for the covariant derivative, it helps to remember that it is a generalization of the more familiar directional derivative. To then give an example, the directional derivative of the vector field
$\vec{V}(x,y,z) = z \hat{z}$
along the $\hat{z}$ direction is always positive, even though the direction of $\vec{V}$ never changes (except at $z=0$).
A: No, I think it's just direction. A vector which is parallel transported will not have its magnitude changed, only its direction. This is mathematically encoded in the statement that the covariant derivative is "metric compatible" i.e. $\nabla_\mu g_{\nu \rho} = 0.$
