Given a tangent vector field $X(x,y,z) = y\frac{\partial}{\partial x} -x\frac{\partial}{\partial y}$ of the sphere $S^2 \subset \mathbb{R}^3$.
Compute the Levi-Civita covariant derivative $\nabla_{v_p}X$ of any tangent vector $v_p$.
Secondly, show that this is a Killing vector field for the sphere.
I am having trouble with the first part, computing the covariant derivative.
Is the easiest way to compute it to use the ambient covariant derivative?