The text I am using states without proof that the tangent vector representing the velocity field due to a rigid rotation about the x-axis is: $$ V= -\sin{\phi}\, \partial_\theta -\cot{\theta} \cos{\phi} \, \partial_\phi $$ where $\theta$ is the standard polar angle (measured from the z-axis) and $\phi$ is the standard azimuthal angle.
I tried to get to get this from what I think is the representation of $V$ in Cartesian coordinates $$ V= V^y \partial_y+ V^z\partial_z = -z \, \partial_y + y \,\partial_z $$
based on the rotation matrix for infinitesimal rotations about the x-axis. I thought I would then calculate the components of $V$ in spherical polar coordinates via $$ V^\theta = \left(\frac{\partial\theta}{\partial y} \right)V^y +\left(\frac{\partial \theta}{\partial z} \right)V^z $$ and $$ V^\phi = \left(\frac{\partial\phi}{\partial y} \right)V^y +\left(\frac{\partial \phi}{\partial z} \right)V^z $$
since that is how coordinates are supposed to transform. However, I don't get the above result.
I think that I have a fundamental misunderstanding here and need a clue as where to start.