Rotational gradient vector

Just reading Rendering with Radiance book and have a question regarding the rotational gradient vector.

In p541(12.8), the grad direction is sum of vk vectors over the hemisphere, where vk said to be the base plane unit vector in the phi + 90 direction.

I assume the gradient of cosine is image where the vector v is spherical unit vector on sphere surface pointing to direction opposite z axis. I was expecting to see this vector instead of vk(base plane unit vector) in the book. Are they same?

I believe the formula in the book is correct. You can get more information from the paper Irradiance Gradients. The latest code no longer uses this formula, as it determines the rotational gradient a la Schwarzhaupt et al.