 Solution

Use the right-handed orthonormal basis $({\mathbf{e}}_r,{\mathbf{e}}_{\theta },{\mathbf{e}}_{\varphi })$. The cross product with ${\mathbf{e}}_r$ rotates tangential components by a right angle; taking its cross product again in the opposite order projects a gradient onto the tangent plane. The radial curl of angular gradient fields then depends only on the rotated-gradient part. All coordinate calculations are made away from the polar coordinate singularities and extend wherever the original fields are smooth.