Is there some computation beforehand of the Jacobian of spherical coordinates map $J^\intercal_\eta \, J_\eta$? Spherical coordinates is the $\mathcal{C}^\infty$ map from $[0, 2\pi] \times [-\pi, \, \pi]$ to $\mathbb{R}^3$given below:
\begin{equation} \eta(u) = \rho \begin{pmatrix} \sin(u_1) \, \cos(u_0) \\ \sin(u_1) \, \sin(u_0)\\ \cos(u_1) \end{pmatrix} \end{equation}
The jacobian matrix is, therefore, equal to $\frac{\partial}{\partial(u_0, \, u_1)} \, \eta(u)$.