\displaystyle\frac{\partial(x,y,z)}{\partial(r,\varphi,\psi)}=
\left[\begin{array}{ccc}
\cos\psi\cos\varphi & -r\cos\psi\sin\varphi & -r\sin\psi\cos\varphi\\
\cos\psi\sin\varphi & r\cos\psi\cos\varphi & -r\sin\psi\sin\varphi\\
\sin\psi & 0 & r\cos\psi
\end{array}\right]