3D polar coordinates. Define \(\Phi:[0, \infty) \times[0,2 \pi) \times[-\pi / 2, \pi / 2) ightarrow \mathbb{R}^{3}\) by

\[\Phi(r, \theta, \omega):=(r \cos \theta \cos \omega, r \sin \theta \cos \omega, r \sin \omega)\]

Show that \(|\operatorname{det} D \Phi(r, \theta, \omega)|=r^{2} \cos \omega\) and find the integral formula for the coordinate change from Cartesian to polar coordinates \((x, y, z) ightsquigarrow(r, \theta, \omega)\).