 Spherical volume element

With polar angle $\theta$ measured from positive $z$ and azimuth $\varphi$, the Cartesian formulas $x=r\sin \theta \cos \varphi$, $y=r\sin \theta \sin \varphi$, $z=r\cos \theta$ have Jacobian determinant $r^2\sin \theta$. For $r\ge 0$ and $0\le \theta \le \pi$ this gives the unsigned volume element shown. The inverse determinant is $1/(r^2\sin \theta )$ off the origin and polar axis.