= Solution
The <volume element> is the absolute <Jacobian determinant> times the coordinate differentials. In the <Cartesian coordinate system>, <cylindrical coordinates> $(r,\theta,z)$ and <spherical coordinates> $(\rho,\vartheta,\varphi)$, respectively,
$$
\boxed{dV=dx\,dy\,dz,\qquad dV=r\,dr\,d\theta\,dz,\qquad dV=\rho^2\sin\vartheta\,d\rho\,d\vartheta\,d\varphi.}
$$
Here $\rho$ is distance from the origin, $\vartheta$ is the polar <angle> from the positive $z$ axis, and $\varphi$ is the azimuthal <angle>. The <spherical coordinates> convention fixes which <angle> enters the <sine> factor.
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