I'm not able to derive the volume of a parallelepiped a in spherical coordinates, In other words it is under the map $G(\rho,\theta, \varphi)=(\rho \sin\varphi \cos \theta, \rho \sin \varphi \sin \theta ,\rho \cos \varphi)$.
I've started by considering a quite simpler region, that is the region lying above a cone and inside a sphere, using the volume of revolution and after some long calculations, I've found that its volume is $$3\pi\rho^3-\frac{2}{3}\pi\rho^3\cos \varphi$$
where $\varphi$ is the angle between the $z$ axis and the outer sides of the cone. if we want to derive the full formula we should add the angle $\theta$ but i don't know how to do that.
Best Answer
To amplify the point made in the comment, the curvilinear box in the diagram has sides $dr$, $r\sin\theta\, d\phi$ (see below), and $r\, d\theta$, so its volume is approximately the product of its sides. In the limit as the differentials decrease to $0$, this approximation becomes proportionally exact—i.e., the error divided by the infinitesimal volume goes to $0$—because the spherical coordinates mapping is differentiable.
By geometry, an angle $\psi$ subtends an arc $\rho\psi$ at distance $\rho$ from the center. The "longitudinal" side subtends an angle $d\phi$ at distance $r\sin\theta$ from the center, and therefore has length $r\sin\theta\, d\phi$.