TITLE: Why does the volume element in spherical polar coordinates contain a sine of the zenith angle?
QUESTION [9 upvotes]: The volume element in Cartesian coordinates is $dx dy dz$, the volume of a rectangular prism with side lengths being the length elements along the three rectangular axes.  In spherical polar coordinates, however, the infinitesimal volume element is $r^2 \sin\phi dr d\theta d\phi$.
Why is the $\phi$ component of the volume differential not just $rd\phi$ just like the $\theta$ component is $rd\theta$?  I know it has something to do with the projection of the radial position vector on the xy-plane (which has magnitude $r\sin\phi$), but since the $\phi$ component is an arc length swept out by $d\phi$, I can't see why the volume element is not just $r^2 dr d\theta d\phi$.

REPLY [14 votes]: It's because all the longitudes of a sphere converge toward the north (and south) pole.

The length $L$ in the picture falls as $\sin(\theta)$ when you move up toward the north pole, i.e. when you decrease $\theta$ from $\pi /2$ to $0$ , you can prove this geometrically by finding the relationship between $L=r\mathrm{d}\phi \times\sin(\frac{\pi}{2})$ and the new length of $L\,$ for a given $\theta <\pi /2$:

The angle $\mathrm{d}\phi$ remains constant as we decrease $\theta$.
The volume element is simply $\,r\sin(\theta)\mathrm{d}\phi\times r\mathrm{d}\theta \times\mathrm{d}r$ .