In spherical coordinates, we have
$ x = r \sin \theta \cos \phi $;
$ y = r \sin \theta \sin \phi $; and
$z = r \cos \theta $; so that
$dx = \sin \theta \cos \phi\, dr + r \cos \phi \cos \theta \,d\theta – r \sin \theta \sin \phi \,d\phi$;
$dy = \sin \theta \sin \phi \,dr + r \sin \phi \cos \theta \,d\theta + r \sin \theta \cos \phi \,d\phi$; and
$dz = \cos \theta\, dr – r \sin \theta\, d\theta$
The above is obtained by applying the chain rule of partial differentiation.
But in a physics book I’m reading, the authors define a volume element $dv = dx\, dy\, dz$, which when converted to spherical coordinates, equals $r \,dr\, d\theta r \sin\theta \,d\phi$. How do the authors obtain this form?
After 3.5 year there needs to be an answer to this for searchers :D First of all there's no need for complicated calculations. You can obtain that expressions just by looking at the picture of a spherical coordinate system. The only thing you have to notice is that there are two definitions for unit vectors of spherical coordinate system. The only difference between these two definitions is that theta and phi angles are replaced by eachother. This common form of element volume you mentioned is based on the uncommon form of coordinate set in which theta is the angle between z axis and the point.like this :
https://www.tf.uni-kiel.de/matwis/amat/elmat_en/kap_3/illustr/spherical_coordinates.gif
While in common mathematics we say theta is the angle in xy plane. The rest should be easy, if there are any more problems please let me know I'd be happy to help with the detail if any need. :)