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?
Best Answer
$dV=dxdydz=|\frac{\partial(x,y,x)}{\partial(r,\theta,\phi)}|drd\theta d\phi$