(a) Verify explicitly the invariance of the volume element \(d \omega\) of the phase space of a single particle under transformation from the Cartesian coordinates \(\left(x, y, z, p_{x}, p_{y}, p_{z}ight)\) to the spherical polar coordinates \(\left(r, \theta, \phi, p_{r}, p_{\theta}, p_{\phi}ight)\).
(b) The foregoing result seems to contradict the intuitive notion of "equal weights for equal solid angles," because the factor \(\sin \theta\) is invisible in the expression for \(d \omega\). Show that if we average out any physical quantity, whose dependence on \(p_{\theta}\) and \(p_{\phi}\) comes only through the kinetic energy of the particle, then as a result of integration over these variables we do indeed recover the factor \(\sin \theta\) to appear with the subelement \((d \theta d \phi)\).