Chapter 8.11, Problem 22P

You may find the spherical coordinate $\delta$ function written as $\delta \left(r-r_0\right)=\delta \left(r-r_0\right)\delta \left(\cos \theta -\cos {\theta }_0\right)\delta \left(\varphi -{\varphi }_0\right)/r^2$.

Show that this equation is equivalent to $(11.22).$ Hints: You want to show that $\delta \left(\cos \theta -\cos {\theta }_0\right)=\delta \left(\theta -{\theta }_0\right)/\sin \theta .$ See (11.19e). Also note that it doesn’t really matter whether we write $r^2\sin \theta$ or $r_0^2\sin {\theta }_0$ in the denominator of (11.22) since the $\delta$ functions are zero unless $r=r_0$ and $\theta ={\theta }_0$.