Dirac delta function in spherical cordinates

Question

I am studying Dirac Delta Function in Spherical Polar Cordinates. I found this expression

$$\delta\ ^3 (\vec r -\vec r_{o}) = \frac{ \delta\ (r -r_{o})\delta\ (θ -θ_{o})\delta\ (φ -φ_{o} )}{r^2 \sin θ }$$

And then they have reduced it to

$$\delta\ ^3 (\vec r -\vec r_{o}) = \frac{ \delta\ (r -r_{o})\delta\ (θ -θ_{o}) }{2πr^2 \sin θ }$$

How do we get this reduced expression from the above expression? Please explain.

Answer 1

It is not true that two expressions are equal. Note that

$$\begin{align} \int_0^{2\pi}f(r,\theta,\phi)\delta(\phi-\phi_0)\,d\phi&=f(r,\theta,\phi_0)\\\\ \int_0^{2\pi}\frac1{2\pi}f(r,\theta,\phi)\,d\phi&\ne f(r,\theta,\phi_0) \end{align}$$

unless $f(r,\theta,\phi)$ is independent of $\phi$.