How to write this Dirac delta function in spherical coordinate?

Question

I have Dirac delta function in an integral as follows $$ \delta(\mathbf r_1 -\mathbf r_2)\; \tag{1} $$ Is right to write it as follows? $$ \delta(\mathbf r_1 -\mathbf r_2)=\frac{1}{r_1^2}\delta(r_1 -r_2)\delta(\cos\theta_1 -\cos\theta_2)\delta(\phi_1 -\phi_2) \tag{2} $$ Unfortunately I couldn't find Dirac delta in spherical coordinate in books such as Arfken's one so I'm not sure about 2. Any help would be appreciated.

Answer 1

We want to determine the distribution $u$ such that for every $f\in C^\infty$ we have $$ \int_{\phi=0}^{2\pi} \int_{\theta=0}^{\pi} \int_{r=0}^\infty u(r,\theta,\phi) \, f(r,\theta,\phi) \, r^2 \sin\theta \, dr \,d\theta \, d\phi = f(r_0, \theta_0, \phi_0). $$

It's clear that we can take $$ u(r,\theta,\phi) = \frac{1}{r^2 \sin\theta} \delta(r-r_0) \delta(\theta-\theta_0) \delta(\phi-\phi_0). $$

Now the composition of $\delta$ with a function $g$ can be written as a sum over the zeros of $g$: $$ \delta(g(x)) = \sum_{g(x_k)=0} \frac{\delta(x-x_k)}{|g'(x_k)|}, $$ Taking $g(\theta)=\cos\theta-\cos\theta_0$ we get, for $\theta,\theta_0\in(0,\pi)$ that $$ \delta(\cos\theta-\cos\theta_0) = \frac{\delta(\theta-\theta_0)}{\sin\theta_0} = \frac{\delta(\theta-\theta_0)}{\sin\theta} , $$ we can also write $$ u(r,\theta,\phi) = \frac{1}{r^2} \delta(r-r_0) \delta(\cos\theta-\cos\theta_0) \delta(\phi-\phi_0). $$