Consider the following identity: $$\delta(\vec r_1-\vec r_2)=\frac{1}{r_1^2}\delta (r_1-r_2)\delta (\cos\theta_1-\cos\theta_2)\delta (\phi_1-\phi_2) $$
I was trying to understand this geometrically.
It seems that the left-hand side implies that when the line joining the vectors becomes zero, a singularity occurs. This would mean that when vector $r_1$ is exactly on top of $r_2$ (happens when $r's$ and the angles are equal simultaneously) then the left side blows up.
The problem then is the right-hand side blows up even when $r_1=r_2$ and the angles do not necessarily have to be equal.
How is this happening?
Wouldn't $\vec r_1-\vec r_2=0$ be when $r_1-r_2=0$ $\ and$ $\theta_1-\theta_2=0$ $\ and $ $\phi_1-\phi_2=0$ ?