Delta function representation from EM theory

[pellman]

Claim:

$$\nabla \cdot \frac{\hat{e}_r}{r^2}=4\pi\delta^3(\vec{x})$$

Anyone know of a proof of this? (or a reference which covers it?) We need to show that

$$\frac{1}{4\pi}\int_0^R{(\nabla \cdot \frac{\hat{e}_r}{r^2})f(r)dr=f(0)$$.



The claimed identity can be seen in the solution for the electric field of a point charge in EM theory, where

$$\vec{E}=\frac{q}{r^2}\hat{e}_r$$

is the solution to

$$\nabla \cdot \vec{E}=4\pi q\delta^3(\vec{x})$$

It is easy to show in this case that $$\nabla \cdot \vec{E}=0$$ everywhere but the origin, but I don't know how to show that the delta function relation holds at the origin.

 

[fresh_42]

This here
http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/coursenotes/modules/guide03.pdf
doesn't exactly contain your formula. but explains everything which is needed including some examples with their calculations.