## Delta function representation from EM theory

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.
 PhysOrg.com science news on PhysOrg.com >> New language discovery reveals linguistic insights>> US official: Solar plane to help ground energy use (Update)>> Four microphones, computer algorithm enough to produce 3-D model of simple, convex room

 Similar Threads for: Delta function representation from EM theory Thread Forum Replies Calculus 3 Calculus & Beyond Homework 1 Quantum Physics 8 Calculus 1 Beyond the Standard Model 4