Question: I just started in Griffith's Introduction to electrodynamics and I stumbled upon the divergence of $\frac{ \hat r}{r^2} \equiv \frac{{\bf r}}{r^3}$, now from the book, Griffiths says:

[![enter image description here](https://i.stack.imgur.com/cxaLe.jpg)](https://i.stack.imgur.com/cxaLe.jpg)

Now what is the paradox, exactly? Ignoring any physical intuition behind this (point charge at the origin) how are we supposed to believe that the source of $\vec v$ is concentrated at the origin mathematically? Or are we forced to believe that because there was a contradiction with the divergence theorem?

Also how would the situation differ if $\vec v$ was the same vector function but not for a point charge? Or is it impossible?

Answer: 
> 
> Now what is the paradox, exactly?
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> 
> 

The paradox is that the vector field $\vec{v}$ considered obviously points away from the origin and hence seems to have a non-zero divergence, however, when you actually calculate the divergence, it turns out to be zero.

[![enter image description here](https://i.stack.imgur.com/MlEimm.jpg)](https://i.stack.imgur.com/MlEimm.jpg)

> 
> How are we supposed to believe that the source of $\vec v$ is concentrated at the origin mathematically?
> 
> 
> 

Most important point to observe is that $\nabla \cdot \vec v = 0$ everywhere except at the origin. The diverging lines appearing are from the origin. Our calculations cannot account for that since $\vec v$ blows up at $r = 0$. Moreover, eq. (1.84) is not even valid for $r = 0$. In other words, $\nabla \cdot \vec v \rightarrow \infty$ at that point.

However, if you apply the divergence theorem, you will find $$\int \nabla \cdot \vec v \ \text{d}V = \oint \vec v \cdot \text{d}\vec a = 4 \pi$$
Irrespective of the radius of a sphere centred at the origin, we must obtain the surface integral as $4 \pi$. The only conclusion is that this must be contributed from the point $r = 0$.

This serves as the motivation to define the Dirac delta function: a function which vanishes everywhere except blowing up at a point and has a finite area under the curve.
The word "paradox" is not really justified, this is just caused by sloppy math, ignoring that our field isn't defined at the origin. The divergence truly **is equal to zero everywhere the function is defined** and thus the calculation is valid. Our calculation doesn't say anything about the origin. There, special care is needed. In terms of "regular functions", you can't say anything, discrete contributions have to be counted as special cases. But with generalization to distributions, you can say it's a delta function, as this gives you a way to describe discrete contributions, too.