Vector analysis and distributions

[LagrangeEuler]

In many books it is just written that $\Delta(\frac{1}{r})=0$. However it is only the case when $r \neq 0$. In general case $\Delta(\frac{1}{r})=-4\pi \delta(\vec{r})$. What abot $\mbox{div}(\frac{\vec{r}}{r^3})$? What is that in case where we include also point $0$?

 

[anuttarasammyak]

Direct calculation gives $-2r^{-3}$ if I do it right. it diverges at the Origin.

 

[Office_Shredder]

You can use the divergence theorem to compute what delta distribution it is around the origin. When you compute the surface integral you get a constant value no matter which surface you use (you can verify this with spheres pretty easily) and that implies a delta distribution at the origin. You can make this more rigorous depending on how you define a delta distribution