# Curl of Coulomb field

For divergence of Coulomb field, we can directly apply the formula of divergence in spherical coordinate to show that $$\nabla\cdot\left(\frac{\hat{\mathbf{r}}}{r^2}\right)=0$$ However, the formula doesn't work for $r=0$. But we can consider a small volume enclosing the origin and show that the divergence is infinite at the origin, and that in fact it is a Dirac delta function.

We can similarly apply the formula of curl in spherical coordinate to show that $$\nabla\times\left(\frac{\hat{\mathbf{r}}}{r^2}\right)=0$$ at all $r\ne 0$.
• You get $4\pi \delta^3(r)$ – Peter Diehr Aug 21 '16 at 1:28