Curl of electrostatic field at the position of the source point charge

Question

We know that divergence of Electrostatic field is

$$ \overrightarrow{\nabla}\cdot\overrightarrow{E}=\dfrac{\rho(\overrightarrow{r})}{\epsilon_0} $$

in case of a point charge, the divergence would be the Dirac-delta function.

What, similarly could be said about the curl at the location of point charge, we know that curl is zero for an electrostatic field , but that's at the points other than the location of the point charge.

If one tries to use Stoke's theorem to say that curl is zero everywhere, but Stoke's theorem requires that curl be define everywhere, how can we presume curl to be defined everywhere.

Answer 1

Use Faraday's Law:

$$\nabla\times\mathbf{E}=\frac{\partial\mathbf{B}}{\partial t}$$

For a point charge at rest, we know that $\mathbf{B}=0$ everywhere, including at the location of the point charge. Thus $\frac{\partial\mathbf{B}}{\partial t}=\nabla\times\mathbf{E}=0$ everywhere, including at the location of the point charge.