# Divergence in spherical polar coordinates

No, it's a distribution, the socalled Dirac $\delta$ distribution. It's a very useful concepts. It's even so useful that the mathematicians built a whole new subfield of mathematics to define this concept rigorously, functional analysis!
 HW Helper P: 6,188 I suspect you're referring to Maxwell's law that says: $$\textrm{div }\vec E = \frac {\rho} {\epsilon_0}$$ The divergence of a spherically symmetric charge is zero everywhere except at the place where the charge actually is. A point charge is a special case, since the charge density would be infinite (actually a Dirac delta-function). In practice the charge would take in a certain volume. The divergence of the electric field is not zero at the place where the charge density is not zero.
 P: 162 So can I go one more step and state that, at the points in space where the charge 'is' the electric field can no longer be defined by the function $\frac{1}{r^2}$$\hat{r}$ ?
 HW Helper P: 6,188 Correct. In the case of a solid sphere with constant charge density within, the electric field is proportional to $r \hat {\boldsymbol r}$ (inside the sphere).