Let the vector field $\vec{F}$ be defined as the gradient of $$V=\frac{1}{x^2+y^2+z^2}.$$
Now, naively, if we wanted to calculate the flux through a surface defined as an intersection between $S_1:z=x^2+y^2$ and $S_2:z=\sqrt{R^2-x^2-y^2}$, we could either manually calculate
$$\Phi = \Phi_1 + \Phi_2 = \iint_{S_1} \left(\nabla V\right) \cdot d\vec{S_1} +\iint_{S_2} \left(\nabla V\right) \cdot d\vec{S_2},$$
with suitably oriented surfaces, or, we could use the divergence theorem,
$$\Phi = \iiint_V \operatorname{div} \vec{F} dV.$$
The issue is, there's a singularity at $x=y=z=0$, which is at the very boundary of our surface.
How does the singularity being at the very boundary affect the applicability of the two approaches? Is this problem even well-posed?