Higher dimensional FTC in electrostatics : Does it has mathematical rigor?

Question

I have two volumes $V$ and $V'$ in space such that:

1. $∄$ point $P$ $\ni$ $[P \in V ∧ P\in V']$
2. $V$ is filled with electric charge $q$
3. $\rho = \dfrac{dq}{dV}$ varies continuously in $V$
4. $V'$ is filled with electric charge $q'$
5. $\rho' = \dfrac{dq'}{dV'}$ varies continuously in $V'$

Let $r$ be the distance between a point $P_1 \in V$ and a point $P_2 \in V'$

In electrostatics, we use the implication:

$\displaystyle \dfrac{d^2\vec{F}}{dq\ dq'}=k\dfrac{\hat{r}}{r^2} \implies \vec{F}=k\int_q \int_{q'}\dfrac{\hat{r}}{r^2} dq'\ dq$

Does this implication has any mathematically rigor?

NOTE: I know:

$G(x)$ is differentiable on interval $[a,b]$

$\land$ $g(x)$ is Riemann integrable function in interval $[a,b]$

$\land$ $\dfrac{d\ G(x)}{dx} = g(x)$ in interval $(a,b)$

$\implies \displaystyle G(b)-G(a) = \int^b_a g(x) dx$

But I don't see why this works for integral over a volume, integral over a surface, etc...