Higher dimensional FTC in electrostatics: Mathematically rigorous or not?

[oliverkahn]

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...

 

[jim mcnamara]

@oliverkahn - what does FTC stand for? Some of us mentors are confused.. I'm one of them.

 

[fresh_42]

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

In case the dimension is higher than one, it is called Stoke's theorem.

 

[oliverkahn]

@oliverkahn - what does FTC stand for? Some of us mentors are confused.. I'm one of them.

Fundamental Theorem of Calculus

 

[berkeman]

Fundamental Theorem of Calculus

Rats. I had "Federal Trade Commission" in the Mentor pool. Oh well...

(It's always best to define your acronyms when you use them when they may be non-obvious...)

 

[hutchphd]

And assume they are never obvious.