A version of Ampère's law

The most common proof that I have found of the fact that Ampère's law is entailed by the Biot-Savart law uses the fact that, if $\boldsymbol{J}:\mathbb{R}^3\to\mathbb{R}^3$, $\boldsymbol{J}\in C_c^2(\mathbb{R}^3)$, is a compactly supported twice continuously differentiable field such that $\nabla\cdot\boldsymbol{J}\equiv 0$ and $\Sigma$ is a smooth surface satisfying the assumptions of Stokes' theorem, then $$\oint_{\partial^+ \Sigma}\left(\frac{\mu_0}{4\pi}\int_{\mathbb{R}^3}\frac{\boldsymbol{J}(\boldsymbol{x})\times(\boldsymbol{r}-\boldsymbol{x})}{\|\boldsymbol{r}-\boldsymbol{x}\|^3}d\mu_{\boldsymbol{x}}\right)\cdot d\boldsymbol{r}=\mu_0\int_\Sigma \boldsymbol{J}\cdot\boldsymbol{N}_e \,d\sigma\quad(1)$$where $\mu_0$ is any constant (the magnetic permeability in the physical interpretation), $\boldsymbol{N}_e$ is the surface's external normal unit vector and $\mu_{\boldsymbol{x}}$ is Lebesgue $3$-dimensional measure.

Nevertheless, common exercises and applications of Ampère's law found in books of physics use current densities $\boldsymbol{J}\notin C_c^2(\mathbb{R}^3)$, one example being $\boldsymbol{J}$ constant on an infinite cylinder and constantly $\mathbf{0}$ outside the infinite cylinder. Do mathematically rigourous formulations of Ampère's law $(1)$ exist under more relaxed assumptions on $\boldsymbol{J}$, like the quoted case of $\boldsymbol{J}$ constant on a (bounded or unbounded) region and null outside of it, and, if they do, how can they be proved? I have thought about approximating such a $\boldsymbol{J}$ with $\boldsymbol{J}_n\in C_c^2(\mathbb{R}^3)$, but it is not easy to see that the required sequence really exists. I heartily thank any answerer!

• You may want to considering this on physics.stackexchange.com as well. Commented May 6, 2016 at 13:30
• @Math1000 Thank you for the advice, but my experience there tells me that questions asking for the (rigourous, by definition) mathematical proof of a law are not welcome... Commented May 6, 2016 at 13:33
• Sounds like a terrible place then :) Commented May 6, 2016 at 13:36
• Mathematicians do not know some physics assumptions. Better ask physics questions there. This question by you, physics.stackexchange.com/questions/227477/… , had an excellent answer (though not complete) by @FaDA for the case that continuous current is bounded to a closed loop. For those closed current bounded to a surfaces or unbounded, you only need to evoke the linear superposition principle. Commented Nov 7, 2019 at 17:04

Consider a wire of length $2R$ with a semicircular return loop of radius $R$. You can prove rigorously that as $R \to \infty$ the field at a point near the middle converges as $C + O(1/R)$ to some $C$ (the contribution of distant parts of the wire become negligible). This limit can be taken to be the mathematical definition of what we mean by an "infinite wire". This definition is fine for any physics experiment.