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## Main Question or Discussion Point

The most common proof that I have found of the fact that Ampère's law is entailed by the Biot-Savart law essentially 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 common 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 ##\infty##-ly thank any answerer!

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 common 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 ##\infty##-ly thank any answerer!