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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!