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*outline of proof*, for more than a month with no result. I have also been looking for a proof not using Dirac's ##\delta##, which complicates things with respect to the use of Riemann or Lebesgue integrals because I am less familiar with its use and the reasons it "pops out" in some proofs, but I have not succeeded until now.

Therefore I would like to ask here about the steps that I do not understand of that

*outline of proof*, which I suppose by default to be correct, also because it also appears in J.D. Jackson's

*Classical Electrodynamics*, which I know to be considered an outstanding text, and I am posting the question in this section because it focusses on the mathematical methods used to produce that derivation of Ampère's law.

My first doubt starts at the very first commutation between the integral and curl signs. I know that $$\mathbf{J}(\mathbf{l})\times\frac{\mathbf{r}-\mathbf{l}}{|\mathbf{r}-\mathbf{l}|^3}=\nabla_r\times\left(\frac{\mathbf{J}(\mathbf{l})}{|\mathbf{r}-\mathbf{l}|}\right)$$ at least provided that we read the components of ##\nabla_r## as ordinary derivatives in the usual sense of elementary multivariable calculus. The

*outline of proof*says that $$\mathbf{B}(\mathbf{r}):=\iiint_V\,d^3l\,\mathbf{J}(\mathbf{l})\times\frac{\mathbf{r}-\mathbf{l}}{|\mathbf{r}-\mathbf{l}|^3}=\nabla_r\times\iiint_V\,d^3l\,\frac{\mathbf{J}(\mathbf{l})}{|\mathbf{r}-\mathbf{l}|}.$$What mathematical result justifies that commutation between integral and curl?

In order to take the problem on, it is obviously needed to understand

**what the integral**and

**the derivatives**in ##\nabla##

**are**, but I am not sure about what they are. Since theorems such as Stokes' are usually applied when integrating ##\nabla\times\mathbf{B}##, I would tempted to believe that the components of ##\nabla_r## are ordinary derivatives of elementary multivariable/vector calculus, but Dirac's ##\delta##, which is a tool of the theory of distributions, pops out at a certain point in the

*outline of proof*, and in the theory of distributions there exist

*derivatives of distributions*which are a very different thing, although they are "taken", as far as I know, "with respect to the variables" written as "variables of integration" in the distribution integral notation, while, here, we start with ##\nabla_r\times \mathbf{B}## with ##r##, while the integral is ##\iiint_V d^3l## with ##l##. As to the integral sign, I would tend to interpretate it as the Lebesgue integral $$\int_V\mathbf{J}(\mathbf{l})\times\frac{\mathbf{r}-\mathbf{l}}{|\mathbf{r}-\mathbf{l}|^3}\,d\mu_{\mathbf{l}}$$ or as the limit of a Riemann integral $$\lim_{\varepsilon\to 0}\iiint_{V\setminus B(\mathbf{r},\varepsilon)}\mathbf{J}(\mathbf{l})\times\frac{\mathbf{r}-\mathbf{l}}{|\mathbf{r}-\mathbf{l}|^3}\,dl_1dl_2dl_3$$but, then, if the commutations between integral and differential operators used in Wikipedia's

*outline of proof*were licit for this interpretation, we would then conclude that ##\iiint_V\,d^3l\, \mathbf{J}(\mathbf{l})\nabla_l^2\left(\frac{1}{|\mathbf{r}-\mathbf{l}|}\right)## ##=\int_V \mathbf{J}(\mathbf{l})\nabla_l^2\left(\frac{1}{|\mathbf{r}-\mathbf{l}|}\right)\,d\mu_{\mathbf{l}}## ##=\int_{V\setminus{\{\mathbf{r}}\}} \mathbf{J}(\mathbf{l})\nabla_l^2\left(\frac{1}{|\mathbf{r}-\mathbf{l}|}\right)\,d\mu_{\mathbf{l}}=\mathbf{0}## even where ##\mathbf{J}(\mathbf{r})\ne\mathbf{0}##, which cannot be the case.

I ##\infty##-ly thank you for any answer!!!