Mathematical meaning of certain integrals in physics

While studying on texts of physics I notice that differentiation under the integral sign is usually introduced without any comment on the conditions permitting to do so. In that case, I take care of thinking about what the author is assuming and the usual assumption made in physics that all the functions are of class $C^\infty$, at least piecewise on compact subsets, often is enough to guarantee the liceity of freely commutating the derivative and integral signs.

While studying the derivation of Ampère's law from the Biot-Savart law, someting has surprised me in this proof which seems to be ubiquitous on line and in cartaceous texts. In fact the magnetic field in a point $\mathbf{x}$ is $$\mathbf{B}(\mathbf{x}):=\frac{\mu_0}{4\pi}\iiint_V\mathbf{J}(\mathbf{l})\times\frac{\mathbf{x}-\mathbf{l}}{\|\mathbf{x}-\mathbf{l}\|^3}d^3l=\frac{\mu_0}{4\pi}\iiint_V\nabla_x\times\left[\frac{\mathbf{J}(\mathbf{l})}{\|\mathbf{x}-\mathbf{l}\|}\right]d^3l$$where I would prove the identity of the integrands at both members by considering the derivatives as... well, ordinary derivatives. I keep Wikipedia's notation except for $\mathbf{x}$, which is more common as a variable, and the norm sign, for which I have always seen $\|\cdot\|$ elsewhere. Then we can notice that the proof uses a differentiation under the integral sign (at $(1)$ below): since $\nabla_x\times\left[\nabla_x\times\left[\frac{\mathbf{J}(\mathbf{l})}{\|\mathbf{x}-\mathbf{l}\|}\right]\right]=\nabla_x\left[\nabla_x\cdot\left[\frac{\mathbf{J}(\mathbf{l})}{\|\mathbf{x}-\mathbf{l}\|}\right]\right]-\nabla_x^2\left[\frac{\mathbf{J}(\mathbf{l})}{\|\mathbf{x}-\mathbf{l}\|}\right]=\nabla_x\left[\mathbf{J}(\mathbf{l})\cdot\nabla_x\left[\frac{1}{\|\mathbf{x}-\mathbf{l}\|}\right]\right]$ $-\nabla^2\left[\frac{1}{\|\mathbf{x}-\mathbf{l}\|}\right]\mathbf{J}(\mathbf{l})$, where I would calculate the derivatives as ordinarily understood, again, we have that$$\nabla_x\times\mathbf{B}(\mathbf{x})=\nabla_x\times\left[\frac{\mu_0}{4\pi}\iiint_V\nabla_x\times\left[\frac{\mathbf{J}(\mathbf{l})}{\|\mathbf{x}-\mathbf{l}\|}\right]d^3l\right]$$$$=\frac{\mu_0}{4\pi}\iiint_V\nabla_x\times\left[\nabla_x\times\left[\frac{\mathbf{J}(\mathbf{l})}{\|\mathbf{x}-\mathbf{l}\|}\right]\right]d^3l\quad(1)$$$$=\frac{\mu_0}{4\pi}\iiint_V\nabla_x\left[\mathbf{J}(\mathbf{l})\cdot\nabla_x\left[\frac{1}{\|\mathbf{x}-\mathbf{l}\|}\right]\right]-\nabla_x^2\left[\frac{1}{\|\mathbf{x}-\mathbf{l}\|}\right]\mathbf{J}(\mathbf{l})\,d^3l$$and then the integral is split as licit for Riemann, and Lebesgue, integrals when both integrands are integrable, and the gradient and integral signs are commutated in the first of the two resulting integrals to get$$\frac{\mu_0}{4\pi}\nabla_x\iiint_V\mathbf{J}(\mathbf{l})\cdot\nabla_x\left[\frac{1}{\|\mathbf{x}-\mathbf{l}\|}\right]d^3l-\frac{\mu_0}{4\pi}\iiint_V\nabla_l^2\left[\frac{1}{\|\mathbf{x}-\mathbf{l}\|}\right]\mathbf{J}(\mathbf{l})\,d^3l$$where the first addend is $\mathbf{0}$ (I do not understand how it is calculated, but that is not the main focus of my question) and where the identity $\nabla_l^2\left[\frac{1}{\|\mathbf{x}-\mathbf{l}\|}\right]=-4\pi\delta(\mathbf{x}-\mathbf{l})$, where the derivatives are this time intended as derivatives of a distribution, is used to get$$-\frac{\mu_0}{4\pi}\iiint_V\nabla_l^2\left[\frac{1}{\|\mathbf{x}-\mathbf{l}\|}\right]\mathbf{J}(\mathbf{l})\,d^3l=\mu_0\mathbf{J}(\mathbf{x}).$$ Everything of my reasoning seemed to me to work by assuming $V\subset\mathbb{R}^3$ to be compact and such that $\mathbf{x}\notin V$ and intending the integral $\iiint...d^3l$ to be a Riemann (or Lebesgue, which, in that case, I think to be the same) integral, but at this last step I see that it was not what I thought.

What are, then, the integrals appearing in such calculations? They cannot be Riemann integrals, as far as I understand, because then it must be $\mathbf{x}\notin V$ and then $\iiint_V\nabla_l^2\left[\frac{1}{\|\mathbf{x}-\mathbf{l}\|}\right]\mathbf{J}(\mathbf{l})\,d^3l=\mathbf{0}$, and they cannot be Lebesgue integrals, because, even with $\mathbf{x}\in V$, then $\iiint_V\nabla_l^2\left[\frac{1}{\|\mathbf{x}-\mathbf{l}\|}\right]\mathbf{J}(\mathbf{l})\,d^3l$ $=\int_{V\setminus\{\mathbf{x}\}}\nabla_l^2\left[\frac{1}{\|\mathbf{x}-\mathbf{l}\|}\right]\mathbf{J}(\mathbf{l})\,d\mu_{\mathbf{l}}$ $=\mathbf{0}$, even if $\mathbf{J}(\mathbf{x})$ is not null.

What else if not Riemann or Lebesgue integrals? Why is the commutation of the integral and differential operators licit and what do the derivatives mean in such a context? If we intend them to represent functionals as in the context of functional analysis (which is the only one that I know of where Dirac's $\delta$ is defined), which function ($\varphi$, to use the notation used here) is the argument of the functional and what does the functional maps it to?

What are the derivatives expressed by $\nabla$? Since theorems such as Stokes' are usually applied when integrating $\nabla\times\mathbf{B}$, I would believe that they are the ordinary derivatives of elementary multivariate calculus, but then the $\delta$, which is a tool of the theory of distributions, pops up in the outline of proof, and in the theory of distributions there exist derivatives of distributions which are a very different thing, but they are taken, as far as I know, with respect to the variables written as "variables of integration" in the distribution integral notation, while we start with $\nabla_r\times \mathbf{B}$ with $r$ , while the integral appears with $d^3l$...

Or is that one of those cases, whose set I have been told not to be empty, where physics methods, at least at the didactic level, are not as rigourous as mathematics would require? I admit that I was rather inclined to think so until a user of PSE told me, without explaining how to interpretate the integrals and justify the steps, that the quoted proof is rigourous. I heartily thank any answerer.

The difference between the way that Mathematicians and Physicists handle such things is based in an underlying assumption of Physics that solutions of Physical equations exist. The Mathematician would say that, if you have a $C^2$ scalar field $F$ in a region of space, then you can reconstruct $F$ if you know $\nabla^2F$. First, for $r \ne r'$, $$ \frac{1}{|r-r'|}\nabla'^2F(r')=\frac{1}{|r-r'|}\nabla'^2F(r')-F(r')\nabla'^2\frac{1}{|r-r'|}\\ = \nabla'\cdot\left[\frac{1}{|r-r'|}\nabla'F(r')-F(r')\nabla'\frac{1}{|r-r'|}\right] $$ The way this is done is with the integration rules of Calculus: $$ \frac{1}{4\pi}\int_{V}\nabla^2 F(r')\frac{1}{|r-r'|}dV(r') \\ =\lim_{\epsilon\downarrow 0}\frac{1}{4\pi}\int_{V\setminus B_{\epsilon}(r)}\nabla^2F(r')\frac{1}{|r-r'|}dV(r')\\ = \lim_{\epsilon\downarrow 0}\left[\frac{1}{4\pi}\int_{\partial(V\setminus B_{\epsilon}(r))}\frac{1}{|r-r'|}\frac{\partial F}{\partial n}dS(r') -\frac{1}{4\pi}\int_{\partial(V\setminus B_{\epsilon})}F(r')\frac{\partial}{\partial n'}\frac{1}{|r-r'|}dS(r')\right] \\ = -F(r)+\frac{1}{4\pi}\int_{\partial V}\frac{1}{|r-r'|}\frac{\partial F}{\partial n}(r')dS(r')-\frac{1}{4\pi}\int_{\partial V}F(r')\frac{\partial}{\partial n'}\frac{1}{|r-r'|}dS(r') $$ Therefore, the scalar field can be reconstructed if $\nabla^2F$ and $\frac{\partial F}{\partial n}$ are know on some nice region $V$: \begin{align} F(r)& =\frac{1}{4\pi}\int_{\partial V}\frac{1}{|r'-r|}\frac{\partial F}{\partial n}(r')dS-\frac{1}{4\pi}\int_{\partial V}F(r')\frac{\partial}{\partial n'}\frac{1}{|r-r'|}dS(r') \\ & -\frac{1}{4\pi}\int_{V}\frac{1}{|r'-r|}\nabla^2F(r')dV(r').\;\;\;\; (*) \end{align} But the Physicist has more information. The Physicist assumes that they're dealing with a scalar field with known volume and surface density functions $\rho$ and $\sigma$. And therefore it must be that $$ \nabla^2\left[F(r)=\frac{1}{4\pi}\int_{\partial V}\frac{1}{|r'-r|}\sigma(s')dS+\frac{1}{4\pi}\int_{V}\frac{1}{|r'-r|}\rho(r')dV(r')+\cdots\right] \\ =\nabla^2\left[\frac{1}{4\pi}\int_{V}\frac{1}{|r'-r|}\rho(r')dV(r')\right]=-\rho(r) $$ So the Mathematician has a left inverse $L\nabla^2F = F$ from which the Physicist obtains a right inverse: $\nabla^2 L\nabla^2F = \nabla^2F$. This is a full right inverse because, based on Physical grounds, every reasonable density $\rho$ can be written as $\rho=-\nabla^2F$. Hence $\nabla^2 L\rho = -\rho$. Another way of saying it: If $\rho$ is a nice density function (continuous or piecewise continuous, for example) for which there exists some $F$ such that $\nabla^2F=-\rho$, then $$ \nabla^2 \left[\frac{1}{4\pi}\int_{V}\frac{\rho(x')}{|x-x'|}dV(x')\right]=-\rho(x) \;\;\;\; (\dagger) $$ Why? Because $F$ has the above integral representation (*) involving $\nabla^2F$ and the values of $F$, $\frac{\partial F}{\partial n}$ on the boundary, which forces $(\dagger)$ by the argument given above. It's not so hard to construct some such $F$, for example, if $\rho$ is smooth: extend $\rho$ to a cube and use Fourier techniques. All you need to know is there exists some $F$, and the precise boundary information does not matter in order to argue that $(\dagger)$ must hold based on the representation $(*)$ for $F$.

The Mathematician would insist that you prove the existence of such a solution, while the Physicist knows how the world works and doesn't have to ask if nature knows how to solve the equations. :) A solution exists--let's go find it. The Physicst's approach is a productive one; you just need to be aware of the assumptions.

The Helmholtz Theorem of Mathematics is this: every smooth vector field $\vec{F}$ on a nice region can be reconstructed from $\nabla\cdot\vec{F}$ and $\nabla\times\vec{F}$: $$ \begin{align} \vec{F}(x,y,z)=&-\nabla\left[\int_{V}\frac{\nabla'\cdot \vec{F}(\vec{x}')}{4\pi|\vec{x}-\vec{x}'|}dV'-\oint_{S}\frac{\vec{F}(\vec{x}')\cdot\hat{n}}{4\pi|\vec{x}-\vec{x}'|}dS'\right] \\ &+\nabla\times\left[\int_{V}\frac{\nabla'\times \vec{F}(\vec{x}')}{4\pi|\vec{x}-\vec{x}'|}dV'+\oint_{S}\frac{\vec{F}(\vec{x}')\times\hat{n}}{4\pi|\vec{x}-\vec{x}'|}dS'\right] \end{align} $$ I'll let you construct your own right inverses from the left inverse given above. The end result is that it looks like you're interchanging differentiation operators with the integrals, but you're really not. You're assuming a solution of an equation and deriving the required form from the vector identities, knowing that you have a solution. You automatically know that when you apply the differential operators to the solution, you get back what you started with. And, of course, you're using operator identities such as $\nabla\cdot\nabla\times \vec{F}=0$ and $\nabla\times \nabla f=0$ to eliminate terms from the above when you apply differential operators.

These powerful techniques generalize to Differential Geometry. But the roots of this subject lie in Heavside's vectorization of Maxwell's Equations. There were a lot of geniuses involved in this work.

I hope I have been able, thanks to Daniel Fischer's proof of this very interesting result, to find a proof of Ampère's law from the Biot-Savart law based on the interpretation of the integral representing the magnetic potential$$\frac{\mu_0}{4\pi}\iiint_Vd^3l\frac{\mathbf{J}(\mathbf{r})}{|\mathbf{r}-\mathbf{l}|}$$ as a Lebesgue integral $$\frac{\mu_0}{4\pi}\int_V\frac{\mathbf{J}(\mathbf{r})}{\|\mathbf{r}-\mathbf{l}\|}\,d\mu_{\mathbf{l}}$$where $\mu_{\mathbf{l}}$ is the usual tridimensional Lebesgue measure defined on $\mathbb{R}^3$.

Following this reasoning, I have reached an interpretation of the integrals and differennt signs of Wikipedia's (and Jackson's Classical Electrodynamics's) outline of proof as Lebesgue integrals and ordinary derivatives, except for the integral where $\nabla^2\left(\frac{1}{|\mathbf{r}-\mathbf{l}|}\right)$ appears: I suppose that $$\iiint_V d^3l\mathbf{J}(\mathbf{l})\nabla^2\left(\frac{1}{|\mathbf{r}-\mathbf{l}|}\right)$$ is to be interpretated, differently from all the other integrals in that outline of proof, as a symbolic notation for the linear operator whose components are the Laplacians $\nabla^2 T_f$ of the linear functional defined by, $f:\mathbf{l}\mapsto\|\mathbf{r}-\mathbf{l}\|^{-1}$, i.e. $$T_f:J_i\mapsto\int_V\frac{J_i(\mathbf{l})}{\|\mathbf{r}-\mathbf{l}\|}d\mu_{\mathbf{l}}.$$Such Laplacian is such that, for $J_i$ compactly supported within $V$, $$\nabla^2T_f(J_i)=\int_V\frac{\nabla_l^2J_i(\mathbf{l})}{\|\mathbf{r}-\mathbf{l}\|}d\mu_{\mathbf{l}}=-4\pi J(\mathbf{r})=\int\delta(\mathbf{x}-\mathbf{r})J_i(\mathbf{x})$$which, as never stressed enough, is not at all, in general, the same as $$\int_V\nabla_r^2\left(\frac{1}{\|\mathbf{r}-\mathbf{l}\|}\right)\mathbf{J}(\mathbf{l})d\mu_{\mathbf{l}}\equiv\mathbf{0}.$$

I admit that requiring the reader to realise read them as Lebesgue integrals except for the integral where $\nabla^2\left(\frac{1}{|\mathbf{r}-\mathbf{l}|}\right)$ appears.

That would be a tremendously confusing notation for me (even more with that performing an integration by parts), and, I suspect, for many other readers, in particular students. I am not sure that the authors using that outline of proof require such a care to distinguish different meanings of the same integral notations, used in the very same equality, from their readers.