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The delta function identity

$$\nabla^2\left(\frac{1}{\lvert\mathbf{x-x'}\rvert}\right)=-4\pi\delta^{(3)}(\mathbf{x-x'})$$

is often casually derived using the divergence theorem, since the divergence of $\nabla(1/r)$ is zero when $r\ne 0$, and the surface integral of $\mathbf{r}/\lvert r\rvert^3$ over a small sphere surrounding the origin (or over any surface enclosing the origin) has a magnitude $4\pi$ by the DT. However, every book I've seen on Stokes' Theorem has required the form in question to be $C^1$, where this is not even continuous. Indeed, it's easy to imagine that the integrals of certain badly behaved forms would not converge at all, even if the surface integral was convergent.

Furthermore, Jackson, among others, go to great lengths to construct well-behaved potentials like

$$\nabla^2\left(\frac{1}{\sqrt{r^2+\eta^2}}\right)=\frac{3\eta^2}{(r^2+\eta^2)^{5/2}}$$

which resemble the form in question as $\eta\rightarrow 0$, and then use test functions and distributional analysis to formally introduce the delta function. I asked a related question here in the Physics Stack Exchange about an application of this principle.

The actual question is in two parts:

  1. Can any permutation of the generalized Stokes' Theorem be applied reliably to (a particular class of) singular vector fields (rigorously, or informally)? And if not, what is required to demonstrate its validity in a particular case?

  2. What is actually required to rigorously prove this delta function identity, especially using the aforementioned $\eta$-potential method? Is the Divergence Theorem valid, or is the test function method necessary for a rigorous result.

There is a closely related discussion elsewhere on the Math Stack Exchange, where the result is derived using both techniques and the validity of the GST is never really resolved. I am also not well enough versed in the techniques and notation the second author uses to be comfortable with it - an explanation or book reference would also be appreciated.

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  • $\begingroup$ Another related question: math.stackexchange.com/questions/1627558/… $\endgroup$
    – John M
    Commented Aug 1, 2016 at 5:56
  • $\begingroup$ Usually, the context in which your question receives a natural answer is provided by Schwartz's theory of distributions. If you haven't studied it, providing an answer would require to present a whole theory first. Oh, and you are not the first to ask about this. $\endgroup$
    – Alex M.
    Commented Aug 19, 2016 at 14:49
  • $\begingroup$ Does $\nabla$ act on the $\mathbf x'$ variables or on $\mathbf x$? $\endgroup$
    – Alex M.
    Commented Aug 20, 2016 at 11:36
  • $\begingroup$ The $\mathbf{x}$ variable. That could just as easily be $\mathbf{r}$, but I was looking at this in the context of electrostatic fields where we're integrating over a charge distribution $\rho(\mathbf{x'})$. $\endgroup$
    – JAustin
    Commented Aug 20, 2016 at 16:14
  • $\begingroup$ See THIS or more generally THIS $\endgroup$
    – Mark Viola
    Commented Mar 11 at 18:58

2 Answers 2

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If you are interested in physics applications, and want to know about the theory of distributions (and you should), then the book Analysis by Lieb and Loss is fantastic!

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  • $\begingroup$ Growing fond of having a math library nearby :) imgur.com/D7dIEXL. Thanks for the recommendation! $\endgroup$
    – JAustin
    Commented Aug 1, 2016 at 20:56
  • $\begingroup$ I appreciate your recommendation, and do not expect a full introduction to the theory of distributions. I can do that on my own. However, perhaps owing only to my ignorance of the subject, I do think it would be possible to provide some exposition on how distributional analysis would in the most general terms answer the question, and it certainly seems possible to answer the first part in concrete terms without proof. $\endgroup$
    – JAustin
    Commented Aug 19, 2016 at 15:24
  • $\begingroup$ Is the GST valid here (as proven by the theory of distributions), and can it be reliably used without consulting distributions in each particular case? Or does the validity of the GST depend in each case on the convergence of the distribution? $\endgroup$
    – JAustin
    Commented Aug 19, 2016 at 15:24
  • $\begingroup$ But I do believe that the two questions linked above provide enough of an answer to start me in the right direction. I am just curious as a practical matter if using Stokes Theorem on fields with singularities will give me the wrong answer. $\endgroup$
    – JAustin
    Commented Aug 19, 2016 at 15:45
  • $\begingroup$ Do you mean a vector field such as $F(x,y) = (-y/(x^2+y^2), x/(x^2+y^2))$, which has a singularity at zero? And you are asking if e.g. Green's theorem would apply here? $\endgroup$
    – John M
    Commented Aug 19, 2016 at 17:07
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What you can do is to regularize your singular potential using convolution. In your example by taking a sequence of mollifiers $\rho_{\epsilon}$ you can apply the Stoke theorem to derive $$\int_{B} \nabla \cdot \nabla (\frac{1}{|r|} \ast \rho_{\epsilon}) = - \int_{\partial B} \nabla \frac{1}{|r|} \ast \rho_{\epsilon}$$ and then take the limit as $\epsilon \to 0$.

The method of $\eta$ potential is a form of regularization by convolution. I am not sure about what you mean by "test function method".

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  • $\begingroup$ The "test function method" is just the method of using Schwartz distributions (since these form the topological dual of the space of test functions endowed with the Schwartz topology). $\endgroup$
    – Alex M.
    Commented Sep 6, 2016 at 11:54

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