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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.

Community
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JAustin
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2 Answers2

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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!

John M
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  • Growing fond of having a math library nearby :) imgur.com/D7dIEXL. Thanks for the recommendation! – JAustin Aug 01 '16 at 20:56
  • 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. – JAustin Aug 19 '16 at 15:24
  • 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? – JAustin Aug 19 '16 at 15:24
  • 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. – JAustin Aug 19 '16 at 15:45
  • 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? – John M Aug 19 '16 at 17:07
  • Correct. In this case, is $$\int_{\partial R}\frac{\mathbf{r}}{r^3}\cdot dS=\int_R\nabla\cdot\frac{\mathbf{r}}{r^3},d^3x$$

    I am wondering if, for instance, the Divergence Theorem or Green's Theorem in $\mathbb{R}^2$ will always (or for certain classes of fields with singularities) yield the correct answer. I know this example is correct via the evaluation of test functions on the Schwarz space, but is this generally true, or only for a certain class of convergent fields.

     – JAustin Aug 19 '16 at 17:30
  • The most familiar statement of Green's theorem, for instance, requires "$P$ and $Q$ to have continuous first order partial derivatives on $R$". The GST in general usually requires the field to be $C^1$. And yet many references will use them to derive things like Ampere's Law or the aforementioned delta function identity without further justification. Clearly they do give the right answer in this particular case (as proven in one of the links). But is this generally true of some particular class of fields? Is there a convergence test of some sort. – JAustin Aug 19 '16 at 17:41
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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".

Ronan
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  • 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). – Alex M. Sep 06 '16 at 11:54