Derivatives in 3D and Dirac Delta

[UVCatastrophe]

For a research project, I have to take multiple derivatives of a Yukawa potential, e.g.

$\partial_i \partial_j ( \frac{e^{-m r}}{r} )$

or another example is

$\partial_i \partial_j \partial_k \partial_\ell ( e^{-mr} )$

I know that, at least in the first example above, there will be a Dirac delta function somewhere in the answer,

$\partial_i \partial_j \{ \ldots \} \supset - 4\pi \delta^{i j} \delta^{(3)} (\mathbf{x} )$

If you do this by naively applying the chain rule and product rule to the functions, unit vectors, and their derivatives, you will miss it. The way to see that the answer contains a delta function is to apply the divergence theorem, realize there's a contradiction, and then add the 3D Dirac delta ad hoc. (Refer to Chapter 1.5 of Griffiths Electrodynamics for instance.)

This is a subtle point: easy to miss, and worrisome that I have to put some of the answer in by hand. Does anyone know if there are other functions, or combinations of functions and unit vectors, that straight up chain/product rule will give incomplete answers? Or can someone assure me that the delta function is the only case? Thanks!

 

[DrDu]

I wouldn't blame the chain and product rules. 1/r diverges at r=0 and r does not have a continuous derivative there, so you have to decide how to interpret this singularity and the derivatives. E.g., you could replace 1/r by something smooth and finite like $1/\sqrt(r^2+a^2)$ and take the limit a to 0 after you did all the derivatives.

 

[MisterX]

You can try the "test function" method. If $\partial_i \partial_j ( \frac{e^{-m r}}{r} )$ is something involving Dirac deltas and their derivatives, Examine it's action by for example

$$\int d^3x f(\mathbf{r}) \partial_i \partial_j ( \frac{e^{-m r}}{r} ) $$

And then appliMaybe that will be helpful. (Oh I see you need see this about the divergence theorem, nevermind)...