Is this how one would express a sphere as a Dirac delta in Cartesian coordinates?
DiracDelta[Sqrt[R0^2-z^2-x^2],Sqrt[R0^2-z^2-y^2],Sqrt[R0^2-x^2-y^2]]
p 32 Barton et al gives the strong definition of the 3D (spherical coordinate) radial Dirac delta as:
$$\delta^3(\vec{r}) = \frac{\delta(r)}{4\pi r^2}\tag{1}$$
and, correspondingly in 2D (polar coordinate):
$$\delta^2(\vec{r}) = \frac{\delta(r)}{2\pi r}.\tag{2}$$
Since $4\pi r^2$ and $2\pi r$ measure the area and length of sphere and circle respectively, and the surface and line integrals of these two Dirac deltas are 1 (by definition) it seems natural to, in appropriate circumstances, use the radial Dirac delta in modeling density distributions, normalized to 1, over radial surfaces and radial lines respectively.
user64494's statement that $\frac{1}{r^2}$ is a "singularity" at the origin is true for both $\delta(r^2)$ and for a "sphere" of radius 0, is, if we define "singularity" for a "sphere" as a "point".
So I don't understand what people are confused about unless it is an unusual use to which I put $\delta$.
– James Bowery Mar 20 '22 at 19:57