# Product of delta distributions

I’m studying Kleinert theory and Delta functions of surfaces and curves, defined as

$$\boldsymbol{\delta}_S(x)=\int_S \delta^{(3)}(x-y) dy$$

Do you know some references about the extension of the Dirac Delta for generic submanifold? Specifically, I would need references related to the delta product.

$$\int_V\boldsymbol{\delta}_S(x)\cdot \boldsymbol{\delta}_M(x) =\int_M \ \boldsymbol{\delta}_S(x) dy$$

I found this work where there is a brief introduction

but it is not clear to me how they get to the result about the crossing and whether it is extendable to submanifolds that have not only a single common point

• Minor comment to the post (v1): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. Commented Oct 6, 2022 at 18:49
• This seems like a math question, more appropriate for MSE. Commented Oct 6, 2022 at 18:57
• Crossposted from math.stackexchange.com/q/4546742/11127 Commented Oct 7, 2022 at 13:05

The function you've defined as $$\boldsymbol\delta_S(x):=\int_S \delta^{(3)}(x-y)dy$$ is simply the indicator function. It returns $$1$$ if $$x\in S$$ and 0 if $$x\not \in S$$.
The integral $$\int_V \boldsymbol\delta_S (x) \boldsymbol\delta_M (x)$$ thus returns the volume of the intersection of $$V$$, $$S$$, and $$M$$, i.e.
$$\int_V \boldsymbol\delta_S (x) \boldsymbol\delta_M (x)=\textrm{vol}\left( V \cap S \cap M\right)$$
The formula you mentioned thus applies when $$V$$ is the entire manifold.
• Thank you for your interpretation, but as u can see in the referente that I have put, in the context of a single intersection the result is equal to $1/\sin\theta$ The angle between the tangent vectores, so it is not the volume Commented Oct 7, 2022 at 6:16
• @AndrewBe I believe it's because the document you provided defines delta-functions differently according to the dimensionality of the submanifold, so that the total integral over that submanifold gives you $1$, rather than the volume which for $p<N$-dimensional submanifolds have zero $N$-volume. Therefore the formulas I wrote in my answer, as well as the formulas you wrote in your question, apply to $3$-dimensional submanifolds living in a $3$-dimensional space. The formula showing $1/\sin (\theta)$ is for the product of delta functions lower-dimensional submanifolds (i.e. lines or surfaces). Commented Oct 7, 2022 at 19:02
• Okay, I see, now your answer is clear, but I need to understand the specific case of a product between two deltas: one of a surface and the other of a curve, in $R^3$. My idea was to start from a discrete intersection, then we have the scalar product in the point of the infinitesimal vector tangent of the curve and normal vector of the surfaces. But following the idea the integral should be equal to $1 \cdot \cos \theta$ with $\theta$ the angle between the two vectors. For this reason, I don't understand the $1/(\sin \theta)$ Commented Oct 8, 2022 at 11:02