Definition of Dirac Delta function on the surface of a unit sphere

Question

I am looking for a definition of a Dirac Delta function which is defined on the 2D unit sphere surface in 3D.

In other words, I am looking for a function which is zero everywhere on the 2D spherical surface except at one point, (ex: (1, 1, 1)), and integral of the function over entire spherical surface is 1.

I assume that this function must be very well defined and studied. I did some preliminary search, but I could not find such a definition.

Can anyone help?

Thank you.

Answer 1

Typically, the Dirac delta "function" (or more aptly, distribution) $\delta(x)$ is defined as a probability distribution supported at the origin. Thus, to define the distribution supported at any other point, simply shift the distribution to the desired point; e.g., $\delta_{x_0}(x)=\delta(x-x_0)$. In your case, whatever point $x_0\in S^2$ you desire.

Here is a decent, more functional-analytic discussion of the delta function.

Answer 2

In polar coordinates, since the spherical surface form is $R\sin \theta d\theta d\phi $, in order for the integration to be one, the dirac delta should be: $1/(R\sin \theta_0) \delta(\theta-\theta_0,\phi -\phi_0)$