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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.

celtschk
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david
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    (Not a big deal, but $(1,1,1)$ isn't on the unit sphere) – charlestoncrabb Jan 08 '16 at 18:28
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    What's your definition of the Dirac delta in the usual setting? (I ask because none of these Dirac deltas are actual functions, so we should like to know what kind of analogous answer you are looking for.) – Unit Jan 08 '16 at 18:32

2 Answers2

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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: $1/(R\sin \theta_0) \delta(\theta-\theta_0,\phi -\phi_0)$

AnandJ
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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.

Community
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charlestoncrabb
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