Integral over a sphere with the dirac delta function

[tim85ruhruniv]

Homework Statement

$$\[<br /> \underset{\left|\underline{\xi}\right|=1}{\int}\delta_{0}\left(\underline{\xi}\cdot\underline{z}\right)dS_{\xi}=\intop_{0}^{2\pi}d\varphi\intop_{-r}^{+r}\delta_{0}\left(\varsigma\right)\frac{d\varsigma}{r}=\frac{2\pi}{r}\]$$

The $$\delta_{0}$$ is the dirac delta function.the following variable substitution has been made,
$$\[<br /> \varsigma=\underline{\xi}\cdot\underline{z}=rcos\theta\]$$

Homework Equations

I am not really sure whether its over the surface of the sphere or the Volume,

the problem and the solution are given above, I want to know how it has been solved.
What is the Jacobian Determinant for the problem ?

The Attempt at a Solution

I always end up with $$2\pi$$

 

[tim85ruhruniv]

hey guys,

thanx a lot but i got it finally.

by the way... i posted this problem in another section too and i don't know how to delete it...

Thanx...