Integral Over a Sphere with dirac delta function

[tim85ruhruniv]

Hi,

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

the problem and the solution are given below, I want to know how it has been solved.
The [tex]\delta_{0}[/tex] is the dirac delta function.

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

the following variable substitution has been made,

[tex]\[
\varsigma=\underline{\xi}\cdot\underline{z}=rcos\theta\]
[/tex]

Thanx a lot.

 

[tim85ruhruniv]

hey,

I Just got it,

I used this substitution.

[tex]
\[
\intop_{0}^{\pi}\intop_{0}^{2\pi}f(cos\varphi sin\theta,sin\varphi sin\theta,cos\theta)sin\theta d\theta d\varphi\][/tex]

the radius that i used in the variable substitution is not the same as the unit radius.