I'm trying to work out how to find the directional average of a velocity distribution (where the input velocity is a 3d vector). It has been quoted as below:
$$f(v) = \oint f(\textbf{v})d\Omega _v $$
How would you go about calculating this? I know how to do a contour integral but involving the solid angle element is confusing me. Would it still be true that $d\Omega=\sin\theta d\theta d\phi $? Any help for resources to use to help work this out or anything would be really helpful.
If it helps the velocity distribution is an exponential with a $\mid \textbf{v} \mid^2 $ so it only really depends on the magnitude of this vector anyway.
Thanks for any help :)