my problem looks apparently easy but I can figure out a solution. I'm going through a paper about the Boltzmann equation and I got stuck with this change of coordinates. The original formula for Q (it's the collision integral but that is not relevant at the moment) is the following:
$$ Q(f,f)(v) = \int\limits_{x\in{B_{R}}} \;\int\limits_{y\in{B_{R}}} \delta(x\cdot{y}) K(|x|,|y|)[f(v+x)f(v+y)-f(v+x+y)f(v)]dxdy\,. $$
Just to let you know, f is a probability distribution and $K$ is the so-called kernel. $ \delta $ is the Dirac delta as one might guess. $ B_{R} $ is the ball of radius $R$ centered in the origin, $d$ is the dimension of $x$ and $y$.
By a change of coordinates, namely $ x=\rho e $ and $ y=\rho' e' $, I "should" get
$$ Q(f,f)(v) = \frac{1}{4} \int\limits_{e\in{\Bbb S^{d-1}}} \; \int\limits_{e'\in{\Bbb S^{d-1}}} \int_{-R}^{R} \int_{-R}^{R} \rho^{d-2} \rho'^{d-2} \delta(e\cdot{e'}) K(\rho,\rho')[f(v+\rho'e')f(v+\rho e)-f(v+\rho e+\rho'e')f(v)]d\rho d\rho'dede'\,. $$
Here $ \Bbb S^{d-1} $ denotes the sphere of unitary radius centered in the origin of dimension $d-1$.
The integral makes perfectly sense and I tried to compute it directly in one case and the two expressions yield the very same result. By applying the usual change of coordinates for an $n$-sphere (same formula as the one given by Wikipedia on the $n$-sphere article) and by substituting the angles by the surface element after having computed the determinant of the Jacobian, I get the same result except for the exponent of $ \rho $. I get $ \rho^{d-1} \rho'^{d-1} $ instead of $ \rho^{d-2} \rho'^{d-2} $.
Then I tried to solve the problem by considering a single vector made up by the two column vectors $x$ and $y$ (the dimension of the problem is doubled) to see whether I could get the right exponent but I couldn't. Could you please suggest a strategy to get that formula (or a reference)? Thank you anyway.
Notice than in the second integral you have $\delta(e\cdot e^\prime)$ and not $\delta(\rho e\cdot \rho^\prime e^\prime)$. But in fact $\delta(\rho e\cdot \rho^\prime e^\prime)=\frac{1}{\rho\rho^\prime}\delta( e\cdot e^\prime)$ thus explaining the powers of $\rho$ you are missing.