Calculating the Jacobian for this change of variables I'm trying to find the normalizing constant of an (unnormalized) probability density function of $x\in\mathbb{R}^p$ and I think it will be helpful to change variables from $x\in\mathbb{R}^p$ to variables $y$ and $U$ as follows: 
$y = 1 + \frac{g}{a}x^Tx$ and $U \in \mathcal{SO(p)}$ is such that $x = \sqrt{\frac{a}{g}(y-1)}U e_1$ where $\mathcal{SO(p)}$ is the special orthogonal group and $e_1$ is the first standard unit vector. 
Then the inverse transformation from $y$ and $U$ to $x$ is just $\sqrt{\frac{a}{g}(y-1)}U e_1.$ 
How do I calculate the Jacobian of this inverse transformation (or its determinant)? 
 A: There are for sure more direct approaches. However, I prefer the following (similar on how one can derive the surface of the unit sphere).
Let us investigate the Gaussian integral
$$I = \int_{\mathbb{R}^p}\! dx\,e^{-x^Tx}.$$
It separated in the $p$-coordinates and thus we  obtain
$$I = \left( \int_\mathbb{R}\!dx e^{-x^2}\right)^p = \pi^{p/2}.$$
In the new coordinates $r=x^T x$ and $U\in SO(p)$ with $x = r U e_1$, we obtain
$$I = \int_0^\infty dr\int\! dU\,J(r) e^{-r^2}.$$
The factor $J$ can only depend on $r$ as the Haar measure is invariant on the group (and so is the integrand of $I$).
Note that I am refraining from calling $J$ a Jacobian determinant as for this we should have a transformation of $p$-coordinates on $p$-coordinates. This would be only the case if we would parameterize $U$. In this case, part of the Jacobian is in the Haar measure and what $J(r)$ is the remaining part depending only on $r$.
Another important property of $J(r)$ is that it is homogeneous of degree $p-1$ that can be shown by the scale transformation $x' = \lambda x$ in the original integral (and then comparing it to the transformed variables). So $J(r) = \alpha r^{p-1}$.
Now, we have $\int dU=1$ and thus, we require that
$$\alpha \int_0^\infty \!dr\,r^{p-1} e^{-r^2} = \frac{\alpha}2  \Gamma(p/2)= \pi^{p/2};$$
this sets $\alpha$.
Instead of $r$, you want $y= 1+ (g/a) r$ as new variables, which leads to another factor $a/g$ from the substitution from $r$ to $y$. In conclusion, we obtain
$$\int_{\mathbb{R}^p}\!dx \,f(x) = \frac{a S_{p-1}}g \int_1^\infty\!dy\int\!dU \, r^{p-1} f(x)$$
with $$S_n = \frac{2 \pi^{(n+1)/2}}{\Gamma((n+1)/2)}$$
the surface of the $n$-sphere.
