I have the following integral \begin{align} \int f \left(U(X)\right) g \left(U(X) \right) dX \end{align}
Where,
- $X,U(X) \in \mathbb{R}^{n \times d}$;
- $f,g: \mathbb{R}^{n \times d} \xrightarrow{} \mathbb{R}$.
- $f$ is a PDF and $g(X)$ is a positive function for every $X$.
The connection between $X$ and $U(X)$ is given as follow: \begin{equation} \label{Eq: relation} U\left( X \right)^{T} U\left( X \right) =X^{T} X + B \end{equation}
I would like to change the integral such that the integration variable will be $V=U(X)$. It should have the general form of \begin{align} \int f \left(U(X)\right) g \left(U(X) \right) dX &=\int f \left(V\right) g \left(V \right) \det (J\left(X,V \right)) dV \end{align} $J\left(X,V \right)$ is some sort of Jacobian matrix, but I can't figure out what should it be and how to calculate its determinant.
Any help will be appreciated.