How to do change of variables of a j.p.d.f with N pdf(s)?

Question

Given that I have a joint probability distribution(jpdf) of: $$P(x_1,...,x_N) = C_N \prod_{j=1}^{N}(1-x_j)^a(1+x_j)^b \prod_{1\leq j <k \leq N} |x_k - x_j|^2$$ where $$\prod_{1\leq j <k \leq N} |x_k - x_j|^2$$ is the determinant of the Vandermonde matrix. Given that I would like to make the change of variable $$x_j = \cos(\theta_j)$$ and equally for k. I have worked out $$\frac{dx}{d\theta} = - \sin(\theta)$$ because well no sh*t, since I was trying to do the 1-d change of variable via $$f(\theta) = f(x)|\frac{dx}{d\theta}|$$, but now when it's n-dimensional I don't have any clue to start with computing the Jacobian. That's where I'm stuck aha.

Answer 1

Hint: Since your change of variable for each $x_j$ involves only $\theta_j$ (same subscript $j$ only), the Jacobian matrix is diagonal. (What are the diagonal entries?)