(This exercise requires knowledge of three-dimensional determinants.) Let (X, Y, Z) be independent standard normal random variables. Let (Φ, Θ, R) be the corresponding spherical coordinates. The correspondence between rectangular and spherical coordinates is given by
for 0 ≤ φ ≤ π, 0 ≤ θ ≤ 2π, and r > 0. Find the joint density of (Φ, Θ, R) and the marginal density of R.