Prove that hyperspherical coordinates are a diffeomorphism, derive Jacobian

Question

The explicit form for the transformation into hyperspherical coordinates is

$$x_1 = r\sin\theta_1 \sin\theta_2 \dotsb \sin \theta_{n-1} \\ x_2 = r\sin\theta_1 \sin\theta_2 \dotsb \cos \theta_{n-1} \\ x_3 = r\sin\theta_1 \dotsb \cos \theta_{n-2}\\ \vdots \\ x_{n} = r \cos\theta_1$$

for $0 \leq \theta_i \leq \pi \;\;(1\leq i \leq n-2)$ and $0\leq \theta_{n-1} \leq 2\pi$. It has Jacobian $r^{n-1} \sin^{n-2}\theta_1 \sin^{n-3}\theta_2 \dotsb \sin{\theta_{n-2}}.$

I wonder if someone could provide me with a reference for an intuitive explanation as to why this indeed is a diffeomorphism from $\mathbb{R}^n\setminus\{0\} \to \mathbb{R}^n \setminus\{0\}$, and why this is the Jacobian. Or perhaps someone could indicate the idea of a proof. Thanks as always

Answer 1

This Jacobian is calculated on the first pages of the book "Linear equations of mathematical physics" written by S.G.Mikhlin.