How to derive differential volume element in terms of spherical coordinates in high-dimensional Euclidean spaces?

Question

How to derive differential volume element in terms of spherical coordinates in high-dimensional Euclidean spaces (explicitly)? A derivation is here but its conclusions seems not right? The expected result, as the note states, should be $d V_n = r^{n-1} \prod_{i=1}^{n-2} (\sin \theta_i)^{n-i-1} dr \; d \theta_1 \cdots \; d \theta_{(n-1)}$. What's wrong?

Answer 1

It is actually much easier to use induction. Let $n\geq 3$. Consider $S^{n-1} \subset \mathbb R^n$. Let $s(\vec\theta)$ be spherical coordinates on $S^{n-2}\subset\mathbb R^{n-1}$. Then, $S^{n-1}$ may be parameterized by $f( \phi, \vec\theta): [0,\pi]\times\text{Domain }s(\vec\theta) \to \mathbb R^n$ defined by $f(\phi, \vec\theta) = (\sin(\phi) s(\vec\theta), \cos(\phi))$.

Then, we have that $\partial_\phi f = (\cos(\phi)s(\vec\theta), -\sin(\phi)),$ and $ \partial_{\theta_i} f = (\sin(\phi)\partial_{\theta_i} s, 0).$ Note that $\partial_{\theta_i}s \perp s,$ since $s$ parameterizes $S^{n-2}$. Hence, $\partial_\phi f \perp \partial_{\theta_i} f$.

Now, spherical coordinates on $\mathbb R^n$ are given by $F(r, \phi, \vec\theta) = r f(\phi,\vec\theta)$. Clearly the radial derivative is perpendicular to the angular derivatives since $f$ parameterizes a sphere. Also, $\|\partial_r F\| =1 $.

Remember that the Jacobian computes the volume of the parallelpiped formed by $\{\partial_r F, \partial_\phi F, \partial_{\theta_i} F\}_{1\leq i\leq n-2}$. From the orthogonality of the coordinate derivates, we get that this volume must be $$\|\partial_r F\|\times\|\partial_\phi F\|\times \text{Volume}(\text{parallelpiped } \{ \partial_{\theta_i}F\}_{1\leq i\leq n-2}),$$ $$ = r^{n-1}\|\partial_\phi f\|\times \text{Volume}(\text{parallelpiped } \{ \partial_{\theta_i}f\}_{1\leq i\leq n-2}).$$

Now, note that $\partial_{\theta_i} f = \sin(\phi) \partial_{\theta_i} s$, and so $$ \text{Volume}(\text{parallelpiped } \{ \partial_{\theta_i}f\}_{1\leq i\leq n-2}) = \left(\sin(\phi)^{n-2}\right) \text{Volume}(\{\partial_{\theta_i}s\}).$$ Also note that $\|\partial_\phi f \| = 1.$

So we have that the Jacobian for spherical coodinates on $\mathbb R^n$ is $$ = r^{n-1}\left(\sin(\phi)^{n-2}\right) \text{Volume}(\text{parallelpiped } \{ \partial_{\theta_i}s\}_{1\leq i\leq n-2}).$$

Now, note that Jacobian factor for $\mathbb R^{n-1}$ is $$ r^{n-2}\text{Volume}(\text{parallelpiped } \{ \partial_{\theta_i}s\}_{1\leq i\leq n-2}).$$ Letting $\theta_{n-1} = \phi$, we get $$ \left|\frac{\partial (x_1,...,x_n)}{\partial(r,\theta_1,...,\theta_{n-1})}\right| = r \left(\sin(\phi_{n-1})^{n-2}\right) \left|\frac{\partial (x_1,...,x_{n-1})}{\partial(r,\theta_1,...,\theta_{n-2})}\right|.$$

You continue the induction until $\left|\frac{\partial (x_1,x_2)}{\partial(r,\theta)}\right| = r$ to get $$\left|\frac{\partial (x_1,...,x_n)}{\partial(r,\theta_1,...,\theta_{n-1})}\right| = r^{n-1} \prod\limits_{i=2}^{n-1} \sin(\theta_i)^{i-1} .$$

Relabeling the $\theta_i$ gives the formula you are looking for.