证明.

\[\text{Vol}(S^{2n-1})=\frac{2\pi^n}{(n-1)!}\]

\[\text{Vol}(S^{2n})=\frac{2^{n+1}\pi^n}{(2n-1)!!}\]

注意.

$S^{2n-1}$ 的球坐标为

\[\begin{cases}x_1&=\cos\theta_1\\ x_2&=\sin\theta_1\cos\theta_2\\ x_3&=\sin\theta_1\sin\theta_2\cos\theta_3\\ &\vdots\\ x_{2n-1}&=\sin\theta_1\sin\theta_2\cdots\sin\theta_{2n-2}\cos\theta_{2n-1}\\ x_{2n}&=\sin\theta_1\sin\theta_2\cdots\sin\theta_{2n-2}\sin\theta_{2n-1}\end{cases}\]

$S^{2n}$ 的球坐标为

\[\begin{cases}x_1&=\cos\theta_1\\ x_2&=\sin\theta_1\cos\theta_2\\ x_3&=\sin\theta_1\sin\theta_2\cos\theta_3\\ &\vdots\\ x_{2n}&=\sin\theta_1\sin\theta_2\cdots\sin\theta_{2n-1}\cos\theta_{2n}\\ x_{2n+1}&=\sin\theta_1\sin\theta_2\cdots\sin\theta_{2n-1}\sin\theta_{2n}\end{cases}\]