求 $S^n(r)$ 的体积形式
在 $\mathbb{R}^{n+1}$ 中, 考虑球面 $S^n(r)$, 它上面的点 $(x_1,x_2,\ldots,x_{n+1})$ 满足方程
\[
x_1^2+x_2^2+\cdots+x_{n+1}^2=r^2.
\]
取球面坐标:
\[
\begin{cases}
x_1&=r\cos\theta_1,\\
x_2&=r\sin\theta_1\cos\theta_2,\\
&\vdots\\
x_i&=r\sin\theta_1\sin\theta_2\cdots\sin\theta_{i-1}\cos\theta_i,\\
x_{i+1}&=r\sin\theta_1\sin\theta_2\cdots\sin\theta_{i-1}\sin\theta_{i}\cos\theta_{i+1},\\
x_{i+2}&=r\sin\theta_1\sin\theta_2\cdots\sin\theta_{i}\sin\theta_{i+1}\cos\theta_{i+2},\\
&\vdots\\
x_n&=r\sin\theta_1\sin\theta_2\cdots\sin\theta_{n-1}\cos\theta_n,\\
x_{n+1}&=r\sin\theta_1\sin\theta_2\cdots\sin\theta_{n-1}\sin\theta_n.\\
\end{cases}
\]
证明 $S^n(r)$ 上的体积形式为
\[
r^n\sin^{n-1}\theta_1\sin^{n-2}\theta_2\cdots\sin^2\theta_{n-2}\sin\theta_{n-1}d\theta_1\theta_2\cdots\theta_n.
\]