## 球的定义

### 多维空间球的参数方程表示

1. 第$i$次分解, 得到第$m=n-(i-1)$个坐标, 坐标数为$OP_{i-1}\sin\theta_i$, 其中$\theta_i$为$\textbf{e}_m$和$\overrightarrow{OP}_{i-1}$的夹角. 其中$P_0=P$.

2. 如果$m>1$, 令$\overrightarrow{OP}_{i-2}$为$\overrightarrow{OP}_{i-1}$的另一个分量. 此即$\overrightarrow{OP}_{i-1}$和自身与$\textbf{e}_m$的投影的向量差. 此后重复步骤1.

3. 如果$m=1$, 停止进行.

[ \begin{cases} \begin{aligned} &x_n=r\sin\theta_1 \\
&x_{n-1}=r\cos\theta_1\sin\theta_2 \\
&x_{n-2}=r\cos\theta_1\cos\theta_2\sin\theta_2 \\
&\cdots\cdots \\
&x_1=r\cos\theta_1\cdots\cos\theta_{n-1} \end{aligned} \end{cases} ]

[ \begin{aligned} \theta_i\in\begin{cases}[-\frac{\pi}{2},\frac{\pi}{2}),&0\leq i< n-1 \\\ [0,2\pi), &i={n-1}\end{cases} \end{aligned} ]

## $\Gamma$函数——计算体积的核心工具

### $\Gamma$函数的第一层认识：阶乘

&=x\int_{0}^{\infty}t^{x-1}e^{-t}dt=x\Gamma(x) \end{aligned} ]

$$\Gamma(n+1)=n!\quad n=1,2,\cdots$$

### $\Gamma$函数的第二层认识: $B$函数和$\Gamma(\frac{1}{2})$

$B(x,y)= \int_{0}^{1}t^{x-1}(1-t)^{y-1}=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$

[ \frac{\Gamma(\frac{1}{2})\Gamma(\frac{1}{2})}{\Gamma(1)}=\Gamma(\frac{1}{2})^2=\pi ]

## 体积的计算

### 体积的推导

&=\int_{0}^{R}\int_{0}^{2\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cdots\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}r^{n-1}\cos^{n-2}\theta_1\cos^{n-3}\theta_2\cdots\cos\theta_{n-2}d\theta_1d\theta_2\cdots d\theta_{n-1}dr \end{aligned} ]

### 体积的计算

[ \int_{0}^{2\pi}d\theta_{n-1}=2\pi ]

&=2\int_{0}^{\frac{\pi}{2}}\cos^{2\frac{m+1}{2}-1}\theta\sin^{2\frac{1}{2}-1}\theta d\theta \\\ &=B(\frac{m+1}{2},\frac{1}{2}) \end{aligned} ]

&=\int_{0}^{R}\int_{0}^{2\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cdots\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}r^{n-1}\cos^{n-2}\theta_1\cos^{n-3}\theta_2\cdots\cos\theta_{n-2}d\theta_1d\theta_2\cdots d\theta_{n-1}dr \\
& = \frac{R^n}{n}\cdot 2\pi\cdot\Pi_{k=1}^{n}B(\frac{n+1}{2},\frac{1}{2}) \end{aligned} ]

## 一个很可爱的结论

[\sum_{n=1}^{\infty}V_{2n}(R)=e^{\pi R^2}]