Questions in category: 几何 (Geometry)
几何

1. 求下列图形中绿色部分的面积.

Posted by haifeng on 2017-05-08 23:40:16 last update 2017-05-12 09:44:08 | Answers (2) | 收藏


 

 

 

 

 

 

 

 

 

 

 

假设正方形的边长是 10, 求图中绿色部分的面积.

 

 

Answer:  单个的绿色部分面积为

\[25\arccos\frac{1}{2\sqrt{2}}-100\arccos\frac{5}{4\sqrt{2}}+\frac{25}{2}\sqrt{7}\]

 


题目来源: Lei Liu (刘磊)

3. 球面的面积

Posted by haifeng on 2012-06-04 13:11:50 last update 2012-06-04 13:17:45 | Answers (0) | 收藏


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

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