方企勤 第六章 多元函数积分学 第6.2题
📝 题目
6.2.14 用广义球坐标求 $n$ 维球的容积,即求 ${x}_{1}^{2} + {x}_{2}^{2} + \cdots + {x}_{n}^{2} \leq {R}^{2}$ 的体积. 所谓广义球坐标即为
$$ \left\{ \begin{array}{ll} {x}_{1} = r\cos {\theta }_{1}, & r \geq 0, \\ {x}_{2} = r\sin {\theta }_{1}\cos {\theta }_{2}, & 0 \leq {\theta }_{i} \leq \pi , \\ \vdots & i = 1,\cdots ,n - 2, \\ {x}_{n - 1} = r\sin {\theta }_{1}\sin {\theta }_{2}\cdots \sin {\theta }_{n - 2}\cos {\theta }_{n - 1}, & 0 \leq {\theta }_{n - 1} \leq {2\pi }. \\ {x}_{n} = r\sin {\theta }_{1}\sin {\theta }_{2}\cdots \sin {\theta }_{n - 2}\sin {\theta }_{n - 1}, & 0 \leq {\theta }_{n - 1} \leq {2\pi }. \end{array}\right. $$
💡 答案解析
6.2.14 $\frac{{\pi }^{\frac{n}{2}}}{\Gamma \left( {\frac{n}{2} + 1}\right) }{R}^{n}$ .
📋 详细解题步骤
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