# 鬼斧神工：求n维球的体积

$V_n(r)=\int_{x_1^2+x_2^2+\dots+x_n^2\leq r^2}\mathrm{d}x_1 \mathrm{d}x_2\dots \mathrm{d}x_n$

$V_n (r)=\int_{-r}^r V_{n-1} \left(\sqrt{r^2-t^2}\right)\mathrm{d}t$

$t=r\sin\theta_1$，就有

$V_n (r)=r\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} V_{n-1} \left(r\cos\theta_1\right)\cos\theta_1 \mathrm{d}\theta_1$

$V_n (r)=r^2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} V_{n-2} \left(r\cos\theta_1\cos\theta_2\right)\cos\theta_1\cos^2\theta_2 \mathrm{d}\theta_1 \mathrm{d}\theta_2$

\begin{align*}V_n (r)=&r^{n-1}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dots\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} V_1\left(r\cos\theta_1\cos\theta_2\dots \cos\theta_{n-1}\right)\times\\ &\cos\theta_1\cos^2\theta_2\dots\cos^{n-1}\theta_{n-1} \mathrm{d}\theta_1 \mathrm{d}\theta_2\dots \mathrm{d}\theta_{n-1}\end{align*}

$V_n (r)=2r^{n}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dots\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos^2\theta_1\cos^3\theta_2\dots\cos^{n}\theta_{n-1} \mathrm{d}\theta_1 \mathrm{d}\theta_2\dots \mathrm{d}\theta_{n-1}$

\begin{align*} G(n)=\int_{-\infty}^{+\infty}\dots\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \exp\left(-x_1^2-x_2^2-\dots-x_n^2\right)\mathrm{d}x_1 \mathrm{d}x_2 \dots \mathrm{d}x_n\tag{1} \end{align*}

$\int_{-\infty}^{+\infty}\exp(-t^2)\mathrm{d}t=\sqrt{\pi}$

$(1)$只不过是这样的$n$个积分的乘积，因此

\begin{align*} G(n)=\pi^{n/2}\tag{2} \end{align*}

\begin{align*} G(n)=\int_{0}^{+\infty}\mathrm{d}r\int_{S_n(r)}\exp\left(-r^2\right)\mathrm{d}S_n\tag{3} \end{align*}

\begin{align*} G(n)=\int_{0}^{+\infty}\mathrm{d}r\exp\left(-r^2\right)S_n(r)\tag{4} \end{align*}

$V_n (r)=V_n(1)r^n$

$S_n (r)=n V_n(1)r^{n-1}$

\begin{align*}G(n)=&n V_n(1)\int_{0}^{+\infty}r^{n-1}\exp\left(-r^2\right)\mathrm{d}r\\ =&\frac{1}{2}n V_n(1)\int_{0}^{+\infty}(r^2)^{n/2-1}\exp\left(-r^2\right)\mathrm{d}(r^2)\\ =&\frac{1}{2}n V_n(1)\int_{0}^{+\infty}z^{n/2-1}\exp\left(-z\right)\mathrm{d}z\quad\left(z=r^2\right)\\ =&\frac{1}{2}n V_n(1)\Gamma\left(\frac{n}{2}\right)\tag{5}\end{align*}

$\pi^{n/2}=G(n)=\frac{1}{2}n V_n(1)\Gamma\left(\frac{n}{2}\right)$

$V_n(1)=\frac{\pi^{n/2}}{\frac{1}{2}n\Gamma\left(\dfrac{n}{2}\right)}=\frac{\pi^{n/2}}{\Gamma\left(\dfrac{n}{2}+1\right)}$

$\Large\boxed{\displaystyle V_n(r)=\frac{\pi^{n/2}}{\Gamma\left(\dfrac{n}{2}+1\right)}r^n}$

$\Large\boxed{\displaystyle S_n(r)=\frac{2\pi^{n/2}}{\Gamma\left(\dfrac{n}{2}\right)}r^{n-1}}$

$\large\boxed{\displaystyle \color{red}{\frac{\pi^{n/2}}{\Gamma\left(\dfrac{n}{2}+1\right)}=2\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dots\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos^2\theta_1\cos^3\theta_2\dots\cos^{n}\theta_{n-1} \mathrm{d}\theta_1 \mathrm{d}\theta_2\dots \mathrm{d}\theta_{n-1}}}$

posted @ 2016-05-27 09:09  Renascence_5  阅读(1849)  评论(0编辑  收藏