Volume della palla n dimensionale

Volume della palla n dim §

$$Vol(B_r)=\frac{(2\pi {)}^{N/2}}{\Gamma (N/2+1)}{R}^{N}Sup(B_r)=\frac{(2\pi {)}^{N/2}}{\Gamma (N/2)}{R}^{N-1}$$

Dim §

Primo fatto:

$$V=C_N{R}^{N}S=C_N^{\prime }{R}^{N-1}$$

Dobbiamo trovare la costante $C_N$.

Derivando:

$$C_N^{\prime }=N{C}_N$$

Trucco incredibile: integrale gassiano n dim:

(2\pi)^{N/2} = \int_{-\infty}^\infty e^{-\sum_i^n x_i^2} dx = \int_0^\infty e^{-\rho^2} C'_n \rho^{n-1}d\rho$$

C_n’ = \frac{(2\pi)^{N/2}}{\Gamma(N/2)}

$$Perilvolumequindi:$$

C_n = \frac{C_n’}{N} = \frac{(2\pi)^{N/2}}{N\Gamma(N/2)} = \frac{(2\pi)^{N/2}}{\Gamma(N/2+1)}

$\square$