Hipersfēra

Dekarta koordinātu sistēmā sfēra ar centru (a₁, a₂, …, a_n) un rādiusu r > 0 sastāv no punktiem ar koordinātēm (x₁, x₂, …, x_n), kas apmierina vienādojumu

${\displaystyle \,\sum _{i=1}^{n}(x_{i}-a_{i})^{2}=r^{2}.}$ 

Sfēriskās koordinātu sistēmas vispārinājums ir hipersfēriskā koordinātu sistēma. Tajā sfēru apraksta vienādojumi

${\displaystyle {\begin{aligned}x_{1}=a_{1}&+r\cos \alpha _{1},\\x_{2}=a_{2}&+r\sin \alpha _{1}\,\cos \alpha _{2},\\x_{3}=a_{3}&+r\sin \alpha _{1}\,\sin \alpha _{2}\,\cos \alpha _{3},\\&{}\,\,\,\vdots \\x_{n-1}=a_{n-1}&+r\sin \alpha _{1}\,\sin \alpha _{2}\,\sin \alpha _{3}\,\cdots \,\sin \alpha _{n-2}\,\cos \alpha _{n-1},\\x_{n}\,\,\,\,\,\,=a_{n}\,\,\,\,\,&+r\sin \alpha _{1}\,\sin \alpha _{2}\,\sin \alpha _{3}\,\cdots \,\sin \alpha _{n-2}\,\sin \alpha _{n-1},\end{aligned}}}$ 

kur parametri ${\displaystyle \alpha _{i}\,}$  apmierina nevienādības

${\displaystyle 0\leq \alpha _{1},\alpha _{2},\dots ,\alpha _{n-2}\leq \pi }$  un ${\displaystyle 0\leq \alpha _{n-1}<2\pi .}$ 

Virsmas laukuma elements sfērai ar rādiusu r hipersfēriskajā koordinātu sistēmā n dimensijās ir

${\displaystyle {\begin{aligned}dS&={\begin{vmatrix}{\frac {\partial x_{1}}{\partial r}}&{\frac {\partial x_{1}}{\partial \alpha _{1}}}&\cdots &{\frac {\partial x_{1}}{\partial \alpha _{n-1}}}\\[3pt]{\frac {\partial x_{2}}{\partial r}}&{\frac {\partial x_{2}}{\partial \alpha _{1}}}&\cdots &{\frac {\partial x_{2}}{\partial \alpha _{n-1}}}\\[3pt]\vdots &\vdots &\ddots &\vdots \\[3pt]{\frac {\partial x_{n}}{\partial r}}&{\frac {\partial x_{n}}{\partial \alpha _{1}}}&\cdots &{\frac {\partial x_{n}}{\partial \alpha _{n-1}}}\end{vmatrix}}\,d\alpha _{1}\,d\alpha _{2}\,\cdots \,d\alpha _{n-1}\\&=r^{n-1}\left(\prod _{i=1}^{n-2}\sin ^{n-1-i}\alpha _{i}\,d\alpha _{i}\right)d\alpha _{n-1}\\&=r^{n-1}\sin ^{n-2}\alpha _{1}\,\sin ^{n-3}\alpha _{2}\,\cdots \,\sin \alpha _{n-2}\,d\alpha _{1}\,d\alpha _{2}\,\cdots \,d\alpha _{n-1}.\end{aligned}}}$ 

Virsmas laukumu sfērai n dimensijās var atrast ar integrāļa palīdzību:

${\displaystyle S=\underbrace {\int _{\alpha _{1}=0}^{\pi }\int _{\alpha _{2}=0}^{\pi }\cdots \int _{\alpha _{n-2}=0}^{\pi }} _{n-2}\int _{\alpha _{n-1}=0}^{2\pi }dS.}$ 

Integrējot iegūst

${\displaystyle S={\frac {2\pi ^{n/2}}{\Gamma ({\frac {n}{2}})}}\,r^{n-1}.}$ 

Daudz vienkāršāk šo formulu ir iegūt pa tiešo (bez virsmas laukuma elementa atrašanas).

Laukums un tilpums lodei n dimensijās

n	S	V
1	${\displaystyle 2\,}$	${\displaystyle 2r\,}$
2	${\displaystyle 2\pi r\,}$	${\displaystyle \pi r^{2}\,}$
3	${\displaystyle 4\pi r^{2}\,}$	${\displaystyle {\frac {4}{3}}\pi r^{3}\,}$
4	${\displaystyle 2\pi ^{2}r^{3}\,}$	${\displaystyle {\frac {1}{2}}\pi ^{2}r^{4}\,}$
5	${\displaystyle {\frac {8}{3}}\pi ^{2}r^{4}\,}$	${\displaystyle {\frac {8}{15}}\pi ^{2}r^{5}\,}$
6	${\displaystyle \pi ^{3}r^{5}\,}$	${\displaystyle {\frac {1}{6}}\pi ^{3}r^{6}\,}$
7	${\displaystyle {\frac {16}{15}}\pi ^{3}r^{6}\,}$	${\displaystyle {\frac {16}{105}}\pi ^{3}r^{7}\,}$
8	${\displaystyle {\frac {1}{3}}\pi ^{4}r^{7}\,}$	${\displaystyle {\frac {1}{24}}\pi ^{4}r^{8}\,}$
9	${\displaystyle {\frac {32}{105}}\pi ^{4}r^{8}\,}$	${\displaystyle {\frac {32}{945}}\pi ^{4}r^{9}\,}$

Tilpumu n-dimensionālai lodei ar rādiusu r atrod integrējot:

${\displaystyle V=\int _{0}^{r}S\,dr={\frac {rS}{n}}.}$ 

Izmantojot Gamma funkcijas īpašību ${\displaystyle {\tfrac {n}{2}}\Gamma ({\tfrac {n}{2}})=\Gamma ({\tfrac {n}{2}}+1)}$ , iegūst

${\displaystyle V={\frac {\pi ^{n/2}}{\Gamma ({\frac {n}{2}}+1)}}\,r^{n}.}$ 

Veseliem un pusveseliem argumentiem Gamma funkciju var izteikt attiecīgi ar faktoriāla un dubultfaktoriāla palīdzību. Tad laukuma S izteiksmi var pārrakstīt šādi:^[2][3]

${\displaystyle S={\begin{cases}{\dfrac {2\pi ^{\frac {n}{2}}}{{\bigl (}{\frac {n}{2}}-1{\bigr )}!}}\,r^{n-1}&n{\mbox{ pāra}},\\[3ex]{\dfrac {2^{\frac {n+1}{2}}\pi ^{\frac {n-1}{2}}}{(n-2)!!}}\,r^{n-1}&n{\mbox{ nepāra}},\end{cases}}}$ 

bet tilpuma V izteiksmi var pārrakstīt šādi:^[2][4]

${\displaystyle V={\begin{cases}{\dfrac {\pi ^{\frac {n}{2}}}{{\bigl (}{\frac {n}{2}}{\bigr )}!}}\,r^{n}&n{\mbox{ pāra}},\\[3ex]{\dfrac {2^{\frac {n+1}{2}}\pi ^{\frac {n-1}{2}}}{n!!}}\,r^{n}&n{\mbox{ nepāra}}.\end{cases}}}$ 

1. nogrieznis, divi punkti
2. riņķis, riņķa līnija
3. lode, sfēra

• Riņķa līnija
• Sfēra

• Eric W. Weisstein, Hypersphere, MathWorld.
• Howard Haber, The volume and surface area of an n-dimensional hypersphere, lekcijas konspekts.
• Al Lehnen, Properties of Spherical Coordinates in n Dimensions.