Hypersphere
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A hypersphere is a higher-dimensional analogue of a sphere. A hypersphere of radius R in n-dimensional Euclidean space consists of all points at distance R from a given fixed point (the centre of the hypersphere).
The "volume" it encloses is
- <math>V_n={\pi^{n/2}R^n\over\Gamma(n/2+1)}<math>
where Γ is the gamma function.
The "surface area" of this hypersphere is
- <math>S_n=\frac{dV_n}{dR}={2\pi^{n/2}R^{n-1}\over\Gamma(n/2)}<math>
The above hypersphere in n-dimensional Euclidean space is an example of an (n−1)-manifold. It is called an (n−1)-sphere and is denoted Sn−1. For example, an ordinary sphere in three dimensions is a 2-sphere.
The interior of a hypersphere, that is the set of all points whose distance from the centre is less than or equal to R, is called an hyperball.
Hyperspherical volume - some examples
For a unit sphere (<math> R = 1 <math> ) the volumes for some values of n are:
<math>V_1\,<math> = <math>2\,<math> <math>V_2\,<math> = <math>\pi\,<math> = <math>3.14159\ldots\,<math> <math>V_3\,<math> = <math>\frac{4 \pi}{3}\,<math> = <math>4.18879\ldots\,<math> <math>V_4\,<math> = <math>\frac{\pi^2}{2}\,<math> = <math>4.93480\ldots\,<math> <math>V_5\,<math> = <math>\frac{8 \pi^2}{15}\,<math> = <math>5.26379\ldots\,<math> <math>V_6\,<math> = <math>\frac{\pi^3}{6}\,<math> = <math>5.16771\ldots\,<math> <math>V_7\,<math> = <math>\frac{16 \pi^3}{105}\,<math> = <math>4.72478\ldots\,<math> <math>V_8\,<math> = <math>\frac{\pi^4}{24}\,<math> = <math>4.05871\ldots\,<math>
If the dimension, n, is not limited to integral values, the hypersphere volume is a continuous function of n with a global maximum for the unit sphere in dimension n = 5.2569464... where the "volume" is 5.277768...
Hyperspherical coordinates
We may define a coordinate system in an n-dimensional Euclidean space which is analogous to the spherical coordinate system defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate r, and n-1 angular coordinates {φ1,φ2...φn-1}. If xi are the Cartesian coordinates, then we may define
- <math>x_1=r\cos(\phi_1)\,<math>
- <math>x_2=r\sin(\phi_1)\cos(\phi_2)\,<math>
- <math>x_3=r\sin(\phi_1)\sin(\phi_2)\cos(\phi_3)\,<math>
- <math>\cdots\,<math>
- <math>x_{n-1}=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\cos(\phi_{n-1})\,<math>
- <math>x_n~~\,=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\sin(\phi_{n-1})\,<math>
The hyperspherical volume element will be found from the Jacobian of the transformation:
- <math>d^nr =
\left|\det\frac{\partial (x_i)}{\partial(r,\phi_i)}\right| dr\,d\phi_1 \, d\phi_2\ldots d\phi_{n-1}<math>
- <math>=r^{n-1}\sin^{n-2}(\phi_1)\sin^{n-3}(\phi_2)\cdots \sin(\phi_{n-2})\,
dr\,d\phi_1 \, d\phi_2\cdots d\phi_{n-1}<math>
and the above equation for the volume of the hypersphere can be recovered by integrating:
- <math>V_n=\int_{r=0}^R \int_{\phi_1=0}^\pi
\cdots \int_{\phi_{n-2}=0}^\pi\int_{\phi_{n-1}=0}^{2\pi}d^nr. \,<math>