hypersphere

Other Definitions
hypersphere (dict)

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

V_n={\pi^{n/2}R^n\over\Gamma(n/2+1)}

where Γ is the gamma function. The "surface area" of this hypersphere is

S_n=\frac{dV_n}{dR}={2\pi^{n/2}R^{n-1}\over\Gamma(n/2)}

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 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...φ_n-1}. If x_i are the Cartesian coordinates, then we may define

x_1=r\cos(\phi_1)\,

x_2=r\sin(\phi_1)\cos(\phi_2)\,

x_3=r\sin(\phi_1)\sin(\phi_2)\cos(\phi_3)\,

\cdots\,

x_{n-1}=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\cos(\phi_{n-1})\,

x_n~~\,=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\sin(\phi_{n-1})\,

The hyperspherical volume element will be found from the Jacobian of the transformation:

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}

=r^{n-1}\sin^{n-2}(\phi_1)\sin^{n-3}(\phi_2)\ldots \sin(\phi_{n-2})\,

dr\,d\phi_1 d\phi_2\ldots d\phi_{n-1} and the above equation for the volume of the hypersphere can be recovered by integrating:

V_n=\int_{r=0}^R \int_{\phi_1=0}^\pi

\ldots \int_{\phi_{n-2}=0}^\pi\int_{\phi_{n-1}=0}^{2\pi}d^nr

Hypersphere

Hyperspherical coordinates

See also