math lessons - Hypersphere

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

$x_1=r\cos(\phi_1)\,$