Chemistry - 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

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

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

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

$\cdots\,$

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

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

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