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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.
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>\ldots\,<math>
- <math>x_{n-1}=r\sin(\phi_1)\ldots\sin(\phi_{n-2})\cos(\phi_{n-1})\,<math>
- <math>x_n~~\,=r\sin(\phi_1)\ldots\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)\ldots \sin(\phi_{n-2})\,
dr\,d\phi_1 d\phi_2\ldots 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
\ldots \int_{\phi_{n-2}=0}^\pi\int_{\phi_{n-1}=0}^{2\pi}d^nr<math>
See also
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