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In particular, for any fixed value of R the volume tends to a limiting value of 0 as n goes to infinity. Which value of n maximizes V n (R) depends upon the value of R; for example, the volume V n (1) is increasing for 0 ≤ n ≤ 5, achieves its maximum when n = 5, and is decreasing for n ≥ 5. [2]
For any natural number , an -sphere of radius is defined as the set of points in (+) -dimensional Euclidean space that are at distance from some fixed point , where may be any positive real number and where may be any point in (+) -dimensional space.
A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a ...
For a general discussion of the number of linear independent vector fields on a n-sphere, see the article vector fields on spheres. There is an interesting action of the circle group T on S 3 giving the 3-sphere the structure of a principal circle bundle known as the Hopf bundle. If one thinks of S 3 as a subset of C 2, the action is given by
The volume of the unit ball in Euclidean -space, and the surface area of the unit sphere, appear in many important formulas of analysis. The volume of the unit n {\displaystyle n} -ball, which we denote V n , {\displaystyle V_{n},} can be expressed by making use of the gamma function .
The volume of phase space , occupied by a system of degrees of freedom is the product of the configuration volume and the momentum space volume. Since the energy is a quadratic form of the momenta for a non-relativistic system, the radius of momentum space will be so that the volume of a hypersphere will vary as giving a phase volume of
hypersphere volume and surface area graphs: Image title: Graphs of volumes and surface areas of n-spheres of radius 1 by CMG Lee. The apparent intersection is an artifact of the differing scales. In the SVG file, hover over a point to see its decimal value. Width: 100%: Height: 100%
The 6-sphere or hypersphere in seven-dimensional Euclidean space is the six-dimensional surface equidistant from a point, e.g. the origin. It has symbol S 6 , with formal definition for the 6-sphere with radius r of