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In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, a unit sphere is simply a sphere of
radius In classical geometry, a radius of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the Latin ''radius'', meaning ray but al ...
one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ball is the closed set of points of distance less than or equal to 1 from a fixed central point. Usually the center is at the origin of the space, so one speaks of "the" unit ball or "the" unit sphere. Special cases are the
unit circle
unit circle
and the unit disk. The importance of the unit sphere is that any sphere can be transformed to a unit sphere by a combination of
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
and scaling. In this way the properties of spheres in general can be reduced to the study of the unit sphere.


Unit spheres and balls in Euclidean space

In
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimensio ...
of ''n'' dimensions, the -dimensional unit sphere is the set of all points (x_1, \ldots, x_n) which satisfy the equation : x_1^2 + x_2^2 + \cdots + x_n ^2 = 1. The ''n''-dimensional open unit ball is the set of all points satisfying the inequality : x_1^2 + x_2^2 + \cdots + x_n ^2 < 1, and the ''n''-dimensional closed unit ball is the set of all points satisfying the inequality : x_1^2 + x_2^2 + \cdots + x_n ^2 \le 1.


General area and volume formulas

The classical equation of a unit sphere is that of the ellipsoid with a radius of 1 and no alterations to the ''x''-, ''y''-, or ''z''- axes: :f(x,y,z) = x^2 + y^2 + z^2 = 1 The volume of the unit ball in ''n''-dimensional Euclidean space, and the surface area of the unit sphere, appear in many important formulas of
analysis Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics Mathematics (from Ancient Greek, Greek: ) includ ...
. The volume of the unit ball in ''n'' dimensions, which we denote ''V''''n'', can be expressed by making use of the
gamma function In mathematics, the gamma function (represented by \Gamma or , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers ...
. It is :V_n = \frac = \begin / & \mathrmn \ge 0\mathrm \\ ~\\ / & \mathrmn \ge 0\mathrm \end where ''n''!! is the double factorial. The hypervolume of the (''n''−1)-dimensional unit sphere (''i.e.'', the "area" of the boundary of the ''n''-dimensional unit ball), which we denote ''A''''n'', can be expressed as :A_n = n V_n = \frac = \frac\,, where the last equality holds only for . The surface areas and the volumes for some values of n are as follows: where the decimal expanded values for ''n'' ≥ 2 are rounded to the displayed precision.


Recursion

The ''A''''n'' values satisfy the recursion: :A_0 = 0 :A_1 = 2 :A_2 = 2\pi :A_n = \frac A_ for n > 2. The ''V''''n'' values satisfy the recursion: :V_0 = 1 :V_1 = 2 :V_n = \frac V_ for n > 1.


Fractional dimensions

The formulae for ''A''''n'' and ''V''''n'' can be computed for any real number ''n'' ≥ 0, and there are circumstances under which it is appropriate to seek the sphere area or ball volume when ''n'' is not a non-negative integer. File:Ball volume in n dimensions.svg, none, 200px, This shows the volume of a ball in ''x'' dimensions as a continuous function of ''x''.


Other radii

The surface area of an (''n''–1)-dimensional sphere with radius ''r'' is ''A''''n'' ''r''''n''−1 and the volume of an ''n''-dimensional ball with radius ''r'' is ''V''''n'' ''r''''n''. For instance, the area is for the surface of the three-dimensional ball of radius ''r''. The volume is for the three-dimensional ball of radius ''r''.


Unit balls in normed vector spaces

More precisely, the open unit ball in a normed vector space V, with the Norm (mathematics), norm \, \cdot\, , is : \ It is the interior (topology), interior of the closed unit ball of (''V'',, , ·, , ): : \ The latter is the disjoint union of the former and their common border, the unit sphere of (''V'',, , ·, , ): : \ The 'shape' of the ''unit ball'' is entirely dependent on the chosen norm; it may well have 'corners', and for example may look like [−1,1]''n'', in the case of the max-norm in ''R''''n''. One obtains a naturally ''round ball'' as the unit ball pertaining to the usual Hilbert space norm, based in the finite-dimensional case on the Euclidean distance; its boundary is what is usually meant by the ''unit sphere''. Let x=(x_1,...x_n)\in \R^n. Define the usual \ell_p-norm for ''p'' ≥ 1 as: :\, x\, _p = (\sum_^n , x_k, ^p)^ Then \, x\, _2 is the usual Hilbert space norm. \, x\, _1 is called the Hamming norm, or \ell_1-norm. The condition ''p'' ≥ 1 is necessary in the definition of the \ell_p norm, as the unit ball in any normed space must be convex set, convex as a consequence of the triangle inequality. Let \, x\, _\infty denote the max-norm or \ell_\infty-norm of x. Note that for the circumferences C_p of the two-dimensional unit balls (n=2), we have: :C_ = 4 \sqrt is the minimum value. :C_ = 2 \pi \,. :C_ = 8 is the maximum value.


Generalizations


Metric spaces

All three of the above definitions can be straightforwardly generalized to a metric space, with respect to a chosen origin. However, topological considerations (interior, closure, border) need not apply in the same way (e.g., in ultrametric spaces, all of the three are simultaneously open and closed sets), and the unit sphere may even be empty in some metric spaces.


Quadratic forms

If ''V'' is a linear space with a real quadratic form ''F'':''V'' → R, then may be called the unit sphereF. Reese Harvey (1990) ''Spinors and calibrations'', "Generalized Spheres", page 42, Academic Press, or hyperboloid#Relation to the sphere, unit quasi-sphere of ''V''. For example, the quadratic form x^2 - y^2, when set equal to one, produces the unit hyperbola which plays the role of the "unit circle" in the plane of split-complex numbers. Similarly, the quadratic form x2 yields a pair of lines for the unit sphere in the dual number plane.


See also

*ball (mathematics), ball *hypersphere * sphere *superellipse *
unit circle
unit circle
* unit disk *unit sphere bundle *unit square


Notes and references

* Mahlon M. Day (1958) ''Normed Linear Spaces'', page 24, Springer-Verlag. *. Reviewed i
''Newsletter of the European Mathematical Society'' 64 (June 2007)
p. 57. This book is organized as a list of distances of many types, each with a brief description.


External links

* {{DEFAULTSORT:Unit Sphere Functional analysis 1 (number) Spheres es:1-esfera