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In Mathematics , a unit Sphere is the set of points of Distance 1 from a fixed central point, where a generalized concept of distance may be used. A '''unit Ball ''' is the region enclosed by a unit sphere. Usually a specific point has been distinguished as the Origin of the space under study and it is understood that a unit sphere or unit ball is centered at that point. Therefore one speaks of "the" unit ball or "the" unit sphere. A unit sphere is simply a Sphere of Radius one. The importance of the unit sphere is that any sphere can be transformed to a unit sphere by a combination of Translation 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 of ''n'' dimensions, the unit sphere is the set of all points which satisfy the equation : and the closed unit ball is the set of all points satisfying the Inequality : General area and volume formulas 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 . The surface area of the unit sphere in ''n'' dimensions, often denoted in the literature, can be expressed by making use of the Gamma Function . It is :. The volume of the unit ball is . Non-general area and volume formulas In three-dimensional Euclidean space, a unit sphere's volume is : and its surface area is : UNIT BALLS IN NORMED VECTOR SPACES | ||
| It Is The | "http://wwwinformationdelightinfo/information/entry/interior" class="copylinks">Interior of the '''closed unit ball''' of (''V'',ยท), |
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| :<math> \{ X\in V: \x\ | 1 \}</math> |
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