 some unit spheres 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. An illustration of vector norms File links The following pages link to this file: Norm (mathematics) Categories: GFDL images ...
Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
A sphere is a perfectly symmetrical geometrical object. ...
personal space, proxemics. ...
The solid interior of a sphere or circle; in mathematics, latter terms refer specifically to the (n-1)-dimensional surface of an n-dimensional solid ball. ...
In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ...
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. A sphere is a perfectly symmetrical geometrical object. ...
In classical geometry, a radius of a circle or sphere is any line segment with one endpoint on the circle (i. ...
In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ...
In Euclidean geometry, scaling is an affine, linear transformation that can enlarge or diminish an object by certain factors. ...
Unit balls in Euclidean space
In Euclidean space of n dimensions, the unit sphere is the set of all points which satisfy the equation In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
and the closed unit ball is the set of all points satisfying the inequality In mathematics, an inequality is a statement about the relative size or order of two objects. ...
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 is, often denoted ωn in the literature, can be expressed by making use of the Gamma function. It is Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ...
The Gamma function along an interval In mathematics, the Gamma function extends the factorial function to the complex numbers. ...
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The volume of the unit ball is ωn / n.
Unit balls in normed vector spaces More precisely, the open unit ball in a normed vector space V, with the norm , is In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...
The word norm coming from the latin word norma which means angle measure or (lawlike) rule, has a number of meanings: A social or sociological norm; see norm (sociology). ...
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It is the interior of the closed unit ball of (V,||·||), In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...
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The latter is the disjoint union of the former and their common border, the unit sphere of (V,||·||), - .
Comments 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 norm l∞ in Rn. The round ball is understood as 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. In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ...
In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ...
In mathematics the Euclidean distance or Euclidean metric is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ...
Generalization to 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. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
In mathematics, an ultrametric space is a special kind of metric space. ...
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