Euclidean Space

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euclidean space

[yü′klid·ē·ən ′spās]
A space consisting of all ordered sets (x1, …, xn ) of n numbers with the distance between (x1, …, xn ) and (y1, …, yn ) being given by the number n is called the dimension of the space.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Euclidean Space


in mathematics, a space whose properties are described by the axioms of Euclidean geometry. In a more general sense, a Euclidean space is an n-dimensional vector space, into which several special Cartesian coordinates can be introduced so that its metric is defined in the following manner: If point M has the coordinates (x1x2, …, xn and point M* has the coordinates (x1*, x2*, …, xn*), then the distance between these points is

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
In the 3-dimensional space [X.sup.(3)], for any pair of vectors, the Cauchy-Schwarz inequality [8] holds, which for the vectors (5) gives the implication:
where [phi](t) is instantaneous angle between vectors of the IV of current and voltage in the arithmetic 3-dimensional space [X.sup.(3)] at time t.
In this article, we investigate the radial part of Laplace's equations in dimensions both lower and higher than three and interpret what their solutions represent in 3-dimensional space. Consider an n-dimensional Euclidean space given by the linear coordinates [x.sub.1], [x.sub.2], [x.sub.3], ..., [x.sub.n] [-[infinity] < [x.sub.1], [x.sub.2], x, ..., [x.sub.n] < [infinity]].
We can now examine the physical examples which the solutions to Laplace's equations represent in our familiar 3-dimensional space. First, for n = 1, Eqs.
Clearly, this case belongs to an infinite line charge distribution in 3-dimensional space, where the electric field diminishes inversely as the radial distance from the line charge.
These are identified as the potential and electric field, respectively, of the electric dipole in 3-dimensional space.
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The user has an additional navigation "remote control," with arrows to steer and change the orientation in 3-dimensional space.
The light pentad contains three matrices corresponding to the coordinates of 3-dimensional space, and two matrices relevant to mass terms--one for the lepton and one for the neutrino of this lepton.
Jiu, Translation surfaces with constant mean curvature in 3-dimensional spaces, J.

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