Euclidean Space


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euclidean space

[yü′klid·ē·ən ′spās]
(mathematics)
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 It might be outdated or ideologically biased.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 ?
Using a Euclidean space, this paper presents an alternative approach to studying uncertainty in NPV and IRR based on fuzzy set theory.
Bracken, "Determination of surfaces in three-dimensional Minkowski and Euclidean spaces based on solutions of the sinh-Laplace equation," International Journal of Mathematics and Mathematical Sciences, vol.
To prove commutativity, we shall essentially follow Dold's argument in the classical case [10] and reduce to working on subspaces of Euclidean spaces.
Even though the Lie algebra space is flat and isomorphic to the Euclidean space, the system is coupled.
According to the mathematical definition of manifold, a manifold can be divided into a number of overlapped patches and each patch is homeomorphic to an open set of Euclidean space. The open sets are called local coordinates of manifold.
Karger and Novak [5] classified the integral curves of a linear vector field in 3-dimensional Euclidean space. They showed that the integral curves of the linear vector field in E3 are helixes, circles or parallel straight lines.
Let X be a Euclidean space [R.sup.d] or a hyperbolic space [H.sup.d].
This is not just an academic exercise--large parts of the modern world function on the basis of higher-dimensional Euclidean spaces. For example, all types of regression analysis utilize higher-dimensional Euclidean space.
We consider a weakly stationary Gaussian process with mean zero in which data are drawn from the regularly spaced grid on d-dimensional Euclidean space, Z = {Z(s); s [member of] [R.sup.d]} [member of] R (see Figure 1).
Let W be a nonempty subset of n-dimensional Euclidean space [R.sup.n]; the diameter of W is defined as
Since the property of a hypersurface to be a canal hypersurface is conformally invariant, canal hypersurfaces in a multidimensional Euclidean space were investigated in many papers, such as [15, 16].
Let M be an n-dimensional submanifold of an (n + m)-dimensional Euclidean space [E.sup.n+m].