Euclidean Space

Euclidean n-space, sometimes called Cartesian space or simply n-space, is the space of all n-tuples of real numbers, (x_1, x_2, ..., x_n). Such n-tuples are sometimes called points, although other nomenclature may be used (see below). The totality of n-space is commonly denoted R^n, although older literature uses the symbol E^n (or actually, its non-doublestruck variant E^n; O'Neill 1966, p. 3).

R^n is a vector space and has Lebesgue covering dimension n. For this reason, elements of R^n are sometimes called n-vectors. R^1=R is the set of real numbers (i.e., the real line), and R^2 is called the Euclidean plane. In Euclidean space, covariant and contravariant quantities are equivalent so e^->^j=e^->_j.

See also

Euclidean Plane, Point, Pseudo-Euclidean Space, Real Line, Real Number, Tensor, Vector Explore this topic in the MathWorld classroom

Portions of this entry contributed by Christopher Stover

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Gray, A. "Euclidean Spaces." §1.1 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 2-5, 1997.O'Neill, B. Elementary Differential Geometry. San Diego, CA: Academic Press, 1966.

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Euclidean Space

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Stover, Christopher and Weisstein, Eric W. "Euclidean Space." From MathWorld--A Wolfram Web Resource.

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