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The Collaborative International Dictionary

Tensor \Ten"sor\, n. [NL. See Tension.]

  1. (Anat.) A muscle that stretches a part, or renders it tense.

  2. (Geom.) The ratio of one vector to another in length, no regard being had to the direction of the two vectors; -- so called because considered as a stretching factor in changing one vector into another. See Versor.

Douglas Harper's Etymology Dictionary

muscle that stretches or tightens a part, 1704, Modern Latin agent noun from tens-, past participle stem of Latin tendere "to stretch" (see tenet).

  1. Of or relating to tensors n. 1 A muscle that stretches a part, or renders it tense. 2 (context mathematics physics English) An image of a tuple under a tensor product map. 3 (context mathematics physics English) A function of several variables 4 (context mathematics physics English) A mathematical object consisting of a set of components with ''n'' index each of which range from 1 to ''m'' where ''n'' is the rank and ''m'' is the dimension of the tensor.Rowland, Todd and Weisstein, Eric W., , Wolfram MathWorld. v

  2. To compute the tensor product of two tensors.

  1. n. a generalization of the concept of a vector

  2. any of several muscles that cause an attached structure to become tense or firm


Tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps. Euclidean vectors, often used in physics and engineering applications, and scalars themselves are also tensors. A more sophisticated example is the Cauchy stress tensor T, which takes a direction v as input and produces the stress T on the surface normal to this vector for output, thus expressing a relationship between these two vectors, shown in the figure (right).

Given a coordinate basis or fixed frame of reference, a tensor can be represented as an organized multidimensional array of numerical values. The (also degree or rank) of a tensor is the dimensionality of the array needed to represent it, or equivalently, the number of indices needed to label a component of that array. For example, a linear map is represented by a matrix (a 2-dimensional array) in a basis, and therefore is a 2nd-order tensor. A vector is represented as a 1-dimensional array in a basis, and is a 1st-order tensor. Scalars are single numbers and are thus 0th-order tensors. Because they express a relationship between vectors, tensors themselves must be independent of a particular choice of coordinate system. The coordinate independence of a tensor then takes the form of a covariant and/or contravariant transformation law that relates the array computed in one coordinate system to that computed in another one. The precise form of the transformation law determines the type (or valence) of the tensor. The tensor type is a pair of natural numbers , where is the number of contravariant indices and is the number of covariant indices. The total order of a tensor is the sum of these two numbers.

Tensors are important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as elasticity, fluid mechanics, and general relativity. Tensors were first conceived by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.

Tensor (intrinsic definition)

In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multi-linear concept. Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.

In differential geometry an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference to coordinates at all. The same is true in general relativity, of tensor fields describing a physical property. The component-free approach is also used heavily in abstract algebra and homological algebra, where tensors arise naturally.

Note: This article assumes an understanding of the tensor product of vector spaces without chosen bases. An overview of the subject can be found in the main tensor article.
Tensor (disambiguation)

Tensor (Latin tensio "tension", a tensor would then be "someone who tenses") may refer to:

Usage examples of "tensor".

No one, I repeatno onehas used Hamiltonian quaternions since 1915 when tensor analysis was invented.

Morey, for instance, would never have developed the autointegral calculus, to say nothing of tensor and spinor calculus, which were developed two hundred years ago, without the knowledge of the problems of space to develop the need.

He marvelled at the way Schwarzschild geodesies flowed into cheek-planes, the spinors transfigured themselves as eyebrows, and the tensor fields spread out and grew into the forehead of this blazing, inner face.

When Melba reached the dining room with a patch on her head and a tensor round her ankle everyone else was half through.

Einstein obtained through Riemannian geometry and gravitational tensors was derived classically by a German called Paul Gerber, in 1898, when Einstein was nine years old.

The difference, however, is that a mirror rephrasing of this sort results in the antisymmetric tensor field Bμv —the real part of the complexified Kähler form on the mirror Calabi-Yau space—vanishing, and this is a far more drastic sort of singularity than that discussed in Chapter 11.

He says (and with more than a touch of the gibber in his voice) it deflowers, rapes, & pillages, breaks & enters Minkowski's Covariant Tensor.

Being only about to finish high school his training had gone no farther than tensor calculus, statistical mechanics, simple transfinities, generalized geometries of six dimensions, and, on the practical side, analysis for electronics, primary cybernetics and robotics, and basic design of analog computers.

He coasted above the floor, finding treasure everywhere -- lamps, cameras, radios, tape recorders, Tensor lamps, television sets, nose drops, spray cans of paint, plastic models, tropical fish tanks, batteries, soap, scouring pads, light bulbs, canned salted peanuts.

In the first place, there are devices that measure overall gravitational intensity, in both scalar and tensor aspects, at any point in space, whether you know the neighborhood or not.

But Haor Chall had set the fuselage on a pedestal and capped it with two enormous wings, presumably kept from sagging by powerful tensor fields.

He informed me in that schoolmaster way of his that it was because those pioneers did not have the tensor calculus, vector analysis, and matrix algebra.

The trouble was, no words could describe being in linkage: creating n-dimensional spaces, and time-variant curvatures for them, and tensors within, and functions and operations that nobody had ever before imagined.

That amplitude is constantly shifting, of course, but if we get it right, then the torpedo should hold together long enough to emit a magneton pulse that will react with a subspace tensor matrix generated by the Enterprise to create an opening in the space-time continuum.

Instead, he retreated as usual into his private world of tensors and twistors, and despite my own respectable scientific background I couldn't follow him there.