# tensor

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## tensor,

in mathematics, quantity that depends linearly on several vectorvector,
quantity having both magnitude and direction; it may be represented by a directed line segment. Many physical quantities are vectors, e.g., force, velocity, and momentum.
variables and that varies covariantly with respect to some variables and contravariantly with respect to others when the coordinate axes are rotated (see Cartesian coordinatesCartesian coordinates
[for René Descartes], system for representing the relative positions of points in a plane or in space. In a plane, the point P is specified by the pair of numbers (x,y
). Tensors appear throughout mathematics, though they were first treated systematically in the calculuscalculus,
branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit—the notion of tending toward, or approaching, an ultimate value.
of differential forms and in differential geometrydifferential geometry,
branch of geometry in which the concepts of the calculus are applied to curves, surfaces, and other geometric entities. The approach in classical differential geometry involves the use of coordinate geometry (see analytic geometry; Cartesian coordinates),
. They play an important role in mathematical physics, particularly in the theory of relativityrelativity,
physical theory, introduced by Albert Einstein, that discards the concept of absolute motion and instead treats only relative motion between two systems or frames of reference.
. Tensors are also important in the theory of elasticity, where they are used to describe stress and strain. The study of tensors was formerly known as the absolute differential calculus but is now called simply tensor analysis.

### Bibliography

See R. Abraham et al., Manifolds, Tensor Analysis, and Applications (1988).

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Tensor

a term in mathematics that came into use in the mid-19th century and has since been employed in two distinct senses. The term is most commonly used in the modern tensor calculus, where it refers to a special type of quantity that transforms according to a special law. In mechanics, particularly elasticity theory, the term is used as a synonym for a linear operator Φ that transforms a vector Φ into the vector Φx and is symmetric in the sense that the scalar product yΦx remains unchanged if the vectors x and y are interchanged. The term originally referred to the small tensile (hence “tensor”) and compressional strains arising in elastic deformation. It was subsequently carried over into other fields of mechanics. Thus, we speak of a deformation tensor, stress tensor, inertia tensor, and so on.

## tensor

[′ten·sər]
(mathematics)
An object relative to a locally euclidean space which possesses a specified system of components for every coordinate system and which changes under a transformation of coordinates.
A multilinear function on the cartesian product of several copies of a vector space and the dual of the vector space to the field of scalars on the vector space.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

## TensorFLow

An open source framework for mathematical processing from Google. TensorFlow is deployed on all popular platforms to develop mathematical layers in neural networks for machine learning. The TensorFlow software library runs on CPUs, GPUs and specialized Tensor Processing Unit (TPU) chips. See deep learning and machine learning.

The Tensor Layer
A "tensor" is a multidimensional array that can hold a variety of data types. The number of dimensions is the "tensor shape."
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References in periodicals archive ?
How do the Einstein second rank tensor equations fit in the theory?
([dagger]) The tensor equations have been normalized, see Kastner [8] and appendix.
The particle quantities like mass, spin, charge, magnetic momentum are integration constants from mentioned tensor equations, and are inserted as parameters into the initial conditions.
The field has distributed stresses which are expressed by an addition to the electromagnetic field stress-tensor (see the Einstein tensor equations).
Looking at the right side of the Einstein equations (35), which determine distributed matter, we see that [rho] and U are included only in the scalar (first) equation and also three tensor equations with the indices 11, 22, 33 (the 5th, 8th, and 10th equations).We can therefore find a formula for U.

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