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).

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.
References in periodicals archive ?
For tensors of the order 0 or 1 this is not a restriction, since we have zero or one index, respectively, which we cannot permutate.
sup][4],[5] Conventional magnetic resonance imaging (MRI) and diffusion tensor imaging (DTI) can noninvasively reveal macroscopic and microscopic changes in muscle fibers, respectively.
2 Order enriched categorical definitions and initial lemmas on tensors
In this scheme, first, multichannel EEG data of hybrid tasks are transformed into multiway tensors representation in multimodes of channel, time, and frequency domain, and for each hybrid task, including individual tasks and simultaneous tasks, the assembled tensors in training dataset are calculated.
Levy, "Symmetric tensors of the second order whose covariant derivatives vanish," The Annals of Mathematics, vol.
Keywords: Lattice Discrete Element Method, Damage evaluation, Acoustic Emission, Moment Tensor
Then, the covariant components of Riemann tensors on S are defined by
We find the Ricci curvature tensors for this spacetimes and put them equal to zero.
The conformational tensor is a micro-structural factor that shows the deformation of polymeric chains during the flow [17, 18].
The relation between stress and strain tensors is expressed by:
In this paper, we study SOPDs for third-order partially symmetric tensors and fourth-order fully and partially symmetric tensors and propose a new method called partial column-wise least squares (PCLS) to compute the SOPD.

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