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.
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 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
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). 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.
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 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),
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. 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.
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. 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 an incompressible and isothermal polymer fluid, Beris and Edwards have derived the following relation for the Equation of evolution of the conformation tensor and the stress tensor [22, 30]:
Where [LAMBDA] is a fourth-order tensor, called the mobility tensor which is essentially the inverse of the relaxation time of the polymer fluids, [rho] is the fluid density, Q is anisotropic viscosity matrix that is related to viscous dissipation, L is coupling parameter between the velocity gradient field and the structural tensor field, C is second-order conformation tensor which is symmetric, and it has nine components as follow:
TeamViewer Tensor allows companies to opt in and out any time to collect a log of all relevant actions (who, what, when, where) during remote control sessions and for activities performed at the management console level with reports only visible to authorized users.
TeamViewer Tensor can easily be scaled to a large number of devices (Windows and Mac) using standard software deployment solutions.
where L denotes the Lie derivative, S is the Ricci tensor and [lambda] is real constant on ik.
In this section, we first introduce tensor operations used in this paper and provide the bistatic MIMO radar signal model.
One then interprets Poisson's equation in metric language as -[[nabla].sup.2][g.sub.00] = [kappa][T.sub.00], where matter energy density is interpreted as the time-time component of the divergence free, symmetric, matter energy momentum tensor [T.sub.ab].
To solve (1), we need to calculate an exponential function about tensor A.
Therefore, this paper firstly uses the data completion method based on the tensor form to complete the missing RTMS data.
defined the primitivity of nonnegative tensors (as Definition 1), extended the theory of nonnegative matrices to nonnegative tensors, and proved the convergence of the NQZ method which is an extension of the Collatz method and can be used to find the largest eigenvalue of any nonnegative irreducible tensor.
Here [A.sub.PN] is an amplitude of P-wave displacement measured on north component; [G.sub.PN] (1) is the P-wave Green's function's derivative for a far-field ray approximation of north component amplitude due to the first component of moment tensor M ; numbers (1)...