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 ?
Additionally, due to the recent result in Pal & Ghosh (2015), we have calculated hidden nonlocality for two-qubit Hirsch states which has led us to report tensorial activation of hidden nonlocality.
In this work, the author proposed a tensorial form of the finite element method for the u-p formulation within TPM framework.
A tensorial approach to computational continuum mechanics using oh)ect-oriented techniques, Computers in Physics, Vol.
Fully tensorial nodal and modal shape functions for triangles and tetrahedra," International Journal for Numerical Methods in Engineering, Vol.
EM mass] = 0 implies the existence of some tensorial gauge transformation (2) (there is no need here to determine this gauge explicitly) which provides the mathematical relationship between both tensors [T.
Las formas de onda observadas son expresadas como una combinacion lineal de las formas de ondas calculadas, en notacion tensorial como,
Kamthan, Infinite Matrices and tensorial transformations, Acta Math.
Some applications of plasmonic behavior can also be tuned by a dc external magnetic field, and the applied magnetic field produces a plasmon with a tensorial permittivity.
Dejando a un lado la introduccion subrepticia de referenciales privilegiados (10) (oculta a menudo bajo nombres aparentemente asepticos, como "dependencia del hiperplano") cabria imaginar una contrapartida tensorial para el calculo de probabilidades, que la hiciese tan independiente del sistema de referencia como son las magnitudes espacio-temporales en la geometria de Minkowski.
Taking into account the tensorial transformation laws of the adapted basis of vector fields on E = [J.
No tensorial formalisms will be followed as the analytical solutions refer to 1-D problems.