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 ?
The relations between the scalar, vectorial and tensorial quantities of this paragraph can be derived from the corresponding relations we proved in the second paragraph, considering that the acceleration and the velocity of the material particle vanish, that is [alpha] = [alpha] (w) = 0 and u = u (w) = 0.
A tensorial model to describe the clamping errors caused by small rigid body displacements and on the other hand clamping errors caused by the elastic strain.
where the operations "()" and "[]" on tensorial indices denote symmetrization and antisymmetrization, respectively, and summation is applied to repeated tensorial indices over all space-time values.
Scalar field is in the role of a variable gravitational "constant", leaving tensorial metric field and its geodesics to act as trajectories of freely falling particles as in GR.
Here we show an explicit basis of the diagonal invariant algebra as a free module over the tensorial invariant algebra of all projective reflection groups G(r, p, q, n).
This expression has a tensorial character as it should be, because by definition both quantities [[GAMMA].
These transformations can be put into the following tensorial form
The governing equations are the momentum equations, without inertia terms, and the continuity equation, which can be written in dimensionless tensorial form as:
Let us investigate the simplest one-dimensional case and reduce the treatment to one component of the above tensorial equation.
Equation 13 has a proper tensorial form and is useful for solving geometrically non-linear problems.
Then we derive the entropy production for fluids with a single tensorial internal variable (dynamic degree of freedom).
Consequently, the tensorial equation (Eq 8) is solved at a local level; it reduces to a (4 x 4) linear algebraic system.