# tensor

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## tensor,

in mathematics, quantity that depends linearly on several vector**vector,**

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.

**.....**Click the link for more information. variables and that varies covariantly with respect to some variables and contravariantly with respect to others when the coordinate axes are rotated (see Cartesian coordinates

**Cartesian 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*

**.....**Click the link for more information. ). Tensors appear throughout mathematics, though they were first treated systematically in the calculus

**calculus,**

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.

**.....**Click the link for more information. of differential forms and in differential geometry

**differential 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),

**.....**Click the link for more information. . They play an important role in mathematical physics, particularly in the theory of relativity

**relativity,**

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.

**.....**Click the link for more information. . 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.