tensor

tensor

tensor, in mathematics, quantity that depends linearly on several vector 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). Tensors appear throughout mathematics, though they were first treated systematically in the calculus of differential forms and in differential geometry. They play an important role in mathematical physics, particularly in the theory of relativity. 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).

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The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.

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.

McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

Tensor chip

An SoC (system-on-chip) from Google that made its first appearance in a smartphone on the Pixel 6 and 6 Pro in 2021. The Tensor chip has an 8-core CPU and 20-core GPU, an advanced image processor for photos and a machine learning processor for AI, known as the Tensor Processing Unit. There is also a context hub for very-low power functions. The Tensor SoC is built on a 5 nm process technology. See Pixel phone and TensorFlow.

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TensorFLow

An open source framework for mathematical processing from Google. TensorFlow is deployed on all popular platforms to develop mathematical layers in neural networks for machine learning. The TensorFlow software library runs on CPUs, GPUs and specialized Tensor Processing Unit (TPU) chips. See deep learning, machine learning and Tensor chip.

The Tensor Layer
A "tensor" is a multidimensional array that can hold a variety of data types. The number of dimensions is the "tensor shape."

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