This article is about tensors on a single vector space. It is not to be confused with Vector field or Tensor field.
The second-order Cauchy stress tensor describes the stress experienced by a material at a given point. For any unit vector , the product is a vector, denoted , that quantifies the force per area along the plane perpendicular to . This image shows, for cube faces perpendicular to , the corresponding stress vectors along those faces. Because the stress tensor takes one vector as input and gives one vector as output, it is a second-order tensor.
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix.
Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), general relativity (stress–energy tensor, curvature tensor, ...), and others. In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".
Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 – continuing the earlier work of Bernhard Riemann, Elwin Bruno Christoffel, and others – as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.[1]
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Kline, Morris (1990). Mathematical Thought From Ancient to Modern Times. Vol. 3. Oxford University Press. ISBN 978-0-19-506137-6.
two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span V ⊗ W {\displaystyle V\otimes W} in the sense...
Stress tensor may refer to: Cauchy stress tensor, in classical physics Stress deviator tensor, in classical physics Piola–Kirchhoff stress tensor, in continuum...
first-generation Tensor chip debuted on the Pixel 6 smartphone series in 2021, and were succeeded by the Tensor G2 chip in 2022 and G3 in 2023. Tensor has been...
tensor (symbol σ {\displaystyle {\boldsymbol {\sigma }}} , named after Augustin-Louis Cauchy), also called true stress tensor or simply stress tensor...
mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the...
electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a...
metric field on M consists of a metric tensor at each point p of M that varies smoothly with p. A metric tensor g is positive-definite if g(v, v) > 0 for...
and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used...
elasticity tensor is a fourth-rank tensor describing the stress-strain relation in a linear elastic material. Other names are elastic modulus tensor and stiffness...
Einstein in 1915 in the form of a tensor equation which related the local spacetime curvature (expressed by the Einstein tensor) with the local energy, momentum...
The tensor fasciae latae (or tensor fasciæ latæ or, formerly, tensor vaginae femoris) is a muscle of the thigh. Together with the gluteus maximus, it acts...
Look up tense in Wiktionary, the free dictionary. Tense may refer to: Tense, a state of muscle contraction Grammatical tense, a property of verbs indicating...
relationship between the Ricci tensor and the matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of...
deformation tensors. In 1839, George Green introduced a deformation tensor known as the right Cauchy–Green deformation tensor or Green's deformation tensor (the...
see IPA § Brackets and transcription delimiters. In phonology, tenseness or tensing is, most broadly, the pronunciation of a sound with greater muscular...
universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. The tensor product of an algebra and...
In multilinear algebra, a tensor decomposition is any scheme for expressing a "data tensor" (M-way array) as a sequence of elementary operations acting...
stapedius. The tensor tympani is supplied by the tensor tympani nerve, a branch of the mandibular branch of the trigeminal nerve. As the tensor tympani is...
May 2019, Google announced TensorFlow Graphics for deep learning in computer graphics. In May 2016, Google announced its Tensor processing unit (TPU), an...
In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold...
Tensor Processing Unit (TPU) is an AI accelerator application-specific integrated circuit (ASIC) developed by Google for neural network machine learning...
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. In components...
Tensor networks or tensor network states are a class of variational wave functions used in the study of many-body quantum systems. Tensor networks extend...
The tensor veli palatini muscle (tensor palati or tensor muscle of the velum palatinum) is a thin, triangular muscle of the head that tenses the soft palate...
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