Copyright © Philip M. Parker, INSEAD. Terms of Use.
Definitions: Tensor |
TensorNoun1. A generalization of the concept of a vector. 2. Any of several muscles that cause an attached structure to become tense or firm. Source: WordNet 1.7.1 Copyright © 2001 by Princeton University. All rights reserved. |
Etymology: Tensor \Ten"sor\, noun. [New Latin expression. See Tension.]. (Websters 1913) |
| Domain | Definitions |
Aerospace | An array of functions which obeys certain laws of transformation. A one-row or one-column tensor array is a vector. The motivation for the use of tensors in some branches of physics is that they are invariants, not depending on the particular coordinate system employed. (references) |
Electrical Engineering | 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. Source: European Union. (references) |
Source: compiled by the editor from various references; see credits. | |
(From Wikipedia, the free Encyclopedia)
In mathematics, a tensor is a certain kind of geometrical entity which generalizes the concepts of scalar, vector (spatial) and linear operator in a way that is independent of any chosen frame of reference. Tensors are of importance in physics and engineering.
Tensors can be represented by arrays of components. The point of having a tensor theory is to explain the further implication of saying that a quantity is a tensor, beyond that specifying it requires a number of indexed components.
This article attempts to provide a non-technical introduction to the idea of tensors, and to provide an introduction to the articles which describe different, complementary treatments of the theory of tensors in detail.
The notation was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential geometry. The tensor calculus achieved broader acceptance with the introduction of Einstein's theory of general relativity, around 1915, which is formulated completely in the language of tensors. But it is used also within continuum mechanics, for example the strain tensor, from where its name orginates (see linear elasticity).
Note that the word "tensor" is often used as a shorthand for tensor field, which a tensor value defined at every point in a manifold. To understand tensor fields, you need to first understand the basic idea of tensors.
There are two ways of approaching the definition of tensors:
Not all relationships in nature are linear, but most are differentiable and so may be locally approximated with sums of multilinear maps. Thus most quantities in the physical sciences can be usefully expressed as tensors.
As a simple example, consider a ship in the water. We want to describe its response to an applied force. Force is a vector, and the ship will respond with an acceleration, which is also a vector. The acceleration will in general not be in the same direction as the force, because of the particular shape of the ship's body. However, it turns out that the relationship between force and acceleration is linear. Such a relationship is described by a tensor of type (1,1) (that is to say, it transforms a vector into another vector). The tensor can be represented as a matrix which when multiplied by a vector results in another vector. Just as the numbers which represent
a vector will change if one changes the coordinate system, the numbers in the matrix that represents the tensor will also change when the coordinate system is changed.
In engineering, the stresses inside a rigid body or fluid are also described by a tensor; the word "tensor" is Latin for something that stretches, i.e. causes tension. If a particular surface element inside the material is singled out, the material on one side of the surface will apply a force on the other side. In general, this force will not be orthogonal to the surface, but it will depend on the orientation of the surface in a linear manner. This is described by a tensor of type (2,0), or more precisely by a tensor field of type (2,0) since the stresses may change from point to point.
Some well known examples of tensors in geometry are quadratic forms, and the curvature tensor. Examples of physical tensors are the energy-momentum tensor and the polarization tensor.
Geometric and physical quantities may be categorized by considering
the degrees of freedom inherent in their description. The scalar
quantities are those that can be represented by a single number ---
speed, mass, temperature, for example. There are also vector-like
quantities, such as force, that require a list of numbers for their
description. Finally, quantities such as quadratic forms
naturally require a multiply indexed array for their representation.
These latter quantities can only be conceived of as tensors.
Actually, the tensor notion is quite general, and applies to all of
the above examples; scalars and vectors are special kinds of
tensors. The feature that distinguishes a scalar from a vector, and
distinguishes both of those from a more general tensor quantity is
the number of indices in the representing array. This number is
called the rank of a tensor. Thus, scalars are rank zero tensors (no
indices at all), and vectors are rank one tensors.
There are equivalent approaches to visualizing and working with tensors; that the content is actually the same may only become apparent with some familiarity with the material.
This idea can then be further generalized to tensor fields, where the elements of the tensor are functions, or even differentials.
The tensor field theory can roughly be viewed, in this approach, as a further extension of the idea of the Jacobian.
This treatment has largely replaced the component-based treatment for advanced study, in the way that the more modern component-free treatment of vectors replaces the traditional component-based treatment after the component-based treatment has been used to provide an elementary motivation for the concept of a vector. You could say that the slogan is 'tensors are elements of some tensor space'.
It is also possible for a tensor field to have a "density". A tensor with density r transforms as an ordinary tensor under coordinate transformations, except that it is also multiplied by the determinant of the Jacobian to the rth power. This is best explained, perhaps, using vector bundles: where the determinant bundle of the tangent bundle is a line bundle that can be used to 'twist' other bundles r times.Background
The choice of approach
Covariant vectors, for instance, are also described as one-forms, or as the elements of the dual space to the contravariant vectors.Examples
Approaches, in detail
The classical approach views tensors as multidimensional arrays that are n-dimensional generalizations of scalars, 1-dimensional vectors and 2-dimensional matrices. The "components" of the tensor are the indices of the array.
The modern (component-free) approach views tensors initially as abstract objects, expressing some definite type of multi-linear concept. Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra.
In the end the same computational content is expressed, both ways. See glossary of tensor theory for a listing of technical terms.
Tensor densities
| Rank | Alias | Element notation | Common transformation* | Geometrical interpretation |
|---|---|---|---|---|
| 0 | Scalar | a | S'=|a|S | · |
| 1 | Vector | ai | V'i=|a|aijVj |  |
| 2 | Matrix | aij | M'ij=|a|aikajlMkl |  |
| 3 | ? | aijk | M'ijk=|a|ailajsakmMlsm |  |
See also:
Source: adapted by the editor from Wikipedia, the free encyclopedia under a copyleft GNU Free Documentation License (GFDL) from the article "Tensor."
Crosswords: Tensor |
| English words defined with "tensor": tensor tympani. (references) |
| Specialty definitions using "tensor": acoustic reflex, auditory reflex, aural reflex ♦ Drucker-Prager criterion ♦ muscle tensor tympani ♦ Palatal Muscles ♦ Reflex, Acoustic ♦ Songbirds, stress tensor ♦ tensor choroideae muscle, tensor product ♦ viscous stresses. (references) |
| Non-English Usage: "Tensor" is also a word in the following languages with English translations in parentheses. Portuguese (strainer, stretcher, tightener), Spanish (chest expander, tensile, tensor). |
| Domain | Title |
Books |
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Source: compiled by the editor from various references; see credits. | |
| "Tensor" is generally used as a noun (singular) -- approximately 100.00% of the time. "Tensor" is used about 207 times out of a sample of 100 million words spoken or written in English. Its rank is based on over 700,000 words used in the English language. Some parts-of-speech are not covered due to the samples used by the British National Corpus. (note: percents less than one-hundredth of one percent have been omitted) |
| Parts of Speech | Percent | Usage per 100 Million Words | Rank in English |
| Noun (singular) | 100% | 207 | 21,147 |
Source: compiled by the editor from several corpora; see credits.
Expressions using "tensor": muscle tensor tympani ♦ tensor choroideae muscle ♦ tensor product ♦ Tensor Tympani. Additional references. | |
| Source: compiled by the editor from various references; see credits. |
| The following statistics estimate the number of searches per day across the major English-language search engines as identified by various trade publications. Hyperlinks lead to commercial use of the expression at Amazon.com. |
| Language | Translations for "tensor"; alternative meanings/domain in parentheses. | ||||||||||||||||||||||||||||||||||||||||
Bulgarian | сгъвач (folder), флексор (flexor), тенсор. (various references) | ||||||||||||||||||||||||||||||||||||||||
Chinese | 量. (various references) | ||||||||||||||||||||||||||||||||||||||||
Danish | tensor, wirestrammer (turnbuckle), vantskruer (turnbuckle). (various references) | ||||||||||||||||||||||||||||||||||||||||
Dutch | tensor. (various references) | ||||||||||||||||||||||||||||||||||||||||
Finnish | tensori, musculus tensor, jännittäjälihas. (various references) | ||||||||||||||||||||||||||||||||||||||||
French | tenseur, tendeur (tensioning device, tenter, twine tensioner). (various references) | ||||||||||||||||||||||||||||||||||||||||
German | Spannmuskel. (various references) | ||||||||||||||||||||||||||||||||||||||||
Greek | εκτείνων μυς (extensor), εκτείνων, τείνων μυς, τανύων μύσ, τανυστής. (various references) | ||||||||||||||||||||||||||||||||||||||||
Hebrew | מתחן. (various references) | ||||||||||||||||||||||||||||||||||||||||
Hungarian | tenzor. (various references) | ||||||||||||||||||||||||||||||||||||||||
Italian | tensore. (various references) | ||||||||||||||||||||||||||||||||||||||||
Japanese Kanji | テロ対策 (10, 99.99999999%, anti-terrorist, temper, tempera, temperament, temperate, tempo, temporary, temporary worker, temptation, tempura, ten, ten key, ten nines, tendency, tenderloin, tenderloin steak, ten-gallon hat, tense, tension, tension people, tent). (various references) | ||||||||||||||||||||||||||||||||||||||||
Japanese Katakana | テンソル . (various references) | ||||||||||||||||||||||||||||||||||||||||
Korean | 텐서. (various references) | ||||||||||||||||||||||||||||||||||||||||
Manx | tensoor. (various references) | ||||||||||||||||||||||||||||||||||||||||
Pig Latin | ensortay tensor (bridle, strainer, stretcher, tensioning device, tightener, turnbuckle, wire strainer). (various references) тензор. (various references) tensor (binder, chest expander, linkage check rod, rigging screw, straightener device, strainer, stretcher, tensile, tensioning device, tightening screw, turnbuckle, twine tensioner, twisting stick, wire strainer). (various references) sträckmuskel. (various references) gerey, geren kas, dansör. (various references) тензор (tenser). (various references) | ||||||||||||||||||||||||||||||||||||||||
| Language | Period | Translations |
| Latin | 500 BCE-Modern | musculus tensor. (various references) |
| Source: compiled by the editor from various references. | ||
Derivations | |
Words beginning with "tensor": tensors. (additional references) | |
Words ending with "tensor": extensor. (additional references) | |
Words containing "tensor": extensors, ostensoria, ostensorium. (additional references) | |
| |
"Tensor" is suggested in spellcheckers for the following: ensor, Nensorb, tendor, tenior, Tenko, tensir, tensire, tenson, tenyor, teso, Tesol, tresoor, tresour. (additional references) | |
| Source: compiled by the editor, based on several corpora (additional references). | |
| Words rhyming with "tensor" (pronounced 'Ten"sor'): Accensor, Aggressor, Antecessor, Antecursor, Censor, Comprehensor, cursor, Defensor, divisor, Elisor, Extensor, Fidejussor, Impulsor, Incensor, Insessor, Intercessor, lessor, Precessor, Precursor, Predecessor, Provisor, Rescussor, scissor, sensor, sponsor, successor, suspensor, Tonsor, Versor, Windsor. (additional references) |
Scrabble® Enable2K-Verified Anagrams | |
Direct Anagrams: nestor, noters, stoner, tenors, toners, trones. | |
| Words within the letters "e-n-o-r-s-t" | |
-1 letter: nerts, noter, notes, onset, rents, roset, rotes, senor, seton, snore, snort, steno, stern, stone, store, tenor, terns, toner, tones, tores, torse, trone. | |
-2 letters: eons, erns, eros, erst, nest, nets, noes, nose, note, ones, ores, orts, rent, rest, rets, roes, rose, rote, rots, sent, snot, sone, sore, sorn, sort, tens, tern, toes, tone, tons, tore, torn, tors. | |
-3 letters: ens, eon, ern, ers, net, nor, nos, not, oes, one, ons, ore, ors, ort, ose, res, ret, roe, rot, sen, ser, set, son, sot, ten, toe, ton, tor. | |
-4 letters: en, er, es, et, ne, no, oe, on, or, os, re, so, to. | |
| Words containing the letters "e-n-o-r-s-t" | |
+1 letter: atoners, cornets, enroots, estrone, frontes, hornets, mentors, monster, nestors, norites, oestrin, orients, postern, rodents, senator, shorten, snorted, snorter, sorbent, stentor, stoners, stonier, tenours, tensors, thrones, tongers, tonners, tonsure, treason. | |
+2 letters: ancestor, antrorse, assentor, baronets, bestrown, bethorns, bornites, brownest, burstone, cointers, concerts, construe, converts, corniest, coronets, counters, crownets, drystone, earstone, enactors, enterons, entresol, estragon, estrogen, estrones, extensor, forspent, fortunes, genitors, honester, horniest, instroke, intoners, introrse, investor, jointers, knotters, mesotron, monstera, monsters, monteros, mordents, mounters, negators, networks, neutrons, northers, notaries, notchers, noticers, notornes, oestrins, oestrone, oneriest, onstream, operants, ornithes, outearns, outsnore, pointers, porniest, portends, portents, posterns, pronates, proteans, proteins, protends, recounts, refronts, remounts, resonant, resonate, retinols, rontgens, roundest, routines, seafront, senators, senorita, serotine, shortens, snootier, snorters, snottier, snoutier, softener, songster, sorbents, southern, starnose, stentors, sternson, stronger, sturgeon, tenoners, tenorist, ternions, tonearms, tonsured, tonsures, torments, torrents, tourneys, treasons, tritones, tropines, trounces, turnsole, tyrosine, unsorted, wrongest. | |
+3 letters: absorbent, accentors, adsorbent, aerations, aeronauts, ancestors, anchorets, anestrous, anointers, anoretics, anterooms, assentors, assertion, atropines, attorneys, baritones, barytones, betatrons, bevatrons, brimstone, bronziest, browniest, buhrstone, burgonets, burrstone, burstones, buttoners, carotenes, centroids, concertos, concretes, confuters, consenter, consorted, construed, construes, consulter, contester, contrives, coparents, copresent, corantoes, cornetist, coronates, corseting, cortisone, countries, courantes, courtesan, creations, creodonts, cretinous, cretonnes, curbstone, daneworts, desertion, dethrones, deuterons, disorient, doctrines, dragonets, dripstone, earstones, electrons, emanators, enthrones, entoderms, entrepots, entresols, entropies, erections, eruptions, erythrons, escorting, estragons, estrogens, exertions, exsertion, extensors, extrusion, firestone, fomenters, foresting, fostering, fourteens, freestone, frontages, frontiers, frontless, frontlets, honeworts, hornstone, horsemint, inceptors, indentors, infectors, injectors, innermost, insertion, inspector, instrokes, intercoms, interiors, interpose, inthrones, introfies, inventors, invertors, investors, ironstone, jointress, jointures, kenotrons, keynoters, marlstone, mesotrons, mestranol, misorient, modernest, modernist, moistener, monastery, monoester, monsteras, motioners, narrowest, nectarous, negatrons, neoterics, nervosity, neurotics, neutrinos, nitrogens, nocturnes, nondesert, nonskater, nonvoters, northeast, northerns, northwest, notarizes, notifiers, observant, obtainers, oestrones, orangiest, ordinates, orientals, ornaments, orneriest, orpiments, otherness, outlearns, outliners, outpreens, outrances, outranges, outsnored, outsnores, overhunts, overspent, overtness, overtones, overturns, oystering, oysterman, oystermen, patentors, patroness, patronise, personate, petronels, plectrons, portances, postponer, progestin, ratooners, reactions, reanoints, rebuttons, relations, remoisten, remotions, renovates, reobtains, reorients, resection, resolvent, resonants, resonated, resonates, resonator, resorting, restoking, restoring, retinoids, roentgens, rondelets, rotenones, rottenest, roundlets, rousement, sclerotin, seafronts, secretion, sectoring, senhorita, seniority, senoritas, serotinal, serotines, serotonin, serration, shortened, shortener, shortness, snakeroot, softeners, songsters, sonneteer, sortieing, southerns, sportsmen, starnoses, stationer, stereoing, sternmost, sternpost, sternsons, stonecrop, stoneware, stonework, stonewort, strenuous, strongest, strongmen, strongyle, sturgeons, sunstroke, taborines, teardowns, tenebrous, tenorists, tenorites, tensioner, terpinols, tetragons, thermions, thorniest, thornless, threnodes, tornadoes, tournedos, transpose, trecentos, tricornes, trombones, trotlines, trouncers, trudgeons, turnovers, turnsoles, turnstone, tyrosines, undermost, undershot, undertows, unstopper, unthrones, uptowners, venturous, wantoners, youngster. | |
| Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro. | |
Hexadecimal (or equivalents, 770AD-1900s) (references)54 65 6E 73 6F 72 |
| Leonardo da Vinci (1452-1519; backwards) (references)
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| American Sign Language (origins from 1620-1817 in Italy and, especially, France) (references)
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| Semaphore (1791, in France) (references)
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| Braille (1829, in France) (references)
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Morse Code (1836) (references)- . -. ... --- .-. |
| Dancing Men (Sir Arthur Conan Doyle, 1903) (references)
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Binary Code (1918-1938, probably earlier) (references)01010100 01100101 01101110 01110011 01101111 01110010 |
HTML Code (1990) (references)T e n s o r |
ISO 10646 (1991-1993) (references)0054 0065 006E 0073 006F 0072 |
| British Sign Language (Fingerspelling, BSL; 1992, British Deaf Association Dictionary of British Sign Language) (references)
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Encryption (beginner's substitution cypher): (references)547180858184 |
| 1. Definition 2. Crosswords 3. Usage: Commercial 4. Usage Frequency | 5. Expressions 6. Expressions: Internet 7. Translations: Modern 8. Translations: Ancient | 9. Derivations 10. Rhymes 11. Anagrams 12. Orthography | 13. Bibliography |
Copyright © Philip M. Parker, INSEAD. Terms of Use.
