Editing Tensor (section)

== History ==
The concepts of later tensor analysis arose from the work of [[Carl Friedrich Gauss]] in [[differential geometry]], and the formulation was much influenced by the theory of [[algebraic form]]s and invariants developed during the middle of the nineteenth century.<ref>{{cite book
 |first=Karin |last=Reich
 |title=Die Entwicklung des Tensorkalküls
 |year=1994
 |publisher=Birkhäuser
 |isbn=978-3-7643-2814-6
 |series=Science networks historical studies |volume=11
 |oclc= 31468174
 |url={{google books |plainurl=y |id=O6lixBzbc0gC}}
}}</ref>  The word "tensor" itself was introduced in 1846 by [[William Rowan Hamilton]]<ref>{{cite journal
 |first=William Rowan |last=Hamilton
 |title=On some Extensions of Quaternions
 |url=http://www.emis.de/classics/Hamilton/ExtQuat.pdf
 |journal=[[Philosophical Magazine]]
 |year=1854–1855
 |pages=492–9, 125–137, 261–9, 46–51, 280–290
 |editor-first=David R.|editor-last=Wilkins
 |issue=7–9
 |issn=0302-7597
}}  From p. 498:  "And if we agree to call the ''square root'' (taken with a suitable sign) of this scalar product of two conjugate polynomes, P and KP, the common TENSOR of each, ...  "</ref> to describe something different from what is now meant by a tensor.<ref group=Note>Namely, the [[norm (mathematics)|norm operation]] in a vector space.</ref> Gibbs introduced [[dyadics]] and [[polyadic algebra]], which are also tensors in the modern sense.<ref name="auto">{{Cite book |last=Guo |first=Hongyu |url=https://books.google.com/books?id=5dM3EAAAQBAJ&q=array+vector+matrix+tensor |title=What Are Tensors Exactly? |date=2021-06-16 |publisher=World Scientific |isbn=978-981-12-4103-1 |language=en}}</ref> The contemporary usage was introduced by [[Woldemar Voigt]] in 1898.<ref name="Voigt1898">{{cite book|first=Woldemar |last=Voigt|title=Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung |trans-title=The fundamental physical properties of crystals in an elementary presentation |url={{google books |plainurl=y |id=QhBDAAAAIAAJ|page=20}}|year=1898|publisher=Von Veit|pages=20–|quote= Wir wollen uns deshalb nur darauf stützen, dass Zustände der geschilderten Art bei Spannungen und Dehnungen nicht starrer Körper auftreten, und sie deshalb tensorielle, die für sie charakteristischen physikalischen Grössen aber Tensoren nennen. [We therefore want [our presentation] to be based only on [the assumption that] conditions of the type described occur during stresses and strains of non-rigid bodies, and therefore call them "tensorial" but call the characteristic physical quantities for them "tensors".]}}</ref>

Tensor calculus was developed around 1890 by [[Gregorio Ricci-Curbastro]] under the title ''absolute differential calculus'', and originally presented in 1892.<ref>{{cite journal
 |first=G. |last=Ricci Curbastro
 |title=Résumé de quelques travaux sur les systèmes variables de fonctions associés à une forme différentielle quadratique
 |url={{google books |plainurl=y |id=1bGdAQAACAAJ}}
 |journal=Bulletin des Sciences Mathématiques
 |volume=2
 |pages=167–189
 |year=1892
 |issue=16
}}</ref>  It was made accessible to many mathematicians by the publication of Ricci-Curbastro and [[Tullio Levi-Civita]]'s 1900 classic text ''Méthodes de calcul différentiel absolu et leurs applications'' (Methods of absolute differential calculus and their applications).{{sfn|Ricci|Levi-Civita|1900}} In Ricci's notation, he refers to "systems" with covariant and contravariant components, which are known as tensor fields in the modern sense.<ref name="auto"/>

In the 20th century, the subject came to be known as ''tensor analysis'', and achieved broader acceptance with the introduction of [[Albert Einstein]]'s theory of [[general relativity]], around 1915.  General relativity is formulated completely in the language of tensors.  Einstein had learned about them, with great difficulty, from the geometer [[Marcel Grossmann]].<ref>{{cite book
 |first=Abraham |last=Pais
 |title=Subtle Is the Lord: The Science and the Life of Albert Einstein
 |publisher=Oxford University Press
 |year=2005
 |isbn=978-0-19-280672-7
 |url={{google books |plainurl=y |id=U2mO4nUunuwC}}
}}</ref>  Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis.  The correspondence lasted 1915–17, and was characterized by mutual respect:

{{blockquote|I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot.|Albert Einstein<ref name="Goodstein">{{cite journal
 |last=Goodstein|first=Judith R.|author-link=Judith R. Goodstein
 |title = The Italian Mathematicians of Relativity
 |journal = Centaurus
 |volume = 26
 |doi = 10.1111/j.1600-0498.1982.tb00665.x
 |pages = 241–261
 |year = 1982
 |bibcode = 1982Cent...26..241G
 |issue = 3
}}</ref>}}

Tensors and [[tensor field]]s were also found to be useful in other fields such as [[continuum mechanics]].  Some well-known examples of tensors in [[differential geometry]] are [[quadratic form]]s such as [[metric tensor]]s, and the [[Riemann curvature tensor]].  The [[exterior algebra]] of [[Hermann Grassmann]], from the middle of the nineteenth century, is itself a tensor theory, and highly geometric, but it was some time before it was seen, with the theory of [[differential form]]s, as naturally unified with tensor calculus.  The work of [[Élie Cartan]] made differential forms one of the basic kinds of tensors used in mathematics, and [[Hassler Whitney]] popularized the [[tensor product]].<ref name="auto"/>

From about the 1920s onwards, it was realised that tensors play a basic role in [[algebraic topology]] (for example in the [[Künneth theorem]]).<ref name="Spanier2012">{{cite book|first=Edwin H. |last=Spanier|title=Algebraic Topology|url={{google books |plainurl=y |id=iKx3BQAAQBAJ&|page=227}}|date=2012|publisher=Springer |isbn=978-1-4684-9322-1|pages=227|quote=the  Künneth formula expressing the homology of the tensor product...}}</ref> Correspondingly there are types of tensors at work in many branches of [[abstract algebra]], particularly in [[homological algebra]] and [[representation theory]].  Multilinear algebra can be developed in greater generality than for scalars coming from a [[field (mathematics)|field]].  For example, scalars can come from a [[ring (mathematics)|ring]].  But the theory is then less geometric and computations more technical and less algorithmic.<ref name="Hungerford2003">{{cite book|first=Thomas W. |last=Hungerford|author-link=Thomas W. Hungerford|title=Algebra|url={{google books |plainurl=y |id=t6N_tOQhafoC|page=168 }}|date=2003|publisher=Springer |isbn=978-0-387-90518-1|page=168 |quote=...the classification (up to isomorphism) of modules over an arbitrary ring is quite difficult...}}</ref>  Tensors are generalized within [[category theory]] by means of the concept of [[monoidal category]], from the 1960s.<ref name="MacLane2013">{{cite book|first=Saunders |last=MacLane|author-link=Saunders Mac Lane|title=Categories for the Working Mathematician|url={{google books |plainurl=y |id=6KPSBwAAQBAJ|page=4}}|date=2013|publisher=Springer |isbn=978-1-4612-9839-7|quote=...for example the monoid M ... in the category of abelian groups, × is replaced by the usual tensor product...|page=4}}</ref>