- Jan Arnoldus Schouten
**Jan Arnoldus Schouten**(28 August 1883 ,Nieuwer-Amstel –20 January 1971 ,Epe ) was a Dutchmathematician . He was an important contributor to the development oftensor calculus and was one of the founders of the Mathematisch Centrum inAmsterdam .Schouten was a descendant of eminent shipping magnates. He started in

electrical engineering , but became fascinated by the power and subtleties of vector analysis.Schouten received his Ph.D. degree from

Delft University of Technology in 1914. His dissertation, written under the direction of J. Cardinaal, applied his "direct analysis," modelled on the vector analysis ofJosiah Willard Gibbs andOliver Heaviside to higher order tensor-like entities he called "affinors." The symmetrical subset of affinors were tensors in the physicists sense ofWoldemar Voigt . Entities such asaxiators , perversors, and deviators appear in this analysis. Just as vector analysis hasdot product s andcross product s, so affinor analysis has different kinds of products for tensors of various levels. However, instead of two kinds of multiplication symbols, Schouten had at least twenty. This made the work a chore to read, although the conclusions were valid. Schouten later said in conversation withHermann Weyl that he would "like to throttle the man who wrote this book." (Karen Reich, in her history of tensor analysis, misattributes this quote to Weyl.) Weyl did, however, say that Schouten's early book has "orgies of formalism that threaten the peace of even the technical scientist." ("Space, Time Matter", p. 54).Roland Weitzenböck wrote of "the terrible book he has committed."Schouten independently discovered in 1915 what is now known as the

Levi-Civita connection . Schouten's derivation is generalized to many dimensions rather than just two, and Schouten's proofs are intrinsic rather than extrinsic, unlikeTullio Levi-Civita 's. Despite this, since Schouten's article appeared almost a year after Levi-Civita's, the latter got the credit. Schouten was unaware of Levi-Civita's work because of poor journal distribution and communication duringWorld War I . Schouten engaged in a losing priority dispute with Levi-Civita. Schouten's colleague L. E. J. Brouwer took sides against Schouten. Once Schouten became aware of Ricci's and Levi-Civita's work, he embraced their simpler and more widely accepted notation. Schouten also developed what is now known as aKähler manifold two years beforeErich Kähler .Fact|date=May 2007 Again he did not receive full recognition for this discovery.Schouten's name appears in various mathematical entities and theorems, such as the

Schouten tensor , theSchouten bracket and theWeyl-Schouten theorem .He wrote "Der Ricci Kalcul" in 1922 surveying the field of tensor analysis.

In 1931 he wrote a treatise on

tensor s anddifferential geometry . The second volume, on applications to differential geometry, was authored by his studentDirk Jan Struik .Schouten collaborated with

Élie Cartan on two articles as well as with many other eminent mathematicians such as Kentaro Yano (with whom he co-authored three papers). Through his student and co-author Dirk Struik his work influenced many mathematicians in theUnited States .In the 1950s Schouten completely rewrote and updated the German version of "Ricci-Kalkül" and this was translated into English as "Ricci Calculus". This covers everything that Schouten considered of value in tensor analysis. This included work on

Lie group s and other topics and that had been much developed since the first edition.Later Schouten wrote "Tensor Analysis for Physicists" attempting to present the subtleties of various aspects of tensor calculus for mathematically inclined physicists. It included

Paul Dirac 's matrix calculus. He still used part of his earlier affinor terminology.Schouten, like Weyl and Cartan, was stimulated by

Albert Einstein 's theory ofgeneral relativity . He co-authored a paper withAlexander Aleksandrovich Friedmann of Petersburg and another withVáclav Hlavatý . He interacted withOswald Veblen ofPrinceton University , and corresponded withWolfgang Pauli on spin space. (See H. Goenner, Living Review link below.)Schouten was an effective university administrator and leader of mathematical societies. During his tenure as professor and as institute head he was involved in various controversies with the topologist and

intuitionist mathematician L. E. J. Brouwer. He was a shrewd investor as well as mathematician and successfully managed the budget of the institute and Dutch mathematical society. He hosted theInternational Congress of Mathematicians in Amsterdam in early 1954, and gave the opening address.**Works by Schouten*** "Grundlagen der Vektor- und Affinoranalysis",

Leipzig : Teubner, 1914.

* "On the Determination of the Principle Laws of Statistical Astronomy", Amsterdam: Kirchner, 1918.

* "Der Ricci-Kalkül",Berlin : Julius Springer, 1924.

* "Einführung in die Neuen Methoden der Differentialgeometrie", 2 vols.,Gröningen : Noordhoff,1935-8.

* "Ricci Calculus" 2d edition thoroughly revised and enlarged,New York : Springer-Verlag, 1954.

* With W. Van der Kulk, "Pfaff's Problem and Its Generalizations", New York: Chelsea Publishing Co., 1969.

* "Tensor Analysis for Physicists" 2d edn., New York: Dover Publications, 1989.**Works about Schouten***

Albert Nijenhuis , "J A Schouten : A Master at Tensors", "Nieuw archief voor wiskunde" 20 (1972), 1-19.

* Karin Reich, "History of Tensor Analysis", [1979] transl. Boston: Birkhauser, 1994.

* Dirk J. Struik, "Schouten, Levi-Civita and the Emergence of Tensor Calculus," in David Rowe and John McCleary, eds., "History of Modern Mathematics", vol. 2, Boston: Academic Press, 1989. 99-105.

* Dirk J. Struik, "J A Schouten and the tensor calculus," "Nieuw Arch. Wisk." (3) 26 (1) (1978), 96-107.

* Dirk J. Struik, [review] Die "Entwicklung des Tensorkalküls. Vom absoluten Differentialkalküt zur Relativitätstheorie" , Karin Reich, "Historia Mathematica", vol 22, 1995, 323-326.

* Albert Nijenhuis, article on Schouten in "Dictionary of Scientific Biography", Charles Coulston Gillispie, ed.-in-chief, New York: Scribner, 1970-1980, 214.

* Dirk van Dalen, "Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer" 2 vols., New York: Oxford U. Press, 2001, 2005. Discusses disputes with Brouwer, such as over publication of early paper and priority to Levi-Civita and conflict over editorial board of "Compositio Mathematica".

* Hubert F. M. Goenner, Living Reviews Relativity, vol 7 (2004) Ch. 9, [*http://relativity.livingreviews.org/open?pubNo=lrr-2004-2&page=articlese9.html "Mutual Influences Among Mathematicians and Physicists?"*]**External links***

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