Jan A. Schouten | |
|---|---|
 J. A. Schouten, 1938–39 | |
| Born | (1883-08-28)28 August 1883 |
| Died | 20 January 1971(1971-01-20) (aged 87) |
| Alma mater | Delft University of Technology |
| Known for | Schouten tensor Schouten–Nijenhuis bracket Weyl–Schouten theorem |
| Scientific career | |
| Fields | Mathematics |
| Institutions | Leiden University |
| Doctoral advisor | Jacob Cardinaal [nl] |
| Doctoral students | Johannes Haantjes [de] Albert Nijenhuis Dirk Struik |
Jan Arnoldus Schouten (28 August 1883 – 20 January 1971) was a Dutchmathematician and Professor at theDelft University of Technology. He was an important contributor to the development oftensor calculus andRicci calculus, and was one of the founders of theMathematisch Centrum inAmsterdam.
Schouten was born inNieuwer-Amstel to a family of eminent shipping magnates. He attended aHogere Burger School, and later he took up studies inelectrical engineering at theDelft Polytechnical School. After graduating in 1908, he worked forSiemens inBerlin and for a public utility inRotterdam before returning to study mathematics in Delft in 1912. During his study he had become fascinated by the power and subtleties ofvector analysis. After a short while in industry, he returned to Delft to study Mathematics, where he received hisPh.D. degree in 1914 under supervision of Jacob Cardinaal with a thesis entitledGrundlagen der Vektor- und Affinoranalysis.
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 andintuitionist mathematicianL. 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. Schouten was one of the founders of theMathematisch Centrum inAmsterdam.
Among his PhD candidates students were Johanna Manders (1919),Dirk Struik (1922), Johannes Haantjes (1933), Wouter van der Kulk (1945), andAlbert Nijenhuis (1952).[1]
In 1933 Schouten became member of theRoyal Netherlands Academy of Arts and Sciences.[2]
Schouten died in 1971 inEpe. His sonJan Frederik Schouten (1910-1980) was Professor at the Eindhoven University of Technology from 1958 to 1978.
Schouten's dissertation applied his "direct analysis", modeled on the vector analysis ofJosiah Willard Gibbs andOliver Heaviside, to higher order tensor-like entities he calledaffinors. The symmetrical subset of affinors were tensors in the physicists' sense ofWoldemar Voigt.
Entities such asaxiators,perversors, anddeviators appear in this analysis. Just as vector analysis hasdot products andcross products, 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." (Karin 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."
In 1906,L. E. J. Brouwer was the firstmathematician to consider theparallel transport of avector for the case of a space ofconstant curvature.[3][4] In 1917,Tullio Levi-Civita pointed out its importance for the case of ahypersurfaceimmersed in aEuclidean space, i.e., for the case of aRiemannian manifold immersed in a "larger" ambient space.[5] In 1918, independently of Levi-Civita, Schouten obtained analogous results.[6] In the same year,Hermann Weyl generalized Levi-Civita's results.[7][8] Schouten's derivation is generalized to many dimensions rather than just two, and Schouten's proofs are completely 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 colleagueL. E. J. Brouwer took sides against Schouten. Once Schouten became aware ofRicci's and Levi-Civita's work, he embraced their simpler and more widely accepted notation. WithDavid van Dantzig, Schouten also developed what is now known as aKähler manifold two years beforeErich Kähler.[9][10] Again he did not receive full recognition for this discovery.
Schouten's name appears in various mathematical entities and theorems, such as theSchouten tensor, theSchouten bracket and theWeyl–Schouten theorem.
He wroteDer Ricci-Kalkül in 1922 surveying the field of tensor analysis.
In 1931 he wrote a treatise ontensors anddifferential geometry. The second volume, on applications to differential geometry, was authored by his studentDirk Jan Struik.
Schouten collaborated withÉlie Cartan on two articles as well as with many other eminent mathematicians such asKentaro 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 ofRicci-Kalkül and this was translated into English asRicci Calculus. This covers everything that Schouten considered of value in tensor analysis. This included work onLie groups and other topics and that had been much developed since the first edition.
Later Schouten wroteTensor Analysis for Physicists attempting to present the subtleties of various aspects of tensor calculus for mathematically inclined physicists. It includedPaul Dirac's matrix calculus. He still used part of his earlier affinor terminology.
Schouten, like Weyl and Cartan, was stimulated byAlbert 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.)
Following is a list of works by Schouten.
Quotations related toJan Arnoldus Schouten at Wikiquote