Jan Arnoldus Schouten
Encyclopedia
Jan Arnoldus Schouten was a Dutch
mathematician
. He was an important contributor to the development of tensor calculus and was one of the founders of the Mathematisch Centrum in Amsterdam
.
Schouten was born in Nieuwer-Amstel to a family 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 of Josiah Willard Gibbs
and Oliver Heaviside
to higher order tensor-like entities he called "affinors." The symmetrical subset of affinors were tensors in the physicists sense of Woldemar Voigt
. Entities such as axiators, perversors, and deviators appear in this analysis. Just as vector analysis has dot product
s and cross 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 with Hermann 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, unlike Tullio 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 during World 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 a Kähler manifold
two years before Erich Kähler
. 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, the Schouten bracket and the Weyl-Schouten theorem.
He wrote Der Ricci-Kalkül in 1922 surveying the field of tensor analysis.
In 1931 he wrote a treatise on tensor
s and differential geometry. The second volume, on applications to differential geometry, was authored by his student Dirk 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 the United 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 of general relativity
. He co-authored a paper with Alexander Aleksandrovich Friedmann of Petersburg and another with Václav Hlavatý
. He interacted with Oswald Veblen
of Princeton University
, and corresponded with Wolfgang 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 the International Congress of Mathematicians
in Amsterdam in early 1954, and gave the opening address. He died in Epe
.
Netherlands
The Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He was an important contributor to the development of tensor calculus and was one of the founders of the Mathematisch Centrum in Amsterdam
Amsterdam
Amsterdam is the largest city and the capital of the Netherlands. The current position of Amsterdam as capital city of the Kingdom of the Netherlands is governed by the constitution of August 24, 1815 and its successors. Amsterdam has a population of 783,364 within city limits, an urban population...
.
Schouten was born in Nieuwer-Amstel to a family of eminent shipping magnates. He started in electrical engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...
, but became fascinated by the power and subtleties of vector analysis.
Schouten received his Ph.D.
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
degree from Delft University of Technology
Delft University of Technology
Delft University of Technology , also known as TU Delft, is the largest and oldest Dutch public technical university, located in Delft, Netherlands...
in 1914. His dissertation, written under the direction of J. Cardinaal, applied his "direct analysis," modelled on the vector analysis of Josiah Willard Gibbs
Josiah Willard Gibbs
Josiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...
and Oliver Heaviside
Oliver Heaviside
Oliver Heaviside was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations , reformulated Maxwell's field equations in terms of electric and...
to higher order tensor-like entities he called "affinors." The symmetrical subset of affinors were tensors in the physicists sense of Woldemar Voigt
Woldemar Voigt
Woldemar Voigt was a German physicist, who taught at the Georg August University of Göttingen. Voigt eventually went on to head the Mathematical Physics Department at Göttingen and was succeeded in 1914 by Peter Debye, who took charge of the theoretical department of the Physical Institute...
. Entities such as axiators, perversors, and deviators appear in this analysis. Just as vector analysis has dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
s and cross product
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
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 with Hermann Weyl
Hermann Weyl
Hermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
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
Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
. Schouten's derivation is generalized to many dimensions rather than just two, and Schouten's proofs are intrinsic rather than extrinsic, unlike Tullio Levi-Civita
Tullio Levi-Civita
Tullio Levi-Civita, FRS was an Italian mathematician, most famous for his work on absolute differential calculus and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus...
'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 during World War I
World War I
World War I , which was predominantly called the World War or the Great War from its occurrence until 1939, and the First World War or World War I thereafter, was a major war centred in Europe that began on 28 July 1914 and lasted until 11 November 1918...
. Schouten engaged in a losing priority dispute with Levi-Civita. Schouten's colleague L. E. J. Brouwer
Luitzen Egbertus Jan Brouwer
Luitzen Egbertus Jan Brouwer FRS , usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis.-Biography:Early in his career,...
took sides against Schouten. Once Schouten became aware of Ricci
Gregorio Ricci-Curbastro
Gregorio Ricci-Curbastro was an Italian mathematician. He was born at Lugo di Romagna. He is most famous as the inventor of the tensor calculus but published important work in many fields....
's and Levi-Civita's work, he embraced their simpler and more widely accepted notation. Schouten also developed what is now known as a Kähler manifold
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
two years before Erich Kähler
Erich Kähler
was a German mathematician with wide-ranging geometrical interests.Kähler was born in Leipzig, and studied there. He received his Ph.D. in 1928 from the University of Leipzig. He held professorial positions in Königsberg, Leipzig, Berlin and Hamburg...
. 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, the Schouten bracket and the Weyl-Schouten theorem.
He wrote Der Ricci-Kalkül in 1922 surveying the field of tensor analysis.
In 1931 he wrote a treatise on tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
s and differential geometry. The second volume, on applications to differential geometry, was authored by his student Dirk Jan Struik
Dirk Jan Struik
Dirk Jan Struik was a Dutch mathematician and Marxian theoretician who spent most of his life in the United States.- Life :...
.
Schouten collaborated with Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
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 the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
.
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
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
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
Paul Dirac
Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...
's matrix calculus. He still used part of his earlier affinor terminology.
Schouten, like Weyl and Cartan, was stimulated by Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...
's theory of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. He co-authored a paper with Alexander Aleksandrovich Friedmann of Petersburg and another with Václav Hlavatý
Václav Hlavatý
Václav Hlavatý was a noted Czech-American mathematician, who wrote on the theory of relativity and corresponded extensively with Albert Einstein on the subject. In particular, Hlavatý solved some very difficult equations relating to Einstein's Unified field theory, which was featured in the news...
. He interacted with Oswald Veblen
Oswald Veblen
Oswald Veblen was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905.-Life:...
of Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
, and corresponded with Wolfgang Pauli
Wolfgang Pauli
Wolfgang Ernst Pauli was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after being nominated by Albert Einstein, he received the Nobel Prize in Physics for his "decisive contribution through his discovery of a new law of Nature, the exclusion principle or...
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 the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
in Amsterdam in early 1954, and gave the opening address. He died in Epe
Epe
' is a municipality and a town in the eastern Netherlands.The town has 15,593 inhabitants, whilst the municipality as a whole has a population of 32,989....
.
Works by Schouten
- Grundlagen der Vektor- und Affinoranalysis, LeipzigLeipzigLeipzig Leipzig has always been a trade city, situated during the time of the Holy Roman Empire at the intersection of the Via Regia and Via Imperii, two important trade routes. At one time, Leipzig was one of the major European centres of learning and culture in fields such as music and publishing...
: Teubner, 1914. - On the Determination of the Principle Laws of Statistical Astronomy, Amsterdam: Kirchner, 1918.
- Der Ricci-Kalkül, BerlinBerlinBerlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
: Julius Springer, 1924. - Einführung in die Neuen Methoden der Differentialgeometrie, 2 vols., GröningenGröningenGröningen is a town in the Börde district in Saxony-Anhalt, Germany. It lies approx. 40 km south-west of Magdeburg, and 10 km east of Halberstadt. It has 4,180 inhabitants . Gröningen is part of the Verbandsgemeinde Westliche Börde....
: Noordhoff,1935-8. - Ricci Calculus 2d edition thoroughly revised and enlarged, New YorkNew YorkNew York is a state in the Northeastern region of the United States. It is the nation's third most populous state. New York is bordered by New Jersey and Pennsylvania to the south, and by Connecticut, Massachusetts and Vermont to the east...
: Springer-VerlagSpringer Science+Business Media- Selected publications :* Encyclopaedia of Mathematics* Ergebnisse der Mathematik und ihrer Grenzgebiete * Graduate Texts in Mathematics * Grothendieck's Séminaire de géométrie algébrique...
, 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 NijenhuisAlbert NijenhuisAlbert Nijenhuis is a Dutch mathematician at the University of Washington and the University of Pennsylvania working on combinatorics and differential geometry, who introduced the Nijenhuis tensor , the Schouten-Nijenhuis bracket , the Frölicher-Nijenhuis bracket , and the Nijenhuis-Richardson...
, "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, "Mutual Influences Among Mathematicians and Physicists?"