Jan Arnoldus Schouten (28 August 1883 – 20 January 1971) was a Dutch
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
and Professor at the
Delft University of Technology
Delft University of Technology ( nl, Technische Universiteit Delft), also known as TU Delft, is the oldest and largest Dutch public technical university, located in Delft, Netherlands. As of 2022 it is ranked by QS World University Rankings among ...
. He was an important contributor to the development of
tensor calculus
In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).
Developed by Gregorio Ricci-Curbastro and his student Tullio Levi ...
and
Ricci calculus
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be cal ...
, and was one of the founders of the
Mathematisch Centrum
The (abbr. CWI; English: "National Research Institute for Mathematics and Computer Science") is a research centre in the field of mathematics and theoretical computer science. It is part of the institutes organization of the Dutch Research Cou ...
in
Amsterdam
Amsterdam ( , , , lit. ''The Dam on the River Amstel'') is the Capital of the Netherlands, capital and Municipalities of the Netherlands, most populous city of the Netherlands, with The Hague being the seat of government. It has a population ...
.
Biography
Schouten was born in
Nieuwer-Amstel
Amstelveen () is a municipality in the province of North Holland, Netherlands with a population of 92.353 as of 2022. It is a suburban part of the Amsterdam metropolitan area.
The municipality of Amstelveen consists of the historical villages o ...
to a family of eminent shipping magnates. He attended a
Hogere Burger School, and later he took up studies in
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
at the
Delft Polytechnical School. After graduating in 1908, he worked for
Siemens
Siemens AG ( ) is a German multinational conglomerate corporation and the largest industrial manufacturing company in Europe headquartered in Munich with branch offices abroad.
The principal divisions of the corporation are ''Industry'', '' ...
in
Berlin
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitue ...
and for a public utility in
Rotterdam
Rotterdam ( , , , lit. ''The Dam on the River Rotte'') is the second largest city and municipality in the Netherlands. It is in the province of South Holland, part of the North Sea mouth of the Rhine–Meuse–Scheldt delta, via the ''"N ...
before returning to study mathematics in Delft in 1912. During his study he had become fascinated by the power and subtleties of
vector analysis
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...
. After a short while in industry, he returned to Delft to study Mathematics, where he received his
Ph.D.
A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...
degree in 1914 under supervision of Jacob Cardinaal with a thesis entitled .
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
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fu ...
mathematician
L. E. J. Brouwer
Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and compl ...
. 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
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the Nevanlinna Prize (to be rename ...
in Amsterdam in early 1954, and gave the opening address. Schouten was one of the founders of the
Mathematisch Centrum
The (abbr. CWI; English: "National Research Institute for Mathematics and Computer Science") is a research centre in the field of mathematics and theoretical computer science. It is part of the institutes organization of the Dutch Research Cou ...
in
Amsterdam
Amsterdam ( , , , lit. ''The Dam on the River Amstel'') is the Capital of the Netherlands, capital and Municipalities of the Netherlands, most populous city of the Netherlands, with The Hague being the seat of government. It has a population ...
.
Among his PhD candidates students were Johanna Manders (1919),
Dirk Struik
Dirk Jan Struik (September 30, 1894 – October 21, 2000) was a Dutch-born American (since 1934) mathematician, historian of mathematics and Marxian theoretician who spent most of his life in the U.S.
Life
Dirk Jan Struik was born in 1 ...
(1922), Johannes Haantjes (1933), Wouter van der Kulk (1945), and
Albert Nijenhuis (1952).
In 1933 Schouten became member of the
Royal Netherlands Academy of Arts and Sciences
The Royal Netherlands Academy of Arts and Sciences ( nl, Koninklijke Nederlandse Akademie van Wetenschappen, abbreviated: KNAW) is an organization dedicated to the advancement of science and literature in the Netherlands. The academy is housed ...
.
Schouten died in 1971 in
Epe. His son
Jan Frederik Schouten
Jan Frederik Schouten (29 May 1910 – 12 August 1980) was a Dutch physicist, and Professor at the Eindhoven University of Technology, known for his contributions to biophysics. H. BoumaLevensbericht J.F. Schouten in: ''Jaarboek, 1980'', Amsterda ...
(1910-1980) was Professor at the Eindhoven University of Technology from 1958 to 1978.
Work
Schouten's dissertation applied his "direct analysis", modeled on the vector analysis of
Josiah Willard Gibbs
Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
and
Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...
, 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 (; 2 September 1850 – 13 December 1919) 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 1 ...
.
Entities such as , , and appear in this analysis. Just as vector analysis has
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
s and
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
s, so 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 Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
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
Roland Weitzenböck (26 May 1885 – 24 July 1955) was an Austrian mathematician working on differential geometry who introduced the Weitzenböck connection. He was appointed professor of mathematics at the University of Amsterdam in 1923 at the ...
wrote of "the terrible book he has committed."
Levi-Civita connection
In 1906,
L. E. J. Brouwer
Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and compl ...
was the first
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
to consider the
parallel transport
In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection (vector bundle), c ...
of a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
for the case of
a space of
constant curvature
In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature i ...
. In 1917,
Levi-Civita
Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made signific ...
pointed out its importance for the case of a
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...
immersed in a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, i.e., for the case of a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
immersed in a "larger" ambient space. In 1918, independently of Levi-Civita, Schouten obtained analogous results. In the same year,
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
generalized
Levi-Civita's results. Schouten's derivation is generalized to many dimensions rather than just two, and Schouten's proofs are completely intrinsic rather than extrinsic, unlike
Tullio Levi-Civita
Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significa ...
'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 (28 July 1914 11 November 1918), often abbreviated as WWI, was one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, the United States, and the Ottoman Empire, with fightin ...
. Schouten engaged in a losing priority dispute with Levi-Civita. Schouten's colleague
L. E. J. Brouwer
Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and compl ...
took sides against Schouten. Once Schouten became aware of
Ricci Ricci () is an Italian surname, derived from the adjective "riccio", meaning curly. Notable Riccis Arts and entertainment
* Antonio Ricci (painter) (c.1565–c.1635), Spanish Baroque painter of Italian origin
* Christina Ricci (born 1980), Ameri ...
'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
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnold ...
two years before
Erich Kähler
Erich Kähler (; 16 January 1906 – 31 May 2000) was a German mathematician with wide-ranging interests in geometry and mathematical physics, who laid important mathematical groundwork for algebraic geometry and for string theory.
Education an ...
. Again he did not receive full recognition for this discovery.
Works by Schouten
Schouten's name appears in various mathematical entities and theorems, such as the
Schouten tensor In Riemannian geometry the Schouten tensor is a second-order tensor introduced by Jan Arnoldus Schouten defined for by:
:P=\frac \left(\mathrm -\frac g\right)\, \Leftrightarrow \mathrm=(n-2) P + J g \, ,
where Ric is the Ricci tensor (defined by ...
, the
Schouten bracket and the
Weyl–Schouten theorem
In the mathematical field of differential geometry, the existence of isothermal coordinates for a ( pseudo-)Riemannian metric is often of interest. In the case of a metric on a two-dimensional space, the existence of isothermal coordinates is uncon ...
.
He wrote ''Der Ricci-Kalkül'' in 1922 surveying the field of tensor analysis.
In 1931 he wrote a treatise on
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
s and
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
. The second volume, on applications to differential geometry, was authored by his student
Dirk Jan Struik
Dirk Jan Struik (September 30, 1894 – October 21, 2000) was a Dutch-born American (since 1934) mathematician, historian of mathematics and Marxian theoretician who spent most of his life in the U.S.
Life
Dirk Jan Struik was born in ...
.
Schouten collaborated with
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
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
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...
.
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
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
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 Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
's matrix calculus. He still used part of his earlier affinor terminology.
Schouten, like Weyl and Cartan, was stimulated by
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
's theory of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and 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 (27 January 1894 – 11 January 1969) 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 diff ...
. He interacted with
Oswald Veblen
Oswald Veblen (June 24, 1880 – August 10, 1960) 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; while this was lon ...
of
Princeton University
Princeton University is a private university, private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial Colleges, fourth-oldest ins ...
, and corresponded with
Wolfgang Pauli
Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics fo ...
on spin space. (See H. Goenner, Living Review link below.)
Publications
Following is a list of works by Schouten.
''Grundlagen der Vektor- und Affinoranalysis'' Leipzig
Leipzig ( , ; Upper Saxon: ) is the most populous city in the German state of Saxony. Leipzig's population of 605,407 inhabitants (1.1 million in the larger urban zone) as of 2021 places the city as Germany's eighth most populous, as wel ...
: Teubner, 1914.
* ''On the Determination of the Principle Laws of Statistical Astronomy'', Amsterdam: Kirchner, 1918.
''Der Ricci-Kalkül'' Berlin
Berlin ( , ) is the capital and largest city of Germany by both area and population. Its 3.7 million inhabitants make it the European Union's most populous city, according to population within city limits. One of Germany's sixteen constitue ...
: Julius Springer, 1924.
* ''Einführung in die neueren Methoden der Differentialgeometrie'', 2 vols.,
Gröningen
Gröningen () 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 3.621 inhabitants (December 2015). Gröningen is part of the ''Verbandsgemein ...
: Noordhoff, 1935–8.
* ''Ricci Calculus'' 2d edition thoroughly revised and enlarged,
New York
New York most commonly refers to:
* New York City, the most populous city in the United States, located in the state of New York
* New York (state), a state in the northeastern United States
New York may also refer to:
Film and television
* '' ...
:
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, 1954.
* With W. Van der Kulk, ''Pfaff's Problem and Its Generalizations'', Clarendon Press, 1949;
2nd edn, New York: Chelsea Publishing Co., 1969.
* ''Tensor Analysis for Physicists'' 2d edn., New York: Dover Publications, 1989.
References
Further reading
*
* Karin Reich, ''History of Tensor Analysis'',
979
Year 979 ( CMLXXIX) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar.
Events
By place Byzantine Empire
* March 24 – Second Battle of Pankaleia: An Ibero-Byzantine expeditionary ...
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,
eviewDie ''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?"
External links
*
*
*
{{DEFAULTSORT:Schouten, Jan Arnoldus
1883 births
1971 deaths
20th-century Dutch mathematicians
Differential geometers
Delft University of Technology alumni
Delft University of Technology faculty
Members of the Royal Netherlands Academy of Arts and Sciences
People from Amstelveen