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 ...

, and was one of the founders of the Mathematisch Centrum in

Amsterdam Amsterdam ( , , , lit. ''The Dam on the River Amstel'') is the capital and most populous city of the Netherlands, with The Hague being the seat of government. It has a population of 907,976 within the city proper, 1,558,755 in the urban ar ...

Biography

Schouten was born in Nieuwer-Amstel 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'', ''E ...

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 constitu ...

and for a public utility in

Rotterdam Rotterdam ( , , , lit. ''The Dam on the River Rotte (river), Rotte'') is the second largest List of cities in the Netherlands by province, city and List of municipalities of the Netherlands, municipality in the Netherlands. It is in the Prov ...

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 differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...

. After a short while in industry, he returned to Delft to study Mathematics, where he received his Ph.D. 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 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 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 rena ...

in Amsterdam in early 1954, and gave the opening address. Schouten was one of the founders of the Mathematisch Centrum in

. 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 Albert Nijenhuis (November 21, 1926 – February 13, 2015) was a Dutch-American mathematician who specialized in differential geometry and the theory of deformations in algebra and geometry, and later worked in combinatorics. His high school s ...

(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 (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 develope ...

, 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 , , 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 alg ...

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 ...

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 asso ...

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."

Levi-Civita connection

In 1906, L. E. J. Brouwer was the first

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 on the tangent b ...

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 1917, Levi-Civita 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 Euclidea ...

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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...

, 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, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...

immersed in a "larger" ambient space. In 1918, independently of Levi-Civita, Schouten obtained analogous results. In the same year,

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 signi ...

'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 List of wars and anthropogenic disasters by death toll, one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, ...

. 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 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 Arn ...

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 ...

, 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 uncond ...

. 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 tensor ...

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 mult ...

. The second volume, on applications to differential geometry, was authored by his student Dirk Jan Struik. 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 geometr ...

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 Continental United States, primarily located in North America. It consists of 50 U.S. state, states, a Washington, D.C., ...

. 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 addi ...

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 theor ...

'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. 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 wa ...

Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...

, 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 ...

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 ...

: Teubner, 1914. * ''On the Determination of the Principle Laws of Statistical Astronomy'', Amsterdam: Kirchner, 1918.
''Der Ricci-Kalkül''

: Julius Springer, 1924. * ''Einführung in die neueren Methoden der Differentialgeometrie'', 2 vols., Gröningen: Noordhoff, 1935–8. * ''Ricci Calculus'' 2d edition thoroughly revised and enlarged,

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, 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

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