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'''Carl Gustav Jacob Jacobi''' (10 December 1804 – 18 February 1851) was a [[Germany|German]] [[mathematician]], who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory. His name is occasionally written as '''Carolo Gustavo Iacobo Iacobi''' in his Latin books, and his first name is sometimes given as Karl. |
'''Carl Gustav Jacob Jacobi''' (10 December 1804 – 18 February 1851) was a [[Germany|German]] [[mathematician]], who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory. His name is occasionally written as '''Carolo Gustavo Iacobo Iacobi''' in his Latin books, and his first name is sometimes given as Karl. |
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Jacobi was the first [[Jewish]] mathematician to be appointed professor at a German university. <ref>[http://www.haaretz.com/weekend/week-s-end/setting-the-record-straight-about-jewish-mathematicians-in-nazi-germany-1.397629 Setting the record straight about Jewish mathematicians in Nazi Germany, [[Haaretz]]]</ref> |
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==Biography== |
==Biography== |
Revision as of 11:36, 1 November 2012
Carl Jacobi | |
---|---|
Born | |
Died | 18 February 1851 | (aged 46)
Nationality | German |
Alma mater | University of Berlin |
Known for | Jacobi's elliptic functions Jacobian Jacobi symbol Jacobi identity Jacobi operator |
Scientific career | |
Fields | Mathematician |
Institutions | Königsberg University |
Doctoral advisor | Enno Dirksen |
Doctoral students | Paul Gordan Otto Hesse, Friedrich Julius Richelot |
Carl Gustav Jacob Jacobi (10 December 1804 – 18 February 1851) was a German mathematician, who made fundamental contributions to elliptic functions, dynamics, differential equations, and number theory. His name is occasionally written as Carolo Gustavo Iacobo Iacobi in his Latin books, and his first name is sometimes given as Karl.
Jacobi was the first Jewish mathematician to be appointed professor at a German university. [1]
Biography
Jacobi was born of Ashkenazi Jewish parentage in Potsdam. From 1816 to 1821 Jacobi went to the Victoria-Gymnasium, where he went to the senior classes right from the beginning, but still had to stay for several years. He studied at Berlin University, where he obtained the degree of Doctor of Philosophy in 1825, his thesis being an analytical discussion of the theory of fractions. In 1827 he became a professor and in 1829, a tenured professor of mathematics at Königsberg University, and held the chair until 1842.
Jacobi suffered a breakdown from overwork in 1843. He then visited Italy for a few months to regain his health. On his return he moved to Berlin, where he lived as a royal pensioner until his death. During the Revolution of 1848 Jacobi was politically involved and unsuccessfully presented his parliamentary candidature on behalf of a Liberal club. This led, after the suppression of the revolution, to his royal grant being cut off – but his fame and reputation were such that it was soon resumed. In 1836, he had been elected a foreign member of the Royal Swedish Academy of Sciences.
Jacobi's grave is preserved at a cemetery in the Kreuzberg section of Berlin, the Friedhof I der Dreifaltigkeits-Kirchengemeinde (61 Baruther Street). His grave is close to that of Johann Encke, the astronomer. The crater Jacobi on the Moon is named after him.
Scientific contributions
One of Jacobi's greatest accomplishments was his theory of elliptic functions and their relation to the elliptic theta function. This was developed in his great treatise Fundamenta nova theoriae functionum ellipticarum (1829), and in later papers in Crelle's Journal. Theta functions are of great importance in mathematical physics because of their role in the inverse problem for periodic and quasi-periodic flows. The equations of motion are integrable in terms of Jacobi's elliptic functions in the well-known cases of the pendulum, the Euler top, the symmetric Lagrange top in a gravitational field and the Kepler problem (planetary motion in a central gravitational field).
He also made fundamental contributions in the study of differential equations and to rational mechanics, notably the Hamilton–Jacobi theory.
It was in algebraic development that Jacobi’s peculiar power mainly lay, and he made important contributions of this kind to many areas of mathematics, as shown by his long list of papers in Crelle’s Journal and elsewhere from 1826 onwards. One of his maxims was: 'Invert, always invert' ('man muss immer umkehren'), expressing his belief that the solution of many hard problems can be clarified by re-expressing them in inverse form.
In his 1835 paper, Jacobi proved the following basic result classifying periodic (including elliptic) functions: If a univariate single-value function is multiply periodic, then such a function cannot have more than two periods, and the ratio of the periods cannot be a real number. He discovered many of the fundamental properties of theta functions, including the functional equation and the Jacobi triple product formula, as well as many other results on q-series and hypergeometric series[disambiguation needed].
The solution of the Jacobi inversion problem for the hyperelliptic Abel map by Weierstrass in 1854 required the introduction of the hyperelliptic theta function and later the general Riemann theta function for algebraic curves of arbitrary genus. The complex torus associated to a genus algebraic curve, obtained by quotienting by the lattice of periods is referred to as the Jacobian variety. This method of inversion, and its subsequent extension by Weierstrass and Riemann to arbitrary algebraic curves, may be seen as a higher genus generalization of the relation between elliptic integrals and the Jacobi, or Weierstrass elliptic functions
Jacobi was the first to apply elliptic functions to number theory, for example proving of Fermat's two-square theorem and Lagrange's four-square theorem, and similar results for 6 and 8 squares. His other work in number theory continued the work of C. F. Gauss: new proofs of quadratic reciprocity and introduction of the Jacobi symbol; contributions to higher reciprocity laws, investigations of continued fractions, and the invention of Jacobi sums.
He was also one of the early founders of the theory of determinants; in particular, he invented the Jacobian determinant formed from the n² differential coefficients of n given functions of n independent variables, and which has played an important part in many analytical investigations. In 1841 he reintroduced the partial derivative ∂ notation of Legendre, which was to become standard.
Students of vector fields and Lie theory often encounter the Jacobi identity, the analog of associativity for the Lie bracket[disambiguation needed] operation.
Planetary theory and other particular dynamical problems likewise occupied his attention from time to time. While contributing to celestial mechanics, he introduced the Jacobi integral (1836) for a sidereal coordinate system. His theory of the last multiplier is treated in Vorlesungen über Dynamik, edited by Alfred Clebsch (1866).
He left many manuscripts, portions of which have been published at intervals in Crelle's Journal. His other works include Commentatio de transformatione integralis duplicis indefiniti in formam simpliciorem (1832), Canon arithmeticus (1839), and Opuscula mathematica (1846–1857). His Gesammelte Werke (1881–1891) were published by the Berlin Academy.
Family
He was a brother of the German engineer and physicist Moritz Hermann von Jacobi.[2]
See also
- Abel-Jacobi theorem, a statement about the Jacobian variety of a curve
- Jacobi's formula
- Jacobi field
- Jacobi polynomials
- Carathéodory–Jacobi–Lie theorem
- Jacobi method
- Jacobi eigenvalue algorithm
- Jacobi rotation
- Jacobi form
- Jacobian matrix and determinant
Notes
- ^ Setting the record straight about Jewish mathematicians in Nazi Germany, Haaretz
- ^ Gilman, D. C.; Peck, H. T.; Colby, F. M., eds. (1905). New International Encyclopedia (1st ed.). New York: Dodd, Mead.
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Publications
- Jacobi, C. G. J. (1829), Fundamenta nova theoriae functionum ellipticarum (in Latin), Königsberg, ISBN 978-1-108-05200-9, Reprinted by Cambridge University Press 2012
- Jacobi, C. G. J. (1969) [1881], Gesammelte Werke, Herausgegeben auf Veranlassung der Königlich Preussischen Akademie der Wissenschaften, vol. I–VIII (2nd ed.), New York: Chelsea Publishing Co., MR0260557
- Jacobi, Carl Gustav Jacob (1839), Canon arithmeticus, sive tabulae quibus exhibentur pro singulis numeris primis vel primorum potestatibus infra 1000 numeri ad datos indices et indices ad datos numeros pertinentes, Berlin: Typis Academicis, Berolini, MR0081559
- Jacobi, Carl Gustav Jacob (1996) [1848], Vorlesungen über analytische Mechanik, Dokumente zur Geschichte der Mathematik [Documents on the History of Mathematics], vol. 8, Freiburg: Deutsche Mathematiker Vereinigung, doi:10.1007/978-3-322-80289-7, ISBN 978-3-528-06692-5, MR1414679
- Jacobi, Carl Gustav Jacob (2007) [1836], Vorlesungen über Zahlentheorie---Wintersemester 1836/37, Königsberg, Algorismus. Studien zur Geschichte der Mathematik und der Naturwissenschaften [Algorismus. Studies in the History of Mathematics and the Natural Sciences], vol. 62, Dr. Erwin Rauner Verlag, Augsburg, ISBN 978-3-936905-25-0, MR2573816
- Jacobi, Carl Gustav Jacob (2009) [1866], Clebsch, A.; Balagangadharan, K.; Banerjee, Biswarup (eds.), Jacobi's lectures on dynamics, Texts and Readings in Mathematics, vol. 51, New Delhi: Hindustan Book Agency, ISBN 9788185931913, MR2569315
- Jacobi, Carl Gustav Jacob (2009) [1866], Ollivier, François; Cohn, Sigismund; Borchardt, C. W.; Clebsch, A. (eds.), "The reduction to normal form of a non-normal system of differential equations", Applicable Algebra in Engineering, Communication and Computing, Translation of De aequationum differentialium systemate non normali ad formam normalem revocando, 20 (1): 33–64, doi:10.1007/s00200-009-0088-2, ISSN 0938-1279, MR2496660
- Jacobi, Carl Gustav Jacob (2009) [1865], Ollivier, François; Cohn, Sigismund; Borchardt., C. W. (eds.), "Looking for the order of a system of arbitrary ordinary differential equations", Applicable Algebra in Engineering, Communication and Computing, Translation of De investigando ordine systematis æquationibus differentialium vulgarium cujuscunque, 20 (1): 7–32, doi:10.1007/s00200-009-0087-3, ISSN 0938-1279, MR2496659
References
- Temple Bell, Eric (1937). Men of Mathematics. New York: Simon and Schuster.
- Moritz Cantor (1905), "Jacobi, Carl Gustav Jacob", Allgemeine Deutsche Biographie (in German), vol. 50, Leipzig: Duncker & Humblot, pp. 598–602
- Dirichlet, P. G. Lejeune (1855), "Gedächtnißrede auf Carl Gustav Jacob Jacobi", Journal für die reine und angewandte Mathematik, 52: 193–217, ISSN 0075-4102, MR1104895
- public domain: Chisholm, Hugh, ed. (1911). "Jacobi, Karl Gustav Jacob". Encyclopædia Britannica (11th ed.). Cambridge University Press. This article incorporates text from a publication now in the
- Koenigsberger, Leo (1904), Carl Gustav Jacob Jacobi, Festschrift zur Feier der hundertsten Wiederkehr seines Geburtstages (in German), Leipzig, B.G. Teubner, Review by Pierpont
- Christoph J. Scriba (1974), "Jacobi, Carl Gustav Jacob", Neue Deutsche Biographie (in German), vol. 10, Berlin: Duncker & Humblot, pp. 233–234; (full text online)
External links
- Jacobi's Vorlesungen über Dynamik
- O'Connor, John J.; Robertson, Edmund F., "Carl Gustav Jacob Jacobi", MacTutor History of Mathematics Archive, University of St Andrews
- Rines, George Edwin, ed. (1920). Encyclopedia Americana.
{{cite encyclopedia}}
: Missing or empty|title=
(help) - Gilman, D. C.; Peck, H. T.; Colby, F. M., eds. (1905). New International Encyclopedia (1st ed.). New York: Dodd, Mead.
{{cite encyclopedia}}
: Missing or empty|title=
(help) - Ripley, George; Dana, Charles A., eds. (1879). The American Cyclopædia.
{{cite encyclopedia}}
: Missing or empty|title=
(help) - Carl Gustav Jacob Jacobi - Œuvres complètes Gallica-Math
- Articles with links needing disambiguation from June 2011
- Articles with links needing disambiguation from October 2012
- 1804 births
- 1851 deaths
- People from Potsdam
- 19th-century mathematicians
- Differential geometers
- German mathematicians
- German Jews
- Number theorists
- People from the Margraviate of Brandenburg
- Humboldt University of Berlin alumni
- University of Königsberg faculty
- Members of the Prussian Academy of Sciences
- Members of the Royal Swedish Academy of Sciences
- Foreign Members of the Royal Society
- Corresponding Members of the St Petersburg Academy of Sciences
- Honorary Members of the St Petersburg Academy of Sciences
- People of the Revolutions of 1848