Carl Jacobi
Carl Gustav Jacob Jacobi
Born	December 10, 1804(1804-12-10) Potsdam, Kingdom of Prussia
Died	February 18, 1851 (aged 46) Berlin, Kingdom of Prussia
Residence	Prussia
Nationality	Prussian
Fields	Mathematician
Institutions	Königsberg University
Alma mater	University of Berlin
Doctoral advisor	Enno Dirksen
Doctoral students	Paul Albert Gordan Otto Hesse
Known for	Jacobi's elliptic functions Jacobian Jacobi symbol Jacobi identity

Carl Gustav Jacob Jacobi (December 10, 1804 – February 18, 1851) was a Prussian mathematician, widely considered to be the most inspiring teacher of his time^[1] and one of the greatest mathematicians of all time.^[2]

Contents 1 Biography 2 Scientific contributions 3 See also 4 Notes 5 References 6 External links

[edit] Biography

He was born of Jewish parentage in Potsdam. 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 extraordinary professor and in 1829 ordinary professor of mathematics at Königsberg University, and this chair he filled 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 was 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.

[edit] Scientific contributions

Jacobi's greatest work was his investigation of elliptic functions, particularly his development of the theta function. Second in importance are his researches in differential equations and 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.

His theory of theta and elliptic functions is given in his great treatise Fundamenta nova theoriae functionum ellipticarum (1829), and in later papers in Crelle's Journal. In his 1835 paper, Jacobi proved the following basic result classifying periodic (including elliptic) functions:

If a univariate single-value function is periodic, then the ratio of the periods cannot be a real number, and that such a function cannot have more than two periods.

He proved the functional equation for the theta function. He proved the Jacobi triple product formula and many other results in q-series and hypergeometric series.

Theta functions are of great importance in mathematical physics because of the need to "integrate second order kinetic energy equations". The motion equations in rotational form are integrable only for the three cases of the pendulum, the symmetric top in a gravitational field, and a freely spinning body, wherein solutions are in terms of Jacobi's elliptic functions.

He applied theta functions to Abelian varieties. The solution of the Jacobi inversion problem for the hyperelliptic Abel map by Weierstrass in 1854 required the introduction of the hyperlliptic theta function and later the general Riemann theta function for algebraic curves of arbitrary genus. The complex torus associated to a genus g 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 the 2 square and four-square theorems of Pierre de Fermat, and similar results for 6 and 8 squares.

His other work in number theory continued the work of K. 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.

Jacobi was one of the early founders of the theory of determinants; in particular, he invented the Jacobian determinant formed of 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 operation.

Planetary theory and other particular dynamical problems likewise occupied his attention from time to time. While contributing to celestial mechanics, Jacobi (1836) introduced the Jacobi integral for a sidereal coordinate system. His theory of the ``last multiplier is treated in Vorlesungen über Dynamik, edited by Alfred Clebsch (1866).

He reduced the general quintic equation to the form:

He left a vast store of 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.

[edit] See also

• Jacobi's formula
• Jacobi field
• Jacobi polynomials
• Carathéodory-Jacobi-Lie theorem
• Jacobi method
• Jacobi eigenvalue algorithm
• Jacobi rotation
• Jacobi form

[edit] Notes

[edit] References

• Temple Bell, Eric (1937). Men of Mathematics. New York: Simon and Schuster. 
• Hestenes, David (1986). New Foundations of Classical Mechanics. Dordrecht: Kluwer Adademic Publishers. 
• This article incorporates text from the Encyclopædia Britannica, Eleventh Edition, a publication now in the public domain.

[edit] External links

Wikiquote has a collection of quotations related to: Carl Gustav Jacob Jacobi

• Jacobi's Vorlesungen über Dynamik
• O'Connor, John J.; Robertson, Edmund F., "Carl Gustav Jacob Jacobi", MacTutor History of Mathematics archive .

Persondata
NAME	Jacobi, Carl
ALTERNATIVE NAMES
SHORT DESCRIPTION	Mathematician
DATE OF BIRTH	December 10, 1804
PLACE OF BIRTH	Potsdam, Germany
DATE OF DEATH	February 18, 1851
PLACE OF DEATH	Berlin, Germany