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Carl Gustav Jacob Jacobi
Königsberg University
Thesis	Disquisitiones Analyticae de Fractionibus Simplicibus (1825)
Doctoral advisor	Enno Dirksen
Doctoral students	Paul Gordan Otto Hesse Friedrich Julius Richelot

Carl Gustav Jacob Jacobi (

books, and his first name is sometimes given as Karl.

Jacobi was the first Jewish mathematician to be appointed professor at a German university.^[3]

Biography

Jacobi was born of

In 1821 Jacobi went to study at

In 1827 he became a professor and in 1829, a tenured professor of

Königsberg University

, and held the chair until 1842.

Jacobi suffered a

.

Jacobi died in 1851 from a

is named after him.

Scientific contributions

One of Jacobi's greatest accomplishments was his theory of

and the Kepler problem (planetary motion in a central gravitational field).

He also made fundamental contributions in the study of differential equations and to classical mechanics, notably the Hamilton–Jacobi theory.

It was in algebraic development that Jacobi's particular power mainly lay, and he made important contributions of this kind in many areas of mathematics, as shown by his long list of papers in Crelle's Journal and elsewhere from 1826 onwards. He is said to have told his students that when looking for a research topic, one should 'Invert, always invert' ('man muss immer umkehren'), reflecting his belief that inverting known results can open up new fields for research, for example inverting elliptical integrals and focusing on the nature of elliptic and theta functions.^[8]

In his 1835 paper, Jacobi proved the following basic result classifying periodic (including elliptic) functions: If a univariate single-valued 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

.

The solution of the Jacobi inversion problem for the hyperelliptic Abel map by

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 ${\displaystyle g}$ algebraic curve, obtained by quotienting ${\displaystyle {\mathbf {C} }^{g}}$ by the lattice of periods is referred to as the 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 Fermat's two-square theorem and Lagrange's four-square theorem,^[9] and similar results for 6 and 8 squares. His other work in number theory continued the work of

.

He was also one of the early founders of the theory of determinants.[10] In particular, he invented the Jacobian determinant formed from the n² partial derivatives of n given functions of n independent variables, which plays an important part in changes of variables in multiple integrals, and in many analytical investigations. In 1841 he reintroduced the partial derivative ∂ notation of Legendre, which was to become standard.

He was one of the first to introduce and study the symmetric polynomials that are now known as

.

Students of

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.

Publications

See also

• Augustin-Louis Cauchy
• Friedrich Wilhelm August Argelander
• Friedrich Wilhelm Bessel
• Jacobi logarithm
• Johann Friedrich Herbart
• List of things named after Carl Gustav Jacob Jacobi

Notes

References

External links

Wikiquote has 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, University of St Andrews
• "Jacobi, Karl Gustav Jakob" . Encyclopedia Americana. 1920.
• "Jacobi, Karl Gustav Jakob" .
  New International Encyclopedia
  . 1905.
• "Jacobi, Karl Gustav Jakob" .
  The American Cyclopædia
  . 1879.
• Carl Gustav Jacob Jacobi - Œuvres complètes Gallica-Math