Carl Gustav Jacob Jacobi

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"Carl Jacobi" redirects here. For the American author, see Carl Richard Jacobi.
Carl Gustav Jacob Jacobi
Königsberg University
ThesisDisquisitiones Analyticae de Fractionibus Simplicibus (1825)
Doctoral advisorEnno Dirksen
Doctoral studentsPaul Gordan
Otto Hesse
Friedrich Julius Richelot

Carl Gustav Jacob Jacobi (

determinants, and number theory. His name is sometimes given as Karl Gustav Jakob.[3]

Biography

Jacobi was born of

radicals.[4][5]

In 1821 Jacobi went to study at

curves and surfaces at the University of Berlin.[6][7]

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

Königsberg University
, and held the chair until 1842. He suffered a
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, thanks to the personal intervention of Alexander von Humboldt
.

Jacobi died in 1851 from a

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

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 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.[3] He is said to have told his students that when looking for a research topic, one should 'Invert, always invert' (German original: "man muss immer umkehren"), reflecting his belief that inverting known results can open up new fields for research, for example inverting elliptic 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

hypergeometric series
.

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.

Carl Gustav Jacob Jacobi

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, and similar results for 6 and 8 squares.

His other work in number theory continued the work of Gauss: new proofs of quadratic reciprocity, and the 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.[9] 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.[3] 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 Schur polynomials, giving the so-called bialternant formula for these, which is a special case of the Weyl character formula, and deriving the Jacobi–Trudi identities. He also discovered the Desnanot–Jacobi formula for determinants, which underlie the Plucker relations for Grassmannians.

Students of

vector fields, Lie theory, Hamiltonian mechanics and operator algebras 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, 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).[3]

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.[3]

Publications

See also

References

Citations

  1. ^ Aldrich 2017.
  2. ^ "Jacobi, Carl Gustav Jacob". Random House Webster's Unabridged Dictionary.
  3. ^ a b c d e f Chisholm 1911.
  4. ^ Koenigsberger 1904.
  5. ^ Pierpont 1906, pp. 261–262.
  6. ^ a b Dirichlet 1855, pp. 193–217.
  7. ^ James 2002, pp. 69–74.
  8. ^ Van Vleck 1916, pp. 1–13.
  9. ^ Jacobi 1841, pp. 285–318.

Sources

External links

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