Karl Gustav Jacob Jacobi (Potsdam December 10, 1804 - Berlin February 18, 1851), was not only a great German mathematician but also considered by many as the most inspiring teacher of his time (Bell, p. 330)
He was born of Jewish parentage in 1804. 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 and in 1829 ordinary professor of mathematics at Königsberg, and this chair he filled until 1842. Jacobi suffered a breakdown from overwork in 1843. He then visited Italy for a few months to recruit his health. On his return he removed to Berlin, where he lived as a royal pensioner till his death.
Jacobi wrote the classic treatise (1829) on elliptic functions, 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 elliptic functions.
Jacobi was also the first mathematician to apply elliptic functions to number theory, for example, proving the polygonal number theorem of Pierre de Fermat. The Jacobi theta functions, so frequently applied in the study of hypergeometric series, were named in his honor.
His investigations in elliptic functions, the theory of which he established upon quite a new basis, and more particularly his development of the theta function, as given in his great treatise Fundamenta nova theoriae functionum ellipticarum (Königsberg, 1829), and in later papers in Crelle's Journal, constitute his grandest analytical discoveries. Second in importance only to these are his researches in differential equations, notably the theory of the last multiplier, which is fully treated in his Vorlesungen über Dynamik, edited by R. F. A. Clebsch (Berlin, 1866).
It was in analytical development that Jacobi’s peculiar power mainly lay, and he made many important contributions of this kind to other departments of mathematics, as a glance at the long list of papers that were published by him in Crelle’s Journal and elsewhere from 1826 onwards will sufficiently indicate. He was one of the early founders of the theory of determinants; in particular, he invented the functional determinant formed of the n2 differential coefficients of n given functions of n independent variables, which now bears his name (Jacobian), and which has played an important part in many analytical investigations.
In his 1835 paper, Jacobi proved the following:
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.
Jacobi reduced the general quintic equation to the form,
x5 - 10q2x = p.
Valuable also are his papers on Abelian transcendents, and his investigations in the theory of numbers, in which latter department he mainly supplements the labours of K. F. Gauss.
The 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.
He left a vast store of manuscript, portions of which have been published at intervals in Crelle's Journal. His other works include Comnienlatio 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. Perhaps his most publicized work is Hamilton-Jacobi theory in rational mechanics.
Students of vector theory often encounter the Jacobi identity. And students of differential equations often encounter the Jacobian determinant.