Carl Gustav Jacob Jacobi (December 10, 1804 - February 18, 1851) was a German mathematician, widely considered as the most inspiring teacher of his time (Bell, p. 330), and by many one of the greatest algorists and mathematicians of all time.

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. See Jacobi's elliptic functions.

Jacobi was also the first mathematician to apply elliptic functions to number theory, for example, proving the 2 square and four-square theorems of Pierre de Fermat. He also proved similar results for 6 and 8 squares. The Jacobi theta functions, so frequently applied in the study of hypergeometric series, were named in his honor.

He proved the functional equation for the theta function.

He proved the Jacobi triple product formula and many other results in q-series.

He gave new proofs of quadratic reciprocity, made contributions to higher reciprocity laws, investigated continued fractions and invented Jacobi sums.

In 1841 he reintroduced the partial derivative ∂ notation of Legendre, which was to become standard.

Karl Gustav Jacob Jacobi. 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 (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 Alfred Clebsch (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 manuscripts, 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, those studying differential equations often encounter the Jacobian determinant, and those working in number theory and cryptography use the Jacobi symbol.

The phrase 'Invert, always invert,' is associated with Jacobi for he believed that it is in the nature of things that many hard problems are best solved when they are addressed backwards.

Source: Wikipedia