Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

who made fundamental contributions to

elliptic function In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are ...

s, dynamics, differential equations, determinants and

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

Biography

Jacobi was born of

Ashkenazi Jew Ashkenazi Jews ( ; also known as Ashkenazic Jews or Ashkenazim) form a distinct subgroup of the Jewish diaspora, that Ethnogenesis, emerged in the Holy Roman Empire around the end of the first millennium Common era, CE. They traditionally spe ...

ish parentage in

Potsdam Potsdam () is the capital and largest city of the Germany, German States of Germany, state of Brandenburg. It is part of the Berlin/Brandenburg Metropolitan Region. Potsdam sits on the Havel, River Havel, a tributary of the Elbe, downstream of B ...

on 10 December 1804. He was the second of four children of a banker, Simon Jacobi. His elder brother, Moritz, would also become known later as an engineer and physicist. He was initially home schooled by his uncle Lehman, who instructed him in the classical languages and elements of mathematics. In 1816, the twelve-year-old Jacobi went to the Potsdam Gymnasium, where students were taught all the standard subjects: classical languages, history, philology, mathematics, sciences, etc. As a result of the good education he had received from his uncle, as well as his own remarkable abilities, after less than half a year Jacobi was moved to the senior year despite his young age. However, as the University would not accept students younger than 16 years old, he had to remain in the senior class until 1821. He used this time to advance his knowledge, showing interest in all subjects, including Latin, Greek, philology, history and mathematics. During this period he also made his first attempts at research, trying to solve the

quintic equation In mathematics, a quintic function is a function of the form :g(x)=ax^5+bx^4+cx^3+dx^2+ex+f,\, where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other word ...

by radicals. In 1821 Jacobi went to study at Berlin University, where he initially divided his attention between his passions for

philology Philology () is the study of language in Oral tradition, oral and writing, written historical sources. It is the intersection of textual criticism, literary criticism, history, and linguistics with strong ties to etymology. Philology is also de ...

and

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

. In philology, he participated in the seminars of Böckh, drawing the professor's attention with his talent. Jacobi did not follow a lot of mathematics classes at the time, finding the level of mathematics taught at Berlin University too elementary. He continued, instead, with his private study of the more advanced works of Euler,

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLaplace. By 1823 he understood that he needed to make a decision between his competing interests and chose to devote all his attention to mathematics. In the same year he became qualified to teach secondary school and was offered a position at the Joachimsthal Gymnasium in Berlin. Jacobi decided instead to continue to work towards a university position. In 1825, he obtained the degree of Doctor of Philosophy with a dissertation on the partial fraction decomposition of rational fractions defended before a commission led by Enno Dirksen. He followed immediately with his

habilitation Habilitation is the highest university degree, or the procedure by which it is achieved, in Germany, France, Italy, Poland and some other European and non-English-speaking countries. The candidate fulfills a university's set criteria of excelle ...

and at the same time converted to Christianity. Now qualifying for teaching university classes, the 21-year-old Jacobi lectured in 1825/26 on the theory of

curves A curve is a geometrical object in mathematics. Curve(s) may also refer to: Arts, entertainment, and media Music * Curve (band), an English alternative rock music group * Curve (album), ''Curve'' (album), a 2012 album by Our Lady Peace * Curve ( ...

and surfaces at the University of Berlin. In 1826, Jacobi became a private lecturer, in the next year an extraordinary professor, and in finally 1829, a tenured professor of

at Königsberg University, and held the chair until 1842. He suffered a breakdown from overwork in 1843. He then visited

Italy Italy, officially the Italian Republic, is a country in Southern Europe, Southern and Western Europe, Western Europe. It consists of Italian Peninsula, a peninsula that extends into the Mediterranean Sea, with the Alps on its northern land b ...

for a few months to regain his health. On his return he moved to Berlin, where he lived as a royal pensioner, apart from a very brief interim, 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, thanks to the personal intervention of

Alexander von Humboldt Friedrich Wilhelm Heinrich Alexander von Humboldt (14 September 1769 – 6 May 1859) was a German polymath, geographer, natural history, naturalist, List of explorers, explorer, and proponent of Romanticism, Romantic philosophy and Romanticism ...

. Jacobi died in 1851 from a

smallpox Smallpox was an infectious disease caused by Variola virus (often called Smallpox virus), which belongs to the genus '' Orthopoxvirus''. The last naturally occurring case was diagnosed in October 1977, and the World Health Organization (W ...

infection. His grave is preserved at a cemetery in the

Kreuzberg Kreuzberg () is a district of Berlin, Germany. It is part of the Friedrichshain-Kreuzberg borough located south of Berlin-Mitte, Mitte. During the Cold War era, it was one of the poorest areas of West Berlin, but since German reunification in ...

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 The Moon is Earth's only natural satellite. It Orbit of the Moon, orbits around Earth at Lunar distance, an average distance of (; about 30 times Earth diameter, Earth's diameter). The Moon rotation, rotates, with a rotation period (lunar ...

is named after him. Jacobi's birth name was Jacques Simon, a French-style name (his father was Simon Jacobi). Later, his name was Germanized to Carl Gustav Jacob Jacobi and published in its Latinized form as Carolus Gustavus Jacobus Jacobi. He is sometimes referred to as C. G. J. Jacobi.

Scientific contributions

One of Jacobi's greatest accomplishments was his theory of

s and their relation to the elliptic theta function. This was developed in his great treatise '' Fundamenta nova theoriae functionum ellipticarum'' (1829), and in later papers in Crelle's Journal. Theta functions are of great importance in mathematical physics because of their role in the inverse problem for periodic and quasi-periodic flows. The equations of motion are integrable in terms of Jacobi's elliptic functions in the well-known cases of the

pendulum A pendulum is a device made of a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate i ...

, the Euler top, the symmetric Lagrange top in a

gravitational field In physics, a gravitational field or gravitational acceleration field is a vector field used to explain the influences that a body extends into the space around itself. A gravitational field is used to explain gravitational phenomena, such as ...

, 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 Classical mechanics is a Theoretical physics, physical theory describing the motion of objects such as projectiles, parts of Machine (mechanical), machinery, spacecraft, planets, stars, and galaxies. The development of classical mechanics inv ...

, 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' (German original: ''"man muss immer umkehren"''), reflecting his belief that inverting known results can open up new fields for research, for example inverting

elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising i ...

s and focusing on the nature of elliptic and theta functions. In his 1835 paper, Jacobi proved the following basic result classifying periodic (including elliptic) functions: He discovered many of the fundamental properties of theta functions, including the functional equation and the Jacobi triple product formula, as well as many other results on q-series and hypergeometric series. The solution of the Jacobi inversion problem for the hyperelliptic Abel map by

Weierstrass Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the " father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school t ...

in 1854 required the introduction of the hyperelliptic theta function and later the general Riemann theta function for

algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...

s of arbitrary

genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

. The complex torus associated to a genus

g

algebraic curve, obtained by quotienting

^g

by the lattice of periods is referred to as the

Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelia ...

. 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 Jacobi2

Jacobi was the first to apply elliptic functions to

, for example proving Fermat's two-square theorem and

Lagrange's four-square theorem Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number, nonnegative integer can be represented as a sum of four non-negative integer square number, squares. That is, the squares form an additive basi ...

, and similar results for 6 and 8 squares. His other work in number theory continued the work of

Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...

: new proofs of

quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...

, 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. In particular, he invented the Jacobian determinant formed from the ''n''² 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 In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). P ...

∂ 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 Plücker relations for

Grassmannian In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a ...

s. Students of vector fields,

Lie theory In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact (mathematics), contact of spheres that have come to be called Lie theory. For instance, ...

Hamiltonian mechanics In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...

and

operator algebra In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study o ...

s often encounter the Jacobi identity, the analog of

associativity In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...

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 Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to ...

, 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 Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Humboldt ...

(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

* * * * * * * * *

Notes

References

Citations

Sources

* * * * * * * * * *

External links

Jacobi's ''Vorlesungen über Dynamik''
* * * *
Carl Gustav Jacob Jacobi - Œuvres complètes
Gallica-Math {{DEFAULTSORT:Jacobi, Carl Gustav Jacob 1804 births 1851 deaths 19th-century German mathematicians Mathematicians from the Kingdom of Prussia Corresponding members of the Saint Petersburg Academy of Sciences Deaths from smallpox Differential geometers Foreign members of the Royal Society 19th-century German Jews Honorary members of the Saint Petersburg Academy of Sciences Humboldt University of Berlin alumni Members of the Prussian Academy of Sciences Members of the Royal Swedish Academy of Sciences German number theorists People from Potsdam People from the Margraviate of Brandenburg German people of the Revolutions of 1848 Recipients of the Pour le Mérite (civil class) Academic staff of the University of Königsberg