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, structure, space, models, and change.
History
On ...
who made fundamental contributions to
elliptic function
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s,
determinants
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
, and
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
. His name is occasionally written as Carolus Gustavus Iacobus Iacobi in his
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
books, and his first name is sometimes given as Karl.
Jacobi was the first
Jewish
Jews ( he, יְהוּדִים, , ) or Jewish people are an ethnoreligious group and nation originating from the Israelites Israelite origins and kingdom: "The first act in the long drama of Jewish history is the age of the Israelites""The ...
mathematician to be appointed professor at a German university.
Biography
Jacobi was born of
Ashkenazi Jew
Ashkenazi Jews ( ; he, יְהוּדֵי אַשְׁכְּנַז, translit=Yehudei Ashkenaz, ; yi, אַשכּנזישע ייִדן, Ashkenazishe Yidn), also known as Ashkenazic Jews or ''Ashkenazim'',, Ashkenazi Hebrew pronunciation: , singu ...
ish parentage in
Potsdam
Potsdam () is the capital and, with around 183,000 inhabitants, largest city of the German state of Brandenburg. It is part of the Berlin/Brandenburg Metropolitan Region. Potsdam sits on the River Havel, a tributary of the Elbe, downstream of B ...
on 10 December 1804. He was the second of four children of banker Simon Jacobi. His elder brother
Moritz von Jacobi
Moritz Hermann or Boris Semyonovich (von) Jacobi (russian: Борис Семёнович Якоби; 21 September 1801, Potsdam – 10 March 1874, Saint Petersburg) was a Prussian and Russian Imperial engineer and physicist of Jewish descent. Jac ...
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 algebra, 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 words, a q ...
Berlin University
Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative o ...
, where he initially divided his attention between his passions for
philology
Philology () is the study of language in oral and writing, written historical sources; it is the intersection of textual criticism, literary criticism, history, and linguistics (with especially strong ties to etymology). Philology is also defin ...
and
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. In
philology
Philology () is the study of language in oral and writing, written historical sources; it is the intersection of textual criticism, literary criticism, history, and linguistics (with especially strong ties to etymology). Philology is also defin ...
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 University, as the low level of mathematics at the University of Berlin at the time rendered them too elementary for him. However, he continued with his private study of the more advanced works of
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
,
Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLaplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
. 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
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
of
rational fraction
In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are \frac and \frac. Algebraic fractions are subject to the same laws as arithmetic fractions.
A rationa ...
s defended before a commission led by
Enno Dirksen
Enno Dirksen (3 January 1788 – 16 July 1850) was a German mathematician.
Early life
He was born in Bedekaspel, Germany. Between 1803 and 1807, he obtained private lessons in mathematics, physics, astronomy and navigation from a teacher at ...
. He followed immediately with his
Habilitation
Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including a ...
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), a 2012 album by Our Lady Peace
* "Curve" (song), a 20 ...
and
surfaces
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space.
Surface or surfaces may also refer to:
Mathematics
*Surface (mathematics), a generalization of a plane which needs not be flat
*Surf ...
at the University of Berlin.
In 1827 Jacobi became a professor and in 1829, a tenured professor of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
at
Königsberg University
Königsberg (, ) was the historic Prussian city that is now Kaliningrad, Russia. Königsberg was founded in 1255 on the site of the ancient Old Prussian settlement ''Twangste'' by the Teutonic Knights during the Northern Crusades, and was named ...
, and held the chair until 1842.
He suffered a
breakdown
Breakdown may refer to:
Breaking down
*Breakdown (vehicle), failure of a motor vehicle in such a way that it cannot be operated
*Chemical decomposition, also called chemical breakdown, the breakdown of a substance into simpler components
*Decompo ...
from overwork in 1843. He then visited
Italy
Italy ( it, Italia ), officially the Italian Republic, ) or the Republic of Italy, is a country in Southern Europe. It is located in the middle of the Mediterranean Sea, and its territory largely coincides with the homonymous geographical re ...
for a few months to regain his health. On his return he moved to Berlin, where he lived as a royal pensioner until his death. During the
Revolution of 1848
The Revolutions of 1848, known in some countries as the Springtime of the Peoples or the Springtime of Nations, were a series of political upheavals throughout Europe starting in 1848. It remains the most widespread revolutionary wave in Europea ...
Jacobi was politically involved and unsuccessfully presented his parliamentary candidature on behalf of a
Liberal
Liberal or liberalism may refer to:
Politics
* a supporter of liberalism
** Liberalism by country
* an adherent of a Liberal Party
* Liberalism (international relations)
* Sexually liberal feminism
* Social liberalism
Arts, entertainment and m ...
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. In 1836, he had been elected a foreign member of the
Royal Swedish Academy of Sciences
The Royal Swedish Academy of Sciences ( sv, Kungliga Vetenskapsakademien) is one of the Swedish Royal Academies, royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special ...
.
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 (WHO) c ...
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 Mitte. During the Cold War era, it was one of the poorest areas of West Berlin, but since German reunification in 1990 it ha ...
Johann Encke
Johann Franz Encke (; 23 September 179126 August 1865) was a German astronomer. Among his activities, he worked on the calculation of the periods of comets and asteroids, measured the distance from the Earth to the Sun, and made observations ...
, the astronomer. The crater
Jacobi Jacobi may refer to:
* People with the surname Jacobi (surname), Jacobi
Mathematics:
* Jacobi sum, a type of character sum
* Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations
* Jacobi eigenva ...
on the
Moon
The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
is named after him.
Scientific contributions
One of Jacobi's greatest accomplishments was his theory of
elliptic function
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those in ...
s and their relation to the elliptic
theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...
Crelle's Journal
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics'').
History
The journal was founded by Augus ...
. 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
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
are
integrable
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
in terms of
Jacobi's elliptic functions In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While t ...
in the well-known cases of the
pendulum
A pendulum is 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 it back toward the ...
, the
Euler top
In classical mechanics, the precession of a rigid body such as a spinning top under the influence of gravity is not, in general, an integrable problem. There are however three (or four) famous cases that are integrable, the Euler, the Lagrange, ...
, the symmetric
Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiagravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
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 physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, 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.
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
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
. ''
He discovered many of the fundamental properties of theta functions, including the functional equation and the
Jacobi triple product
In mathematics, the Jacobi triple product is the mathematical identity:
:\prod_^\infty
\left( 1 - x^\right)
\left( 1 + x^ y^2\right)
\left( 1 +\frac\right)
= \sum_^\infty x^ y^,
for complex numbers ''x'' and ''y'', with , ''x'', < 1 and ''y ...
formula, as well as many other results on
q-series
In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product
(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^),
with (a;q)_0 = 1.
It is a ''q''-analog of the Pochhammer symb ...
and hypergeometric series.
The solution of the Jacobi inversion problem for the hyperelliptic Abel map by
Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university without a degree, ...
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
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 abelian vari ...
. This method of inversion, and its subsequent extension by
Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university without a degree, ...
and
Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
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
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777 ...
Lagrange's four-square theorem
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares. That is, the squares form an additive basis of order four.
p = a_0^2 + a_1^2 + a_2^2 + a_ ...
, and similar results for 6 and 8 squares.
His other work in number theory continued the work of C. F. Gauss: 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 introduction of the
Jacobi symbol
Jacobi symbol for various ''k'' (along top) and ''n'' (along left side). Only are shown, since due to rule (2) below any other ''k'' can be reduced modulo ''n''. Quadratic residues are highlighted in yellow — note that no entry with a ...
; contributions to higher reciprocity laws, investigations of
continued fractions
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer pa ...
, and the invention of
Jacobi sum
In mathematics, a Jacobi sum is a type of character sum formed with Dirichlet characters. Simple examples would be Jacobi sums ''J''(''χ'', ''ψ'') for Dirichlet characters ''χ'', ''ψ'' modulo a prime number ''p'', defined by
: J(\chi,\psi) = ...
s.
He was also one of the early founders of the theory of determinants. In particular, he invented the
Jacobian determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...
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). Part ...
∂ 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 polynomial
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in ''n'' variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In re ...
s, giving the so-called bialternant formula for these, which is a special case of the
Weyl character formula
In mathematics, the Weyl character formula in representation theory describes the character theory, characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related fo ...
, and deriving the Jacobi–Trudi identities. He also discovered the Desnanot–Jacobi formula for determinants, which underlie the Plucker relations for
Grassmannians
In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective ...
Lie theory
In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is L ...
,
Hamiltonian mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
and
operator algebras
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 of ...
often encounter the
Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the asso ...
, the analog of associativity for the
Lie bracket
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...
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
In celestial mechanics, Jacobi's integral (also known as the Jacobi integral or Jacobi constant) is the only known conserved quantity for the circular restricted three-body problem.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 Berlin. ...
(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
In mathematics, the ''Canon arithmeticus'' is a table of indices and powers with respect to primitive roots for prime powers less than 1000, originally published by . The tables were at one time used for arithmetical calculations modulo prime pow ...
'' (1839), and ''Opuscula mathematica'' (1846–1857). His ''Gesammelte Werke'' (1881–1891) were published by the Berlin Academy.
Publications
*
*
*
*
*
*
*
*
*
See also
*
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...
*
Friedrich Wilhelm August Argelander
Friedrich Wilhelm August Argelander (22 March 1799 – 17 February 1875) was a German astronomer. He is known for his determinations of stellar brightnesses, positions, and distances.
Life and work
Argelander was born in Memel in the Kingd ...
*
Friedrich Wilhelm Bessel
Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist, and geodesist. He was the first astronomer who determined reliable values for the distance from the sun to another star by the method ...
*
Jacobi logarithm
Zech logarithms are used to implement addition in finite fields when elements are represented as powers of a generator \alpha.
Zech logarithms are named after Julius Zech, and are also called Jacobi logarithms, after Carl Gustav Jacob Jacobi, Car ...
*
Johann Friedrich Herbart
Johann Friedrich Herbart (; 4 May 1776 – 14 August 1841) was a German philosopher, psychologist and founder of pedagogy as an academic discipline.
Herbart is now remembered amongst the post-Kantian philosophers mostly as making the greatest c ...