Karl Gustav Jacob Jacobi

Carl Gustav Jacob Jacobi (December 10, 1804 - 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). Carl Jacobi (1804-1851), picture obviously in the public domain This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... Carl Jacobi (1804-1851), picture obviously in the public domain This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... December 10 is the 344th day (345th in leap years) of the year in the Gregorian calendar. ... 1804 was a leap year starting on Sunday (see link for calendar). ... February 18 is the 49th day of the year in the Gregorian Calendar. ... 1851 was a common year starting on Wednesday (see link for calendar). ... Leonhard Euler is considered by many people to be one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is mathematics. ...

He was born of Jewish parentage in Potsdam. 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 University, and this chair he filled until 1842. Jacobi suffered a breakdown from overwork in 1843. He then visited Italy for a few months to regain his health. On his return he removed to Berlin, where he lived as a royal pensioner until his death. Jews (Hebrew: יהודים, Yehudim) are followers of Judaism or, more generally, members of the Jewish people (also known as the Jewish nation, or the Children of Israel), an ethno-religious group descended from the ancient Israelites and converts who joined their religion. ... Potsdam is the capital city of the state of Brandenburg in Germany. ... Alternative meaning: Humboldt State University, located in Arcata, California German Humboldt-Universität zu Berlin) is the successor to Berlins oldest university, the Friedrich Wilhelm University (Friedrich-Wilhelms-Universität), founded in 1810 by the liberal Prussian educational reformer and linguist Wilhelm von Humboldt whose university model has strongly... Doctor of Philosophy, or Ph. ... 1825 was a common year starting on Saturday (see link for calendar). ... In mathematics, a fraction is a way of expressing a quantity based on an amount that is divided into a number of equal-sized parts. ... Naval Battle of Navarino by Carneray 1827 was a common year starting on Monday (see link for calendar). ... Johann Wolfgang von Goethe 1829 was a common year starting on Thursday (see link for calendar). ... A professor (Latin: one who claims publicly to be an expert) (or prof for short) is a senior teacher, lecturer and/or researcher usually employed by a college or university. ... Euclid, detail from The School of Athens by Raphael. ... The Königsberg University Albertina was opened in 1544 by Albrecht or Albert of Brandenburg Prussia, first duke of a Protestant Prussia. ... 1842 was a common year starting on Saturday (see link for calendar). ... 1843 was a common year starting on Sunday (see link for calendar). ... Berlin is the capital city and a single state of the Federal Republic of Germany. ...

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. Johann Wolfgang von Goethe 1829 was a common year starting on Thursday (see link for calendar). ... In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ... In mathematical physics, Jacobis integral represents a solution to the circular restricted three-body problem of celestial mechanics. ... Simple gravity pendulum assumes no air resistance and no friction of/at the nail/screw. ... The gravitational field is a field that causes bodies with mass to attract each other. ...

Jacobi was also the first mathematician to apply elliptic functions to number theory, for example, proving the 2 square and 4 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. Leonhard Euler is considered by many people to be one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is mathematics. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, Pierre de Fermats theorem on sums of two squares states that an odd prime number p is expressible as with x and y integers, if and only if For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and... Lagranges four-square theorem, also known as Bachets conjecture, was proved in 1770 by Joseph Louis Lagrange. ... Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, southern France, and a mathematician who is given credit for the development of modern calculus. ... In mathematics, theta functions are special functions of several complex variables. ... In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. ...

He proved the functional equation for the theta function.

He proved the Jacobi triple product formula and many other results in q-series. In mathematics, the Jacobi triple product is a relation that re-expresses the Jacobi theta function, normally written as a series, as a product. ... In mathematics, a q-series, also sometimes called a q-shifted factorial, is defined as It is usually considered first as a formal power series; it is also an analytic function of q, in the unit disc. ...

He gave new proofs of quadratic reciprocity, made contributions to higher reciprocity laws, investigated continued fractions and invented Jacobi sums. In mathematics, in number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. ... In mathematics, a continued fraction is an expression such as where a0 is some integer and all the other numbers an are positive integers. ... In mathematics, a Jacobi sum is a type of character sum formed with one or more Dirichlet characters. ...

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). In mathematics, theta functions are special functions of several complex variables. ... Johann Wolfgang von Goethe 1829 was a common year starting on Thursday (see link for calendar). ... Crelles Journal, or just Crelle, is the common name for the Journal für die reine und angewandte Mathematik founded by August Leopold Crelle. ... 1866 (MDCCCLXVI) is a common year starting on Monday of the Gregorian calendar or a common year starting on Wednesday of the 12-day-slower Julian calendar. ...

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. The oldest surviving photograph, Nicéphore Niépce, circa 1826 1826 (MDCCCXXVI) was a common year starting on Sunday (see link for calendar) of the Gregorian calendar (or a common year starting on Tuesday of the 12-day-slower Julian calendar). ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

In his 1835 paper, Jacobi proved the following: | Come and take it, slogan of the Texas Revolution 1835 was a common year starting on Thursday (see link for calendar). ...

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, In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. ...

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. For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (Gauß) (April 30, 1777 - February 23, 1855) was a legendary German mathematician, astronomer and physicist with a very wide range of contributions; he is considered to be one of the greatest mathematicians of all time. ...

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. Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. ... Charles Darwin 1836 was a leap year starting on Friday (see link for calendar). ... In mathematical physics, Jacobis integral represents a solution to the circular restricted three-body problem of celestial mechanics. ...

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 ever. 1832 was a leap year starting on Sunday (see link for calendar). ... 1839 was a common year starting on Tuesday (see link for calendar). ... 1846 was a common year starting on Thursday (see link for calendar). ... 1857 was a common year starting on Thursday (see link for calendar). ... 1881 (MDCCCLXXXI) was a common year starting on Saturday (see link for calendar). ... 1891 (MDCCCXCI) was a common year starting on Thursday (see link for calendar). ... In physics and mathematics, the Hamilton-Jacobi equation (HJE) is a particular canonical transformation of the classical Hamiltonian which results in a first order, non-linear differential equation whose solution describes the behavior of the system. ... Classical mechanics is a model of the physics of forces acting upon bodies. ...

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. In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. ... Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... The German Lorenz cipher machine Cryptography or cryptology is a field of mathematics and computer science concerned with information security and related issues, particularly encryption and authentication. ... The Jacobi symbol is used by mathematicians in the area of number theory. ...

Jacobi crater, on the Moon, is named after him. Jacobi is a lunar crater that is located in the southern highlands on the near side of the Moon. ... Bulk composition of the moons mantle and crust estimated, weight percent Oxygen 42. ...

See also

• Jacobian
• Jacobi identity
• Jacobi's formula
• Jacobi symbol
• Jacobi integral
• Jacobi polynomials
• Carathéodory-Jacobi-Lie theorem
• Jacobi method

In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ... In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. ... In matrix calculus, Jacobis formula expresses the differential of the determinant of a matrix A in terms of the adjugate of A and the differential of A. The formula is It is named after the mathematician C.G.J. Jacobi. ... The Jacobi symbol is used by mathematicians in the area of number theory. ... In mathematical physics, Jacobis integral represents a solution to the circular restricted three-body problem of celestial mechanics. ... In mathematics, Jacobi polynomials are a class of orthogonal polynomials. ... The Carathéodory-Jacobi-Lie theorem is a theorem in symplectic topology which generalizes Darbouxs theorem. ... The Jacobi method is an algorithm in linear algebra for determining the solutions of a system of linear equations with largest absolute values in each row and column dominated by the diagonal element. ...

References

• Men of Mathematics, Eric Temple Bell, Simon and Schuster, New York, 1937.
  
• New Foundations of Classical Mechanics, David Hestenes, Kluwer Adademic Publishers, Dordrecht, 1986.
• This article incorporates text from the Encyclopædia Britannica Eleventh Edition, a publication now in the public domain.

Encyclopædia Britannica, the 11th edition The Encyclopædia Britannica Eleventh Edition (1910–1911) is perhaps the most famous edition of the Encyclopædia Britannica. ... The public domain comprises the body of all creative works and other knowledge—writing, artwork, music, science, inventions, and others—in which no person or organization has any proprietary interest. ...

External link

• John J. O'Connor and Edmund F. Robertson. Carl Gustav Jakob Jacobi at the MacTutor History of Mathematics archive.

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Carl Gustav Jakob Jacobi