NationMaster - Encyclopedia: Bernhard Riemann

Georg Friedrich Bernhard Riemann (pronounced REE mahn or in IPA: ['ri:man]; September 17, 1826 – July 20, 1866) was a German mathematician who made important contributions to analysis and differential geometry, some of them paving the way for the later development of general relativity. Download high resolution version (903x986, 119 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (903x986, 119 KB) Wikipedia does not have an article with this exact name. ... Articles with similar titles include the NATO phonetic alphabet, which has also informally been called the â€œInternational Phonetic Alphabetâ€. For information on how to read IPA transcriptions of English words, see IPA chart for English. ... September 17 is the 260th day of the year (261st in leap years) in the Gregorian calendar. ... The oldest surviving photograph, NicÃ©phore NiÃ©pce, 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). ... July 20 is the 201st day (202nd in leap years) of the year in the Gregorian calendar, with 164 days remaining. ... 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. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

Influence

Riemann was arguably the most influential mathematician of the middle of the nineteenth century. His published works are a small volume only, but opened up research areas combining analysis with geometry.

These would subsequently be major parts of the theories of Riemannian geometry, algebraic geometry and complex manifold theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly Adolf Hurwitz. This area of mathematics was foundational in topology, and in the twenty-first century is still being applied in novel ways to mathematical physics. In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... Felix Christian Klein (April 25, 1849, DÃ¼sseldorf, Germany â€“ June 22, 1925, GÃ¶ttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ... Adolf Hurwitz Adolf Hurwitz (26 March 1859- 18 November 1919) was a German mathematician, and one of the most important figures in mathematics in the second half of the nineteenth century (according to Jean-Pierre Serre, always something good in Hurwitz). He was born in a Jewish family in Hildesheim... A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories1. ...

Riemann worked in real analysis, where he is also a major figure. Besides defining the Riemann integral, by means of Riemann sums, he developed a theory of trigonometric series that are not Fourier series, a first step in generalized function theory, and studied the Riemann-Liouville differintegral. Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ... In mathematics, a Riemann sum is a method for approximating the values of integrals. ... In mathematics, a Fourier series of a periodic function, named in honor of Joseph Fourier (1768-1830), represents the function as a sum of periodic functions of the form where e is Eulers number and i the imaginary unit. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ... In mathematics, generalized functions are objects generalizing the notion of functions. ... In mathematics, the combined differentiation/integration operator used in fractional calculus is called the differintegral, and it has a few different forms which are all equivalent, provided that they are initialized (used) properly. ...

He made some of the most famous contributions to modern analytic number theory. In a single short paper (the only one he published on the subject of number theory), he introduced the Riemann zeta function and established its importance for understanding the distribution of prime numbers. He made a series of conjectures about properties of the zeta function, one of which is the well-known Riemann hypothesis. Analytic number theory is the branch of number theory that uses methods from mathematical analysis. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... In mathematics, a prime number, or prime for short, is a natural number greater than one and whose only distinct positive divisors are 1 and itself. ... Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always Â½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ...

He applied the Dirichlet principle from variational calculus to great effect; this was later seen to be a powerful heuristic, rather than a rigorous method, and its justification took at least a generation. His work on monodromy and the hypergeometric function in the complex domain made a great impression, and established a basic way of working with functions, by consideration only of their singularities. In mathematics, Dirichlets principle in potential theory states that the harmonic function on a domain with boundary condition on can be obtained as the minimizer of the Dirichlet integral amongst all functions such that on , provided only that there exists one such function making the Dirichlet integral finite. ... Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... Look up Heuristic in Wiktionary, the free dictionary. ... In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and differential geometry behave as they run round a singularity. ... In mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients an/an-1 is a rational function of n. ... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ...

Biography

Early life

Riemann was born in Breselenz, a village near Dannenberg in the Kingdom of Hanover in what is today Germany. His father Friedrich Bernhard Riemann was a poor Lutheran pastor in Breselenz. Friedrich Riemann fought in the Napoleonic Wars. Georg's mother also died before her children were grown. Bernhard was the second of six children. He was a shy boy and suffered from numerous nervous breakdowns. From a very young age, Riemann exhibited his exceptional skills, such as fantastic calculation abilities, but suffered from timidity and had a fear of speaking in public. Jameln is a municipality in the district LÃ¼chow-Dannenberg, in Lower Saxony, Germany. ... Dannenberg is a town and a municipality in the district LÃ¼chow-Dannenberg, in Lower Saxony, Germany. ... Capital Hanover Head of State King of Hanover Hanover (German: ) was a historical territory in todays Germany, at various times a principality, an electorate of the Holy Roman Empire, a kingdom and a province of Prussia and of Germany. ... Lutheranism describes those churches within Christianity that were reformed according to the theological insights of Martin Luther in the 16th century. ... Combatants Austria[1] Portugal Prussia[1] Russia[2] Spain[3] Sweden United Kingdom[4] Ottoman Empire[5] Holy Roman Empire[6] French Empire Holland Kingdom of Italy Kingdom of Naples Duchy of Warsaw Bavaria[7] Saxony[8] Denmark [9] Commanders Archduke Charles Prince Schwarzenberg Karl Mack von Leiberich Gebhard von...

Middle life

In high school, Riemann studied the Bible intensively. But his mind often drifted back to mathematics and he even tried to prove mathematically the correctness of the book of Genesis. His teachers were amazed by his genius and by his ability to solve extremely complicated mathematical operations. He often outstripped his instructor's knowledge. In 1840 Bernhard went to Hanover to live with his grandmother and visit the Lyceum. After the death of his grandmother in 1842 he went to the Johanneum in Lüneburg. In 1846, at the age of 19, he started studying philology and theology, in order to become a priest and help with his family's finances. This Gutenberg Bible is displayed by the United States Library of Congress. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ... Genesis (Hebrew: , Greek: Î“ÎÎ½ÎµÏƒÎ¹Ï‚, meaning birth, creation, cause, beginning, source or origin) is the first book of the Torah, the Tanakh, and the Old Testament. ... 1840 is a leap year starting on Wednesday (link will take you to calendar). ... Hanover (German: , IPA: ), on the river Leine, is the capital of the federal state of Lower Saxony (Niedersachsen), Germany. ... 1842 was a common year starting on Saturday (see link for calendar). ... LÃ¼neburg (English: Lunenburg) is a city in Lower Saxony, Germany, about 50km southeast of Hamburg. ... 1846 was a common year starting on Thursday (see link for calendar). ... Philology, etymologically, is the love of words. ... At Wikiversity you can learn more and teach others about Theology at: The School of Theology Theology finds its scholars pursuing the understanding of and providing reasoned discourse of religion, spirituality and God or the gods. ...

In 1847 his father, after scraping together enough money to send Riemann to university, allowed him to stop studying theology and start studying mathematics. He was sent to the renowned University of Göttingen, where he first met Carl Friedrich Gauss, and attended his lectures on the method of least squares. 1847 was a common year starting on Friday (see link for calendar). ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... The Georg-August University of GÃ¶ttingen (Georg-August-UniversitÃ¤t GÃ¶ttingen, often called the Georgia Augusta) was founded in 1734 by George II, King of Great Britain and Elector of Hanover, and opened in 1737. ... Johann Carl Friedrich Gauss or GauÃŸ ( ; Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy, and optics. ... Least squares is a mathematical optimization technique that attempts to find a best fit to a set of data by attempting to minimize the sum of the squares of the differences (called residuals) between the fitted function and the data. ...

In 1847 he moved to Berlin, where Jacobi, Dirichlet and Steiner were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849. 1847 was a common year starting on Friday (see link for calendar). ... Location of Berlin within Germany / EU Coordinates Time zone CET/CEST (UTC+1/+2) Administration Country NUTS Region DE3 City subdivisions 12 boroughs Governing Mayor Klaus Wowereit (SPD) Governing parties SPD / Left. ... 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. ... Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 â€“ May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ... Jakob Steiner (18 March 1796 â€“ April 1, 1863) was a Swiss mathematician. ... 1849 was a common year starting on Monday (see link for calendar). ...

Later life

Riemann held his first lectures in 1854, which not only founded the field of Riemannian geometry but set the stage for Einstein's general relativity. There was an unsuccessful attempt to promote Riemann to extraordinary professor status at the University of Göttingen in 1857, but from that attempt Riemann was finally granted a regular salary. In 1859, following Dirichlet's death he was promoted to head the Mathematics department at Göttingen. He was also the first to propose the theory of higher dimensions^{[citation needed]}, which highly simplified the laws of physics. In 1862 he married Elise Koch and had a daughter. He died of tuberculosis on his third journey to Italy in Selasca (now a hamlet of Ghiffa on Lake Maggiore). 1854 (MDCCCLIV) was a common year starting on Sunday (see link for calendar). ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... â€œEinsteinâ€ redirects here. ... General relativity (GR) (aka general theory of relativity (GTR)) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... The Georg-August University of GÃ¶ttingen (Georg-August-UniversitÃ¤t GÃ¶ttingen, often called the Georgia Augusta) was founded in 1734 by George II, King of Great Britain and Elector of Hanover, and opened in 1737. ... 1857 was a common year starting on Thursday (see link for calendar). ... Year 1859 (MDCCCLIX) was a common year starting on Saturday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Thursday of the 12-day slower Julian calendar). ... Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 â€“ May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ... Higher dimension in mathematics refers to any number of dimensions greater than three. ... 1862 was a common year starting on Wednesday (see link for calendar). ... Tuberculosis (abbreviated as TB for Tubercle Bacillus) is a common and deadly infectious disease that is caused by mycobacteria, primarily Mycobacterium tuberculosis. ... Country Italy Region Piedmont Province Province of Verbano-Cusio-Ossola (VB) Mayor Elevation 201 m Area 13. ... Lake Maggiore (in Italian: Lago Maggiore or lago Verbano) is the most westerly of the three large prealpine lakes of Europe and the second largest after Lake Garda. ...

Euclidean geometry versus Riemannian geometry

Gauss asked his student Riemann in 1853 to prepare a Habilitationsschrift on the foundations of geometry. Over many months, Riemann developed his theory of higher dimensions. When he finally delivered his lecture in 1854, the mathematical public received it with enthusiasm. Image File history File links Hypercube. ... Image File history File links Hypercube. ... A square A projection of a cube (into a two-dimensional image) A projection of a hypercube (into a two-dimensional image) In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). ... Habilitation is a term used within the university system in Germany, Austria, and some other European countries such as the German-speaking part of Switzerland, in Poland, the Czech Republic, Slovakia, Hungary, Slovenia, Russia, and other countries of former Soviet Union, such as Armenia, Azerbaijan, Moldova, Kirgizstan, Kazakhstan, Uzbekistan, etc. ... Higher dimension in mathematics refers to any number of dimensions greater than three. ...

The subject founded by this work is Riemannian geometry. Riemann had found the correct way to extend into n dimensions the differential geometry of surfaces, for which Gauss himself had proved his theorema egregium. The fundamental object is what is now called the Riemann curvature tensor. For the surface case, this can be reduced to a number (scalar), positive, negative or zero, the non-zero and constant cases being models of the known non-Euclidean geometries. In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... The Theorema Egregium (Remarkable Theorem) is an important theorem of Carl Friedrich Gauss concerning the curvature of surfaces. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ...

Higher dimensions

Riemann's idea was to introduce a collection of numbers at every point in space that would describe how much it was bent or curved. Riemann found that in four spatial dimensions, one needs a collection of ten numbers at each point to describe the properties of a manifold, no matter how distorted it is. This is the famous metric tensor. Space has been an interest for philosophers and scientists for much of human history. ... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...

See also

Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always Â½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ... In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ... In mathematics, a Riemann sum is a method for approximating the values of integrals. ... The Riemann-Lebesgue lemma states that the integral of a function like the above is small. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... The Riemann mapping theorem in complex analysis states the following: if U is a simply connected open subset of the complex number plane C which is not all of C, then there exists a bijective holomorphic conformal map f : U -> D, where D = { z in C : |z| < 1 } denotes the... The twenty-first problem of the 23 Hilbert problems, from the celebrated list put forth in 1900 by David Hilbert, was phrased like this (English translation from 1902). ... In mathematics, the Riemann-Hurwitz formula describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. ... In mathematics, the Riemann-von Mangoldt formula, named for Bernhard Riemann and Hans Carl Friedrich von Mangoldt, states that the number N(T) of zeros of the Riemann zeta function with imaginary part greater than 0 and less than or equal to T satisfies The formula was stated by Riemann... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... In mathematics, specifically in complex analysis and algebraic geometry, the Riemann-Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. ... In mathematics, theta functions are special functions of several complex variables. ... In mathematics, the Riemann-Siegel theta function is the function It has an asymptotic formula which is strongly convergent for The interest in the Riemann-Siegel theta function is in studying the Riemann zeta function and defining the Z function. ... In mathematics, Riemanns differential equation is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0,1, and âˆž. // Definition The differential equation is given by The regular singular points are a, b and c. ... In mathematics, the Schottky problem is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties. ... A rendering of the Riemann Sphere. ... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... In mathematics, the Cauchy-Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, are two partial differential equations which provide a necessary but not sufficient condition for a function to be holomorphic. ... In mathematics, the Hirzebruch-Riemann-Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruchs 1954 result contributing to the Riemann-Roch problem for complex algebraic varieties of all dimensions. ... The Riemann-Lebesgue lemma states that the integral of a function like the above is small. ... In mathematics, the Riemann-Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. ... In mathematics, the combined differentiation/integration operator used in fractional calculus is called the differintegral, and it has a few different forms which are all equivalent, provided that they are initialized (used) properly. ... In mathematics, the Riemann series theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series is conditionally convergent, then its terms can be arranged in a permutation so that the series converges to any given value, or even diverges. ... On the Number of Primes Less Than a Given Magnitude (or Über die Anzahl der Primzahlen unter einer gegebenen Grösse) is a seminal 8-page paper by Bernhard Riemann published in the November 1859 edition of the Monthly Reports of the Berlin Academy. ... Prime Obsession is a historical book on mathematics by John Derbyshire, detailing the history of the Riemann hypothesis and some of its applications. ...

Writings in English

Bibliography

John Derbyshire (born June 3, 1945) is a British-born author who lives in the United States and became a naturalized citizen in 2002. ... Prime Obsession is a historical book on mathematics by John Derbyshire, detailing the history of the Riemann hypothesis and some of its applications. ...

External links

Categories: Articles with unsourced statements since February 2007 | All articles with unsourced statements | 1826 births | 1866 deaths | Deaths by tuberculosis | German mathematicians | 19th century mathematicians | Differential geometers | People from Lower Saxony | Christian people Image File history File links Commons-logo. ... The Wikimedia Commons (also called Wikicommons) is a repository of free content images, sound and other multimedia files. ... The MacTutor history of mathematics archive is a website hosted by University of St Andrews in Scotland. ... ScienceWorld, also known as Eric Weissteins World of Science, is a web site that opened to the general public in January 2002. ...

Contents

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