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Symmetry Of Second Derivatives
In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) is the fact that exchanging the order of partial derivatives of a multivariate function :f\left(x_1,\, x_2,\, \ldots,\, x_n\right) does not change the result if some continuity conditions are satisfied (see below); that is, the second-order partial derivatives satisfy the identities :\frac \left( \frac \right) \ = \ \frac \left( \frac \right). In other words, the matrix of the second-order partial derivatives, known as the Hessian matrix, is a symmetric matrix. Sufficient conditions for the symmetry to hold are given by Schwarz's theorem, also called Clairaut's theorem or Young's theorem. In the context of partial differential equations, it is called the Schwarz integrability condition. Formal expressions of symmetry In symbols, the symmetry may be expressed as: :\frac \left( \frac \right) \ = \ \frac \left( \frac \right) \qquad\text\qquad \frac \ = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Function
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; that is, a function of class C^k is a function that has a th derivative that is continuous in its domain. A function of class C^\infty or C^\infty-function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous). Generally, the term smooth function refers to a C^-function. However, it may also mean "sufficiently differentiable" for the problem under consideration. Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Matthieu Hermann Laurent
Paul Matthieu Hermann Laurent (2 September 1841 in Luxembourg City – 19 February 1908 in Paris, France) was a French mathematician. Despite his large body of works, Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansio ... expansions for complex functions were ''not'' named after him, but after Pierre Alphonse Laurent. Publications * * * * * * References External links * * * (not born in Echternach) 1841 births 1908 deaths People from Echternach 19th-century French mathematicians 20th-century French mathematicians French mathematical analysts {{France-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''. Biography Jordan was born in Lyon and educated at the École polytechnique. He was an engineer by profession; later in life he taught at the École polytechnique and the Collège de France, where he had a reputation for eccentric choices of notation. He is remembered now by name in a number of results: * The Jordan curve theorem, a topological result required in complex analysis * The Jordan normal form and the Jordan matrix in linear algebra * In mathematical analysis, Jordan measure (or ''Jordan content'') is an area measure that predates measure theory * In group theory, the Jordan–Hölder theorem on composition series is a basic result. * Jordan's theorem on finite linear groups Jordan's work did much to bring Galois theory into the mainstream. He also investigated the Mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ulisse Dini
Ulisse Dini (14 November 1845 – 28 October 1918) was an Italian mathematician and politician, born in Pisa. He is known for his contributions to real analysis, partly collected in his book "''Fondamenti per la teorica delle funzioni di variabili reali''". Life and academic career Dini attended the Scuola Normale Superiore in order to become a teacher. One of his professors was Enrico Betti. In 1865, a scholarship enabled him to visit Paris, where he studied under Charles Hermite as well as Joseph Bertrand, and published several papers. In 1866, he was appointed to the University of Pisa, where he taught algebra and geodesy. In 1871, he succeeded Betti as professor for Mathematical analysis, analysis and geometry. From 1888 until 1890, Dini was ''rettore'' of the Pisa University, and of the ''Scuola Normale Superiore'' from 1908 until his death in 1918. He was also active as a politician: in 1871 he was voted into the Pisa city council and in 1880 became a member of the Parlia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorenz Leonard Lindelöf
Lorenz Leonard Lindelöf (13 November 1827, Karvia, Finland – 3 March 1908, Helsinki) was a Finnish mathematician, astronomer and politician. Biography Lindelöf grew up in a poor family. He learned German and French and studied astronomy and mathematics at the University of Helsinki. He initially specialized in astronomy at the graduate level and was at Pulkovo Observatory in 1855–1856. After his completion of his PhD (Promotierung), Lindelöf from 1857 to 1874 held the professorial chair of mathematics in Helsinki and from 1869 to 1872 was the rector of the university. He then resigned his professorial chair in favor of Mittag-Leffler and turned to politics. Lindelöf was from 1874 to 1902 minister of state education in Finland and also did actuarial work for Kaleva Mutual Insurance Company. In 1883 he was knighted and in 1888 was a member of the State Council. He was in the Finnish Parliament, served on many committees and was in 1900 District Marshal. Lindelöf publishe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Bertrand
Joseph Louis François Bertrand (; 11 March 1822 – 5 April 1900) was a French mathematician whose work emphasized number theory, differential geometry, probability theory, economics and thermodynamics. Biography Joseph Bertrand was the son of physician Alexandre Jacques François Bertrand and the brother of archaeologist Alexandre Bertrand. His father died when Joseph was only nine years old; by that time he had learned a substantial amount of mathematics and could speak Latin fluently. At eleven years old he attended the course of the École Polytechnique as an auditor. From age eleven to seventeen, he obtained two bachelor's degrees, a license and a PhD with a thesis concerning the mathematical theory of electricity, and was admitted to the 1839 entrance examination of the École Polytechnique. Bertrand was a professor at the École Polytechnique and Collège de France, and was a member of the Paris Academy of Sciences of which he was the permanent secretary for twenty-si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacques Charles François Sturm
Jacques Charles François Sturm (29 September 1803 – 15 December 1855) was a French mathematician, who made a significant addition to equation theory with his work, Sturm's theorem. Early life Sturm was born in Geneva, France in 1803. The family of his father, Jean-Henri Sturm, had emigrated from Strasbourg around 1760—about 50 years before Charles-François's birth. His mother's name was Jeanne-Louise-Henriette Gremay. In 1818, he started to follow the lectures of the academy of Geneva. The death of his father forced Sturm to give lessons to children of the rich in order to support his own family the following year. In 1823, he became tutor to the son of Madame de Staël. At the end of that year, Sturm stayed in Paris for a short time following the family of his student. He resolved, with his school-fellow Jean-Daniel Colladon, to try his fortune in Paris, and obtained employment on the ''Bulletin universel''. Discovery In 1829, he discovered the theorem that bears h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Marie Duhamel
Jean-Marie Constant Duhamel (; ; 5 February 1797 – 29 April 1872) was a French mathematician and physicist. His studies were affected by the troubles of the Napoleonic era. He went on to form his own school ''École Sainte-Barbe''. Duhamel's principle, a method of obtaining solutions to inhomogeneous linear evolution equations, is named after him. He was primarily a mathematician but did studies on the mathematics of heat, mechanics, and acoustics. He also did work in calculus using infinitesimals. Duhamel's theorem for infinitesimals says that the sum of a series of infinitesimals is unchanged by replacing the infinitesimal with its principal part. In 1853 he published about an early recording device he called a vibroscope. Like other similar devices, the vibroscope was a type of measuring device similar to an oscilloscope, and could not play back the etchings it recorded. Honours * 19617 Duhamel, asteroid An asteroid is a minor planet—an object larger than a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy
Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real analysis), pioneered the field complex analysis, and the study of permutation groups in abstract algebra. Cauchy also contributed to a number of topics in mathematical physics, notably continuum mechanics. A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated: : "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)." Cauchy was a prolific worker; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics. Biography Youth and education Cauchy was the son of Loui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaJoseph-Louis Lagrange, comte de l’Empire ''Encyclopædia Britannica'' or Giuseppe Ludovico De la Grange Tournier; 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia, was an Italian and naturalized French mathematician, physicist and astronomer. He made significant contributions to the fields of mathematical analysis, analysis, number theory, and both classical mechanics, classical and celestial mechanics. In 1766, on the recommendation of Leonhard Euler and Jean le Rond d'Alembert, d'Alembert, Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, Prussia, where he stayed for over twenty y ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
