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Elwin Bruno Christoffel
Elwin Bruno Christoffel (; 10 November 1829 – 15 March 1900) was a German mathematician and physicist. He introduced fundamental concepts of differential geometry, opening the way for the development of tensor calculus, which would later provide the mathematical basis for general relativity. Life Christoffel was born on 10 November 1829 in Montjoie (now Monschau) in Prussia in a family of cloth merchants. He was initially educated at home in languages and mathematics, then attended the Jesuit Gymnasium and the Friedrich-Wilhelms Gymnasium in Cologne. In 1850 he went to the University of Berlin, where he studied mathematics with Gustav Dirichlet (which had a strong influence over him) among others, as well as attending courses in physics and chemistry. He received his doctorate in Berlin in 1856 for a thesis on the motion of electricity in homogeneous bodies written under the supervision of Martin Ohm, Ernst Kummer and Heinrich Gustav Magnus. After receiving his doctorat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monschau
Monschau (; french: Montjoie, ; wa, Mondjoye) is a small resort town in the Eifel region of western Germany, located in the Aachen district of North Rhine-Westphalia. Geography The town is located in the hills of the North Eifel, within the Hohes Venn – Eifel Nature Park in the narrow valley of the Rur river. The historic town center has many preserved half-timbered houses and narrow streets have remained nearly unchanged for 300 years, making the town a popular tourist attraction nowadays. An open-air, classical music festival is staged annually at Burg Monschau. Historically, the main industry of the town was cloth-mills. History On the heights above the city is Monschau castle, which dates back to the 13th century — the first mention of Monschau was made in 1198. Beginning in 1433, the castle was used as a seat of the dukes of Jülich. In 1543, Emperor Charles V besieged it as part of the Guelders Wars, captured it and plundered the town. However, the castle staye ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gregorio Ricci-Curbastro
Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician. He is most famous as the discoverer of tensor calculus. With his former student Tullio Levi-Civita, he wrote his most famous single publication, a pioneering work on the calculus of tensors, signing it as Gregorio Ricci. This appears to be the only time that Ricci-Curbastro used the shortened form of his name in a publication, and continues to cause confusion. Ricci-Curbastro also published important works in other fields, including a book on higher algebra and infinitesimal analysis, and papers on the theory of real numbers, an area in which he extended the research begun by Richard Dedekind. Early life and education Completing privately his high school studies at only 16 years of age, he enrolled on the course of philosophy-mathematics at Rome University (1869). The following year the Papal State fell and so Gregorio was called by his father to the city of his birth, Lugo di Romagna. Subsequently he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gymnasium (Germany)
''Gymnasium'' (; German plural: ''Gymnasien''), in the German education system, is the most advanced and highest of the three types of German secondary schools, the others being ''Hauptschule'' (lowest) and '' Realschule'' (middle). ''Gymnasium'' strongly emphasizes academic learning, comparable to the British sixth form system or with prep schools in the United States. A student attending ''Gymnasium'' is called a ''Gymnasiast'' (German plural: ''Gymnasiasten''). In 2009/10 there were 3,094 gymnasia in Germany, with students (about 28 percent of all precollegiate students during that period), resulting in an average student number of 800 students per school.Federal Statistical office of Germany, Fachserie 11, Reihe 1: Allgemeinbildende Schulen – Schuljahr 2009/2010, Wiesbaden 2010 Gymnasia are generally public, state-funded schools, but a number of parochial and private gymnasia also exist. In 2009/10, 11.1 percent of gymnasium students attended a private gymnasium. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kingdom Of Prussia
The Kingdom of Prussia (german: Königreich Preußen, ) was a German kingdom that constituted the state of Prussia between 1701 and 1918. Marriott, J. A. R., and Charles Grant Robertson. ''The Evolution of Prussia, the Making of an Empire''. Rev. ed. Oxford: Clarendon Press, 1946. It was the driving force behind the unification of Germany in 1871 and was the leading state of the German Empire until its dissolution in 1918. Although it took its name from the region called Prussia, it was based in the Margraviate of Brandenburg. Its capital was Berlin. The kings of Prussia were from the House of Hohenzollern. Brandenburg-Prussia, predecessor of the kingdom, became a military power under Frederick William, Elector of Brandenburg, known as "The Great Elector". As a kingdom, Prussia continued its rise to power, especially during the reign of Frederick II, more commonly known as Frederick the Great, who was the third son of Frederick William I.Horn, D. B. "The Youth of Fre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time or four-dimensional spacetime. In particular, the ' is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of second order partial differential equations. Newton's law of universal gravitation, which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Calculus
In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita, it was used by Albert Einstein to develop his general theory of relativity. Unlike the infinitesimal calculus, tensor calculus allows presentation of physics equations in a form that is independent of the choice of coordinates on the manifold. Tensor calculus has many applications in physics, engineering and computer science including elasticity, continuum mechanics, electromagnetism (see mathematical descriptions of the electromagnetic field), general relativity (see mathematics of general relativity), quantum field theory, and machine learning. Working with a main proponent of the exterior calculus Elie Cartan, the influential geometer Shiing-Shen Chern summarizes the role of tensor calculus:In our subject of differenti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate causes of phenomena, and usually frame their understanding in mathematical terms. Physicists work across a wide range of research fields, spanning all length scales: from sub-atomic and particle physics, through biological physics, to cosmological length scales encompassing the universe as a whole. The field generally includes two types of physicists: experimental physicists who specialize in the observation of natural phenomena and the development and analysis of experiments, and theoretical physicists who specialize in mathematical modeling of physical systems to rationalize, explain and predict natural phenomena. Physicists can apply their knowledge towards solving practical problems or to developing new technologies (also known as app ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarz–Christoffel Mapping
In complex analysis, a Schwarz–Christoffel mapping is a conformal map of the upper half-plane or the complex unit disk onto the interior of a simple polygon. Such a map is guaranteed to exist by the Riemann mapping theorem (stated by Bernhard Riemann in 1851); the Schwarz–Christoffel formula provides an explicit construction. They were introduced independently by Elwin Christoffel in 1867 and Hermann Schwarz in 1869. Schwarz–Christoffel mappings are used in potential theory and some of its applications, including minimal surfaces, hyperbolic art, and fluid dynamics. Definition Consider a polygon in the complex plane. The Riemann mapping theorem implies that there is a biholomorphic mapping ''f'' from the upper half-plane : \ to the interior of the polygon. The function ''f'' maps the real axis to the edges of the polygon. If the polygon has interior angles \alpha,\beta,\gamma, \ldots, then this mapping is given by : f(\zeta) = \int^\zeta \frac \,\mathrmw where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Curvature Tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is ''flat'', i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection. It is a central mathematical tool in the theory of general relativity, the modern theory of gravity, and the curvature of spacetime is in principle observable via the geodesic deviation equation. The curvature tensor represents the tidal force experienced by a rigid body moving ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christoffel–Darboux Formula
In mathematics, the Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by and . It states that : \sum_^n \frac = \frac \frac where ''f''''j''(''x'') is the ''j''th term of a set of orthogonal polynomials of squared norm ''h''''j'' and leading coefficient ''k''''j''. There is also a "confluent form" of this identity: \sum_^n \frac = \frac \left _'(x)f_(x) - f_'(x) f_(x)\right Version for Associated Legendre Polynomials In the case of associated Legendre polynomials, the relation takes the form: : \begin (\mu-\mu')\sum_^L\,(2l+1)\frac\,P_(\mu)P_(\mu')=\qquad\qquad\qquad\qquad\qquad\\\frac\big _(\mu)P_(\mu')-P_(\mu)P_(\mu')\big\end See also *Turán's inequalities In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by (and first published by ). There are many generalizations to other polynomials, often called Turán's inequalities, given by and other authors. If is ... * Sturm Chain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
