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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 doctorate, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monschau
Monschau (; , ; ) 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. 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 stayed with Jülich until 1609, when it became part of Palatinate-Neuburg. In 1795, the French capt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christoffel–Darboux Formula
In mathematics, the Christoffel–Darboux formula or Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by and . There is also a "confluent form" of this identity by taking y\to x limit: Proof Specific cases Hermite The Hermite polynomials are orthogonal with respect to the gaussian distribution. The H polynomials are orthogonal with respect to \frac e^, and with k_n = 2^n.\sum_^n \frac = \frac\,\frac.The He polynomials are orthogonal with respect to \frac e^, and with k_n = 1.\sum_^n \frac = \frac\,\frac. Laguerre The Laguerre polynomials L_n are orthonormal with respect to the exponential distribution e^, \quad x \in (0, \infty), with k_n = (-1)^n/n!, so\sum_^n L_k(x) L_k(y)=\frac\left _n(x) L_(y)-L_n(y) L_(x)\right/math> Legendre Associated Legendre polynomials: : \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 Christoff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as a foundational paper of analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time. Early years Riemann was born on 17 September 1826 in Breselenz, a village near Danne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electricity
Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwell's equations. Common phenomena are related to electricity, including lightning, static electricity, electric heating, electric discharges and many others. The presence of either a positive or negative electric charge produces an electric field. The motion of electric charges is an electric current and produces a magnetic field. In most applications, Coulomb's law determines the force acting on an electric charge. Electric potential is the Work (physics), work done to move an electric charge from one point to another within an electric field, typically measured in volts. Electricity plays a central role in many modern technologies, serving in electric power where electric current is used to energise equipment, and in electronics dealing w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In analysis, he advanced the theory of Fourier series and was one of the first to give the modern formal definition of a function. In mathematical physics, he studied potential theory, boundary-value problems, and heat diffusion, and hydrodynamics. Although his surname is Lejeune Dirichlet, he is commonly referred to by his mononym Dirichlet, in particular for results named after him. Biography Early life (1805–1822) Gustav Lejeune Dirichlet was born on 13 February 1805 in Düren, a town on the left bank of the Rhine which at the time was part of the First French Empire, reverting to Prussia after the Congress of Vienna in 1815. His father Johann Arnold Lejeune Dirichlet was the postmaster, merchant, and city councilor. His paternal grandfather had come to Düren from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cologne
Cologne ( ; ; ) is the largest city of the States of Germany, German state of North Rhine-Westphalia and the List of cities in Germany by population, fourth-most populous city of Germany with nearly 1.1 million inhabitants in the city proper and over 3.1 million people in the Cologne Bonn Region, Cologne Bonn urban region. Cologne is also part of the Rhine-Ruhr metropolitan region, the List of EU metropolitan regions by GDP#2021 ranking of top four German metropolitan regions, second biggest metropolitan region by GDP in the European Union. Centered on the left bank of the Rhine, left (west) bank of the Rhine, Cologne is located on the River Rhine (Lower Rhine), about southeast of the North Rhine-Westphalia state capital Düsseldorf and northwest of Bonn, the former capital of West Germany. The city's medieval Cologne Cathedral () was the History of the world's tallest buildings#Churches and cathedrals: Tallest buildings between the 13th and 20th century, world's talles ... [...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 grammar school system or with university preparatory school, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kingdom Of Prussia
The Kingdom of Prussia (, ) was a German state that existed from 1701 to 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 played a significant role in the unification of Germany in 1871 and was a major constituent of the German Empire until its German Revolution of 1918–1919, dissolution in 1918. Although it took its name from the Prussia (region), region called Prussia, it was based in the Margraviate of Brandenburg. Its capital was Berlin. The list of monarchs of Prussia, kings of Prussia were from the House of Hohenzollern. The polity of 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 the Great, Frederick II "the Great".Horn, D. B. "The Youth of Frederick the Great 1712–30." ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General theory of relativity, 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 in physics, time, or four-dimensional spacetime. In particular, the ''curvature of spacetime'' is directly related to the energy and momentum of whatever is present, including matter and radiation. 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 gravity in classical mechanics, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Calculus
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics ( stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, ...), electrodynamics ( electromagnetic ten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, 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 ... [...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 Phenomenon, phenomena, and usually frame their understanding in mathematical terms. They work across a wide range of Physics#Research fields, research fields, spanning all length scales: from atom, sub-atomic and particle physics, through biological physics, to physical cosmology, cosmological length scales encompassing the universe as a whole. The field generally includes two types of physicists: Experimental physics, experimental physicists who specialize in the observation of natural phenomena and the development and analysis of experiments, and Theoretical physics, theoretical physicists who specialize in mathematical modeling of physical systems to rationalize, explain and predict natural phenomena. Physicists can apply their k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
