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Smooth Manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space [...More...]  "Smooth Manifold" on: Wikipedia Yahoo Parouse 

Tropic Of Cancer Coordinates: 23°26′14″N 0°0′0″W / 23.43722°N 0.00000°E / 23.43722; 0.00000 (Prime Meridian)World map showing the Tropic of CancerCarretera 83 (Vía Corta) ZaragozaVictoria, km 27+800. Of all crossings of the Tropic of Cancer Tropic of Cancer with Mexican federal highways, this is the only place where the latitude is marked with precision and where the annual drift between the years 2005 and 2010 can be appreciated.The Tropic of Cancer, also referred to as the Northern Tropic, is currently 23°26′12.9″ (or 23.43692°) north of the Equator. It is the most northerly circle of latitude on Earth Earth at which the Sun Sun can be directly overhead [...More...]  "Tropic Of Cancer" on: Wikipedia Yahoo Parouse 

Riemann Surface In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a onedimensional complex manifold. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. The main point of Riemann surfaces is that holomorphic functions may be defined between them [...More...]  "Riemann Surface" on: Wikipedia Yahoo Parouse 

James Clerk Maxwell James Clerk Maxwell James Clerk Maxwell FRS FRSE (/ˈmækswɛl/;[2] 13 June 1831 – 5 November 1879) was a Scottish[3][4] scientist in the field of mathematical physics.[5] His most notable achievement was to formulate the classical theory of electromagnetic radiation, bringing together for the first time electricity, magnetism, and light as different manifestations of the same phenomenon. Maxwell's equations Maxwell's equations for electromagnetism have been called the "second great unification in physics"[6] after the first one realised by Isaac Newton. With the publication of "A Dynamical Theory of the Electromagnetic Field" in 1865, Maxwell demonstrated that electric and magnetic fields travel through space as waves moving at the speed of light [...More...]  "James Clerk Maxwell" on: Wikipedia Yahoo Parouse 

Gregorio RicciCurbastro Gregorio RicciCurbastro (Italian: [ɡreˈɡɔːrjo ˈrittʃi kurˈbastro]; 12 January 1853 – 6 August 1925) was an Italian mathematician born in Lugo di Romagna. He is most famous as the inventor of tensor calculus, but also published important works in other fields. With his former student Tullio LeviCivita, he wrote his most famous single publication,[1] a pioneering work on the calculus of tensors, signing it as Gregorio Ricci [...More...]  "Gregorio RicciCurbastro" on: Wikipedia Yahoo Parouse 

Tullio LeviCivita Tullio LeviCivita, FRS[1][2] /ˈtʊlioʊ ˈlɛvi ˈtʃɪvɪtə/ (29 March 1873 – 29 December 1941; Italian pronunciation: [ˈtullio ˈlɛːvi ˈtʃiːvita]) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio RicciCurbastro, the inventor of tensor calculus [...More...]  "Tullio LeviCivita" on: Wikipedia Yahoo Parouse 

Tensor Analysis In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space Euclidean space or manifold). Tensor Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, like length) and a vector (a geometrical arrow in space), a tensor field is a generalization of a scalar field or vector field that assigns, respectively, a scalar or vector to each point of space. Many mathematical structures called "tensors" are tensor fields [...More...]  "Tensor Analysis" on: Wikipedia Yahoo Parouse 

General Covariance In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the form of physical laws under arbitrary differentiable coordinate transformations. The essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws.Contents1 Overview 2 Remarks 3 See also 4 Notes 5 References 6 External linksOverview[edit] A physical law expressed in a generally covariant fashion takes the same mathematical form in all coordinate systems,[1] and is usually expressed in terms of tensor fields [...More...]  "General Covariance" on: Wikipedia Yahoo Parouse 

Coordinate Transformation In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.[1][2] The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the xcoordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring [...More...]  "Coordinate Transformation" on: Wikipedia Yahoo Parouse 

Einstein Albert Einstein Albert Einstein (14 March 1879 – 18 April 1955) was a Germanborn theoretical physicist[5] who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics).[4][6]:274 His work is also known for its influence on the philosophy of science.[7][8] He is best known by the general public for his mass–energy equivalence formula E = mc2 (which has been dubbed "the world's most famous equation").[9] He received the 1921 Nobel Prize in Physics Nobel Prize in Physics "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect",[10] a pivotal step in the evolution of quantum theory. Near the beginning of his career, Einstein thought that Newtonian mechanics was no longer enough to reconcile the laws of classical mechanics with the laws of the electromagnetic field [...More...]  "Einstein" on: Wikipedia Yahoo Parouse 

Equivalence Principle In the theory of general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is the same as the pseudoforce experienced by an observer in a noninertial (accelerated) frame of reference.Contents1 Einstein's statement of the equality of inertial and gravitational mass 2 Developmen [...More...]  "Equivalence Principle" on: Wikipedia Yahoo Parouse 

Hermann Weyl Hermann Klaus Hugo Weyl, ForMemRS[2] (German: [vaɪl]; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland Switzerland and then Princeton, he is associated with the University of Göttingen University of Göttingen tradition of mathematics, represented by David Hilbert David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines including number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years.[5][6][7] Weyl published technical and some general works on space, time, matter, philosophy, logic, symmetry and the history of mathematics [...More...]  "Hermann Weyl" on: Wikipedia Yahoo Parouse 

Atlas (mathematics) In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an atlas has its more common meaning. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fibre bundles.Contents1 Charts 2 Formal definition of atlas 3 Transition maps 4 More structure 5 See also 6 References 7 External linksCharts[edit] See also: Manifold Manifold § Charts The definition of an atlas depends on the notion of a chart [...More...]  "Atlas (mathematics)" on: Wikipedia Yahoo Parouse 

Bernhard Riemann Georg Friedrich Bernhard Riemann Bernhard Riemann (German: [ˈʀiːman] ( listen); 17 September 1826 – 20 July 1866) was a German mathematician who made 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 famous 1859 paper on the primecounting function, containing the original statement of the Riemann hypothesis, is regarded as one of the most influential papers in analytic number theory [...More...]  "Bernhard Riemann" on: Wikipedia Yahoo Parouse 

Second Countable In topology, a secondcountable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space T displaystyle T is secondcountable if there exists some countable collection U = U i i = 1 ∞ displaystyle mathcal U = U_ i _ i=1 ^ infty of open subsets of T displaystyle T such that any open subset of T displaystyle T can be written as a union of elements of some subfamily of U displaystyle mathcal U . A secondcountable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being secondcountable restricts the number of open sets that a space can have. Many "wellbehaved" spaces in mathematics are secondcountable [...More...]  "Second Countable" on: Wikipedia Yahoo Parouse 

Hausdorff Space In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters. Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology [...More...]  "Hausdorff Space" on: Wikipedia Yahoo Parouse 