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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
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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) Zaragoza-Victoria, 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
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Riemann Surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional 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
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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
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Gregorio Ricci-Curbastro
Gregorio Ricci-Curbastro (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 Levi-Civita, he wrote his most famous single publication,[1] a pioneering work on the calculus of tensors, signing it as Gregorio Ricci
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Tullio Levi-Civita
Tullio Levi-Civita, 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 Ricci-Curbastro, the inventor of tensor calculus
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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
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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
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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 x-coordinate". 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
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Einstein
Albert Einstein
Albert Einstein
(14 March 1879 – 18 April 1955) was a German-born 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
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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 pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference.Contents1 Einstein's statement of the equality of inertial and gravitational mass 2 Developmen
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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
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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
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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 prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as one of the most influential papers in analytic number theory
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Second Countable
In topology, a second-countable 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 second-countable 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 second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have. Many "well-behaved" spaces in mathematics are second-countable
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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
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