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Breather Surface
In differential geometry, a breather surface is a one-parameter family of mathematical surfaces which correspond to breather solutions of the sine-Gordon equation, a differential equation appearing in theoretical physics. The surfaces have the remarkable property that they have constant curvature -1, where the curvature is well-defined. This makes them examples of generalized pseudospheres. Mathematical background There is a correspondence between embedded surfaces of constant curvature -1, known as pseudospheres, and solutions to the sine-Gordon equation. This correspondence can be built starting with the simplest example of a pseudosphere, the tractroid. In a special set of coordinates, known as asymptotic coordinates, the Gauss–Codazzi equations, which are consistency equations dictating when a surface of prescribed first and second fundamental form can be embedded into three-dimensional space with the flat metric, reduce to the sine-Gordon equation. In the corresponde ... [...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 structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three-dimensional Space
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position (geometry), position of an element (i.e., Point (mathematics), point). This is the informal meaning of the term dimension. In mathematics, a tuple of Real number, numbers can be understood as the Cartesian coordinates of a location in a -dimensional Euclidean space. The set of these -tuples is commonly denoted \R^n, and can be identified to the -dimensional Euclidean space. When , this space is called three-dimensional Euclidean space (or simply Euclidean space when the context is clear). It serves as a model of the physical universe (when relativity theory is not considered), in which all known matter exists. While this space remains the most compelling and useful way to model the world as it is experienced, it is only one example of a large variety of spaces in three dimensions called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surfaces
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. Surface or surfaces may also refer to: Mathematics *Surface (mathematics), a generalization of a plane which needs not be flat *Surface (differential geometry), a differentiable two-dimensional manifold *Surface (topology), a two-dimensional manifold * Algebraic surface, an algebraic variety of dimension two *Coordinate surfaces *Fractal surface, generated using a stochastic algorithm *Polyhedral surface * Surface area *Surface integral Arts and entertainment * Surface (band), an American R&B and pop trio ** ''Surface'' (Surface album), 1986 *Surfaces (band), American musical duo * ''Surface'' (Circle album), 1998 * "Surface" (Aero Chord song), 2014 * ''Surface'' (2005 TV series), an American science fiction show, 2005–2006 * ''Surface'' (2022 TV series), an American psychological thriller miniseries that began streaming in 2022 *'' The Surface'', an American film, 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Breather Surface
In differential geometry, a breather surface is a one-parameter family of mathematical surfaces which correspond to breather solutions of the sine-Gordon equation, a differential equation appearing in theoretical physics. The surfaces have the remarkable property that they have constant curvature -1, where the curvature is well-defined. This makes them examples of generalized pseudospheres. Mathematical background There is a correspondence between embedded surfaces of constant curvature -1, known as pseudospheres, and solutions to the sine-Gordon equation. This correspondence can be built starting with the simplest example of a pseudosphere, the tractroid. In a special set of coordinates, known as asymptotic coordinates, the Gauss–Codazzi equations, which are consistency equations dictating when a surface of prescribed first and second fundamental form can be embedded into three-dimensional space with the flat metric, reduce to the sine-Gordon equation. In the corresponde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Scattering Method
In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solve many linear partial differential equations. The name "inverse scattering method" comes from the key idea of recovering the time evolution of a potential from the time evolution of its scattering data: inverse scattering refers to the problem of recovering a potential from its scattering matrix, as opposed to the direct scattering problem of finding the scattering matrix from the potential. The inverse scattering transform may be applied to many of the so-called exactly solvable models, that is to say completely integrable infinite dimensional systems. Overview The inverse scattering transform was first introduced by for the Korteweg–de Vries equation, and soon extended to the nonlinear Schrödinger equation, the Sine-Gordon equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kuen Surface
KUEN, virtual channel 9 (UHF digital channel 36), is an educational independent television station serving Salt Lake City, Utah, United States that is licensed to Ogden. The station is owned by the Utah State Board of Regents, and is operated by the Utah Education Network on behalf of higher education ( USHE) as well as public education (USOE). KUEN's studios are located at the Eccles Broadcast Center on the University of Utah campus, and its transmitter is located at Farnsworth Peak in the Oquirrh Mountains, southwest of Salt Lake City. The station has a large network of broadcast translators that extend its over-the-air coverage throughout Utah, as well as portions of Colorado. History On March 21, 1984, the Federal Communications Commission (FCC) granted an original construction permit to Weber State College (now Weber State University) for a full-power educational television station to serve Ogden and the Salt Lake City area on VHF channel 9. The station's original call ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bäcklund Transformation
Backlund is a Swedish surname. Notable people with the surname include: * Albert Victor Bäcklund (1845-1922), mathematician * Bengt Backlund (1926–2006), Swedish flatwater canoer * Bob Backlund (born 1949), American professional wrestler * Filip Backlund (born 1990), Swedish motorcycle road racer * Göran Backlund (born 1957), Swedish sprint canoer * Gordon Backlund (born 1940), American politician and electrical engineer * Gösta Backlund (1893—1918), Swedish footballer * Gotthard Backlund, Swedish chess master * Ivar Backlund (1892—1969), Swedish officer * Johan Backlund (born 1981), Swedish ice hockey goaltender * Jukka Backlund (born 1982), Finnish music producer * Kaj Backlund (1945–2013), Finnish jazz trumpeter, composer, and bandleader * Mikael Backlund (born 1989), Swedish ice hockey player * Nils Backlund (1896–1964), Swedish water polo player * Oskar Backlund Johan Oskar Backlund (28 April 1846 – 29 August 1916) was a Swedish-Russian astronomer. His name ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dini's Surface
In geometry, Dini's surface is a surface with constant negative curvature that can be created by twisting a pseudosphere. It is named after Ulisse Dini and described by the following parametric equations: : \begin x&=a \cos u \sin v \\ y&=a \sin u \sin v \\ z&=a \left(\cos v +\ln \tan \frac \right) + bu \end Another description is a generalized helicoid constructed from the tractrix In geometry, a tractrix (; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a pulling point (the ''tractor'') that moves at a right angl .... See also * Breather surface References {{reflist Surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentz Boost
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz. The most common form of the transformation, parametrized by the real constant v, representing a velocity confined to the -direction, is expressed as \begin t' &= \gamma \left( t - \frac \right) \\ x' &= \gamma \left( x - v t \right)\\ y' &= y \\ z' &= z \end where and are the coordinates of an event in two frames with the origins coinciding at 0, where the primed frame is seen from the unprimed frame as moving with speed along the -axis, where is the speed of light, and \gamma = \left ( \sqrt\right )^ is the Lorentz factor. When speed is much smaller than , the Lorentz factor is negligibly different from 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentz Invariance
In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of vectors, or from contracting tensors of the theory. While the components of vectors and tensors are in general altered under Lorentz transformations, Lorentz scalars remain unchanged. A Lorentz scalar is not always immediately seen to be an invariant scalar in the mathematical sense, but the resulting scalar value is invariant under any basis transformation applied to the vector space, on which the considered theory is based. A simple Lorentz scalar in Minkowski spacetime is the ''spacetime distance'' ("length" of their difference) of two fixed events in spacetime. While the "position"-4-vectors of the events change between different inertial frames, their spacetime distance remains invariant under the corresponding Lorentz transformation. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point of is a bilinear form defined on the tangent space at (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric tensor on consists of a metric tensor at each point of that varies smoothly with . A metric tensor is ''positive-definite'' if for every nonzero vector . A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying ''infinitesimal'' distance on the manifold. On a Riemannian manifold , the length of a smooth curve between two points and can be defined by integration, and the distance between and can be defined as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Fundamental Form
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold. Surface in R3 Motivation The second fundamental form of a parametric surface in was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, , and that the plane is tangent to the surface at the origin. Then and its partial derivatives with respect to and vanish at (0,0). Therefore, the Taylor expansion of ''f'' at (0,0) starts with quadratic terms: : z=L\frac + Mxy + N\frac + \text\,, and the second fundamental form at the origin in the coordinates is the qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
