|
Parabolic Coordinates
Parabolic coordinates are a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal parabolas. A three-dimensional version of parabolic coordinates is obtained by rotating the two-dimensional system about the symmetry axis of the parabolas. Parabolic coordinates have found many applications, e.g., the treatment of the Stark effect and the potential theory of the edges. Two-dimensional parabolic coordinates Two-dimensional parabolic coordinates (\sigma, \tau) are defined by the equations, in terms of Cartesian coordinates: : x = \sigma \tau : y = \frac \left( \tau^ - \sigma^ \right) The curves of constant \sigma form confocal parabolae : 2y = \frac - \sigma^ that open upwards (i.e., towards +y), whereas the curves of constant \tau form confocal parabolae : 2y = -\frac + \tau^ that open downwards (i.e., towards -y). The foci of all these parabolae are located at the origin. The Cartesian coordinates x and y can be converted to para ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabolic Coords
Parabolic usually refers to something in a shape of a parabola, but may also refer to a parable. Parabolic may refer to: *In mathematics: **In elementary mathematics, especially elementary geometry: **Parabolic coordinates ** Parabolic cylindrical coordinates ** parabolic Möbius transformation ** Parabolic geometry (other) ** Parabolic spiral ** Parabolic line **In advanced mathematics: *** Parabolic cylinder function *** Parabolic induction *** Parabolic Lie algebra ***Parabolic partial differential equation *In physics: **Parabolic trajectory *In technology: **Parabolic antenna ** Parabolic microphone **Parabolic reflector **Parabolic trough - a type of solar thermal energy collector ** Parabolic flight - a way of achieving weightlessness ** Parabolic action, or parabolic bending curve - a term often used to refer to a progressive bending curve in fishing rods. *In commodities and stock markets: ** Parabolic SAR - a chart pattern in which prices rise or fall with an incr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabolic Cylindrical Coordinates
In mathematics, parabolic cylindrical coordinates are a three-dimensional orthogonal coordinates, orthogonal coordinate system that results from projecting the two-dimensional parabolic coordinates, parabolic coordinate system in the perpendicular z-direction. Hence, the Coordinate system#Coordinate surface, coordinate surfaces are confocal parabola, parabolic cylinders. Parabolic cylindrical coordinates have found many applications, e.g., the potential theory of edges. Basic definition The parabolic cylindrical coordinates are defined in terms of the Cartesian coordinates by: :\begin x &= \sigma \tau \\ y &= \frac \left( \tau^2 - \sigma^2 \right) \\ z &= z \end The surfaces of constant form confocal parabolic cylinders : 2 y = \frac - \sigma^2 that open towards , whereas the surfaces of constant form confocal parabolic cylinders : 2 y = -\frac + \tau^2 that open in the opposite direction, i.e., towards . The foci of all these parabolic cylinders are located along ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henry Margenau
Henry Margenau (April 30, 1901 – February 8, 1997) was a German-American physicist and philosopher of science. Biography Early life Born in Bielefeld, Germany, Margenau obtained his bachelor's degree from Midland Lutheran College, Nebraska before his M.Sc. from the University of Nebraska in 1926, and PhD from Yale University in 1929. World War II Margenau worked on the theory of microwaves and the development of duplexing systems that enabled a single radar antenna both to transmit and receive signals. He also worked on spectral line broadening, a technique used to analyse and review the dynamics of the atomic bombing of Hiroshima. Philosophy and history of science Margenau wrote extensively on science, his works including: ''Ethics and Science'', ''The Nature of Physical Reality'', ''Quantum Mechanics'' and ''Integrative Principles of Modern Thought''. He wrote in 1954 the important introduction for the classic book of Hermann von Helmholtz, On the Sensations of Tone. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herman Feshbach
Herman Feshbach (2 February 1917 – 22 December 2000) was an American physicist. He was an Institute Professor Emeritus of physics at MIT. Feshbach is best known for Feshbach resonance and for writing, with Philip M. Morse, ''Methods of Theoretical Physics''. Background Feshbach was born in New York City and graduated from the City College of New York in 1937. He was a member of the same family as Murray Feshbach, the Sovietologist and retired Georgetown University professor. He then went on to receive his Ph.D. in physics from MIT in 1942. Feshbach attended the Shelter Island Conference of 1947. Career Feshbach was invited to stay at MIT after he received his doctorate. He remained on the physics faculty for over fifty years. From 1967 to 1973, he was the director of MIT's Center for Theoretical Physics, and from 1973 to 1983, he was chairman of the physics department. In 1983, Feshbach was named as an institute professor, the highest faculty honor at MIT. Activism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philip M
Philip, also Phillip, is a male name derived from the Macedonian Old Koine language, Greek (''Philippos'', lit. "horse-loving" or "fond of horses"), from a compound of (''philos'', "dear", "loved", "loving") and (''hippos'', "horse"). Prominent Philips who popularized the name include List of kings of Macedonia, kings of Macedonia and one of the apostles of early Christianity. ''Philip'' has #Philip in other languages, many alternative spellings. One derivation often used as a surname is Phillips (surname), Phillips. The original Greek spelling includes two Ps as seen in Philippides (other), Philippides and Philippos, which is possible due to the Greek endings following the two Ps. To end a word with such a double consonant—in Greek or in English—would, however, be incorrect. It has many diminutive (or even hypocorism, hypocoristic) forms including Phil, Philly (other)#People, Philly, Phillie, Lip (other), Lip, and Pip (other), Pip. There ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvilinear Coordinates
In geometry, curvilinear coordinates are a coordinate system for Euclidean space in which the coordinate lines may be curved. These coordinates may be derived from a set of Cartesian coordinates by using a transformation that is invertible, locally invertible (a one-to-one map) at each point. This means that one can convert a point given in a Cartesian coordinate system to its curvilinear coordinates and back. The name ''curvilinear coordinates'', coined by the French mathematician Gabriel Lamé, Lamé, derives from the fact that the coordinate surfaces of the curvilinear systems are curved. Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R3) are Cylindrical coordinate system, cylindrical and spherical coordinates, spherical coordinates. A Cartesian coordinate surface in this space is a coordinate plane; for example ''z'' = 0 defines the ''x''-''y'' plane. In the same space, the coordinate surface ''r'' = 1 in spherical coordinates i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Coordinate System
In mathematics, orthogonal coordinates are defined as a set of coordinates \mathbf q = (q^1, q^2, \dots, q^d) in which the coordinate hypersurfaces all meet at right angles (note that superscripts are indices, not exponents). A coordinate surface for a particular coordinate is the curve, surface, or hypersurface on which is a constant. For example, the three-dimensional Cartesian coordinates is an orthogonal coordinate system, since its coordinate surfaces constant, constant, and constant are planes that meet at right angles to one another, i.e., are perpendicular. Orthogonal coordinates are a special but extremely common case of curvilinear coordinates. Motivation While vector operations and physical laws are normally easiest to derive in Cartesian coordinates, non-Cartesian orthogonal coordinates are often used instead for the solution of various problems, especially boundary value problems, such as those arising in field theories of quantum mechanics, fluid f ... [...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 field 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] |
Riemannian Manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifold, manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them. Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport. Any smooth surface in three-dimensional Eucl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parabolic Coordinates 3D
Parabolic usually refers to something in a shape of a parabola, but may also refer to a parable. Parabolic may refer to: *In mathematics: **In elementary mathematics, especially elementary geometry: **Parabolic coordinates **Parabolic cylindrical coordinates ** parabolic Möbius transformation ** Parabolic geometry (other) ** Parabolic spiral ** Parabolic line **In advanced mathematics: *** Parabolic cylinder function ***Parabolic induction ***Parabolic Lie algebra ***Parabolic partial differential equation *In physics: **Parabolic trajectory *In technology: **Parabolic antenna **Parabolic microphone **Parabolic reflector **Parabolic trough - a type of solar thermal energy collector **Parabolic flight - a way of achieving weightlessness ** Parabolic action, or parabolic bending curve - a term often used to refer to a progressive bending curve in fishing rods. *In commodities and stock markets: ** Parabolic SAR - a chart pattern in which prices rise or fall with an increasin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Coordinates
In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ..., orthogonal coordinates are defined as a set of coordinates \mathbf q = (q^1, q^2, \dots, q^d) in which the Coordinate system#Coordinate surface, coordinate hypersurfaces all meet at right angles (note that superscripts are Einstein notation, indices, not exponents). A coordinate surface for a particular coordinate is the curve, surface, or hypersurface on which is a constant. For example, the three-dimensional Cartesian coordinate system, Cartesian coordinates is an orthogonal coordinate system, since its coordinate surfaces constant, constant, and constant are planes that meet at right angles to one another, i.e., are perpendicular. Orthogonal coordinates are a special but extremely common case of curvilinear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the nabla operator), or \Delta. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian of a function at a point measures by how much the average value of over small spheres or balls centered at deviates from . The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
