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Conical Coordinates
Conical coordinates, sometimes called sphero-conal or sphero-conical coordinates, are a three-dimensional orthogonal coordinate system consisting of concentric spheres (described by their radius ) and by two families of perpendicular elliptic cones, aligned along the - and -axes, respectively. The intersection between one of the cones and the sphere forms a spherical conic. Basic definitions The conical coordinates (r, \mu, \nu) are defined by : x = \frac : y = \frac \sqrt : z = \frac \sqrt with the following limitations on the coordinates : \nu^ < c^ < \mu^ < b^. Surfaces of constant are spheres of that radius centered on the origin : whereas surfaces of constant and are mutually perpendicular cones : and : In this coordinate system, both |
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] |
Coordinate System
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are not interchangeable; they are commonly distinguished by their position in an ordered tuple, or by a label, such 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. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and ''vice versa''; this is the basis of analytic geometry. Common coordinate systems Number line The simplest example of a coordinate system is the identification of points on a line with real numbers using the '' number line''. In this system, an arbitrary point ''O'' (the ''origin'') is chosen on a given line. The coordinate o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Conic
In mathematics, a spherical conic or sphero-conic is a curve on the sphere, the intersection of the sphere with a concentric elliptic cone. It is the spherical analog of a conic section (ellipse, parabola, or hyperbola) in the plane, and as in the planar case, a spherical conic can be defined as the locus (mathematics), locus of points the sum or difference of whose great-circle distances to two focus (geometry), foci is constant. By taking the antipodal point to one focus, every spherical ellipse is also a spherical hyperbola, and vice versa. As a space curve, a spherical conic is a quartic function, quartic, though its orthogonal projections in three Principal axis theorem, principal axes are planar conics. Like planar conics, spherical conics also satisfy a "reflection property": the great-circle arcs from the two foci to any point on the conic have the tangent and normal to the conic at that point as their angle bisectors. Many theorems about conics in the plane extend to sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace's Equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties in 1786. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nabla \cdot \nabla = \nabla^2 is the Laplace operator,The delta symbol, Δ, is also commonly used to represent a finite change in some quantity, for example, \Delta x = x_1 - x_2. Its use to represent the Laplacian should not be confused with this use. \nabla \cdot is the divergence operator (also symbolized "div"), \nabla is the gradient operator (also symbolized "grad"), and f (x, y, z) is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function, h(x, y, z), we have \Delta f = h This is called Poisson's equation, a generalization of Laplace's equation. Laplace's equation and Poisson's equation are the simp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmholtz Equation
In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation: \nabla^2 f = -k^2 f, where is the Laplace operator, is the eigenvalue, and is the (eigen)function. When the equation is applied to waves, is known as the wave number. The Helmholtz equation has a variety of applications in physics and other sciences, including the wave equation, the diffusion equation, and the Schrödinger equation for a free particle. In optics, the Helmholtz equation is the wave equation for the electric field. The equation is named after Hermann von Helmholtz, who studied it in 1860. from the Encyclopedia of Mathematics. Motivation and uses The Helmholtz equation often arises in the study of physical problems involving par ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Coordinates
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are * the radial distance along the line connecting the point to a fixed point called the origin; * the polar angle between this radial line and a given ''polar axis''; and * the azimuthal angle , which is the angle of rotation of the radial line around the polar axis. (See graphic regarding the "physics convention".) Once the radius is fixed, the three coordinates (''r'', ''θ'', ''φ''), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle) is called the ''reference plane'' (sometimes '' fundamental plane''). Terminology The radial distance from the fixed point of origin is also called the ''radius'', or ''radial line'', or ''radial coord ... [...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] |
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] |
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] |
Theresa M
Teresa (also Theresa, Therese; ) is a feminine given name. It originates in the Iberian Peninsula in late antiquity. Its derivation is uncertain, it may be derived from Greek θερίζω (''therízō'') "to harvest or reap", or from θέρος (''theros'') "summer". Another origin of the name is from Latin word "Terra" which means earth. Terra mother Earth. It is first recorded in the form ''Therasia'', the name of Therasia of Nola, an aristocrat of the 4th century. Its popularity outside of Iberia increased because of saint Teresa of Ávila, and more recently Thérèse of Lisieux and Mother Teresa. In the United States it was ranked as the 852nd most popular name for girls born in 2008, down from 226th in 1992 (it ranked 65th in 1950, and 102nd in 1900). Spelled "Teresa," it was the 580th most popular name for girls born in 2008, down from 206th in 1992 (it ranked 81st in 1950, and 220th in 1900). People Aristocracy *Teresa of Portugal (other) ** Theresa, C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three-dimensional Coordinate Systems
In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), point. Most commonly, it is the three-dimensional Euclidean space, that is, the Euclidean space of dimension three, which models physical space. More general three-dimensional spaces are called ''3-manifolds''. The term may also refer colloquially to a subset of space, a ''three-dimensional region'' (or 3D domain (mathematical analysis), domain), a ''solid figure''. Technically, 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 pair formed by a -dimensional Euclidean space and a Cartesian coordinate system. When , this space is called the three-dimensional Euclidean space (or simply "E ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
