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Elliptic Cylindrical Coordinates
Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular z-direction. Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae. The two foci F_ and F_ are generally taken to be fixed at -a and +a, respectively, on the x-axis of the Cartesian coordinate system. Basic definition The most common definition of elliptic cylindrical coordinates (\mu, \nu, z) is : x = a \ \cosh \mu \ \cos \nu : y = a \ \sinh \mu \ \sin \nu : z = z where \mu is a nonnegative real number and \nu \in , 2\pi/math>. These definitions correspond to ellipses and hyperbolae. The trigonometric identity : \frac + \frac = \cos^ \nu + \sin^ \nu = 1 shows that curves of constant \mu form ellipses, whereas the hyperbolic trigonometric identity : \frac - \frac = \cosh^ \mu - \sinh^ \mu = 1 shows that curves of constant \nu form hyperbolae. Scale factors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Cylindrical Coordinates
Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular z-direction. Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae. The two foci F_ and F_ are generally taken to be fixed at -a and +a, respectively, on the x-axis of the Cartesian coordinate system. Basic definition The most common definition of elliptic cylindrical coordinates (\mu, \nu, z) is : x = a \ \cosh \mu \ \cos \nu : y = a \ \sinh \mu \ \sin \nu : z = z where \mu is a nonnegative real number and \nu \in , 2\pi/math>. These definitions correspond to ellipses and hyperbolae. The trigonometric identity : \frac + \frac = \cos^ \nu + \sin^ \nu = 1 shows that curves of constant \mu form ellipses, whereas the hyperbolic trigonometric identity : \frac - \frac = \cosh^ \mu - \sinh^ \mu = 1 shows that curves of constant \nu form hyperbolae. Scale factors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmholtz Equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplacian"), 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, including the wave equation and the diffusion equation, and it has uses in other sciences. Motivation and uses The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis. For example, consider the wave equation \left(\nabla^2-\frac\frac\right) u(\mathbf,t)=0. Separation of variables begins by assumi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theresa M
Teresa (also Theresa, Therese; french: Thérèse) 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". 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 In aristocracy: *Teresa of Portugal (other) ** Theresa, Countess of Portugal (1080–1130), mother of Afonso Henriques, the first K ... [...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 Hermann Ludwig Ferdinand von ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herman Feshbach
Herman Feshbach (February 2, 1917, in New York City – 22 December 2000, in Cambridge, Massachusetts) 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 Dr. 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 Pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philip M
Philip, also Phillip, is a male given name, derived from the 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 kings of Macedonia and one of the apostles of early Christianity. ''Philip'' has many alternative spellings. One derivation often used as a surname is Phillips. It was also found during ancient Greek times with two Ps as Philippides and Philippos. It has many diminutive (or even hypocoristic) forms including Phil, Philly, Lip, Pip, Pep or Peps. There are also feminine forms such as Philippine and Philippa. Antiquity Kings of Macedon * Philip I of Macedon * Philip II of Macedon, father of Alexander the Great * Philip III of Macedon, half-brother of Alexander the Great * Philip IV of Macedon * Philip V of Macedon New Testament * Philip the Apostle * Philip the Evangelist Others * Philippus of Croton (c. 6th centur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass and is its velocity (also a vector quantity), then the object's momentum is : \mathbf = m \mathbf. In the International System of Units (SI), the unit of measurement of momentum is the kilogram metre per second (kg⋅m/s), which is equivalent to the newton-second. Newton's second law of motion states that the rate of change of a body's momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any inertial frame it is a ''conserved'' quantity, meaning that if a closed system is not affected by external forces, its total linear momentum does not change. Momentum is also conserved in special relativity (with a modified formula) and, in a modified form, in electrodynamics, quantum mechanics, quan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathieu Differential Equation
In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation : \frac + (a - 2q\cos(2x))y = 0, where a and q are parameters. They were first introduced by Émile Léonard Mathieu, who encountered them while studying vibrating elliptical drumheads.Morse and Feshbach (1953).Brimacombe, Corless and Zamir (2021) They have applications in many fields of the physical sciences, such as optics, quantum mechanics, and general relativity. They tend to occur in problems involving periodic motion, or in the analysis of partial differential equation boundary value problems possessing elliptic symmetry.Gutiérrez-Vega (2015). Definition Mathieu functions In some usages, ''Mathieu function'' refers to solutions of the Mathieu differential equation for arbitrary values of a and q. When no confusion can arise, other authors use the term to refer specifically to \pi- or 2\pi-periodic solutions, which exist only for special valu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. Single mechanical or electromagnetic waves propagating in a pre-defined direction can also be described with the first-order one-way wave equation which is much easier to solve and also valid for inhomogenious media. Introduction The (two-way) wave equation is a second-order partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions of a time variable (a variable repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electric Field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field for a system of charged particles. Electric fields originate from electric charges and time-varying electric currents. Electric fields and magnetic fields are both manifestations of the electromagnetic field, one of the four fundamental interactions (also called forces) of nature. Electric fields are important in many areas of physics, and are exploited in electrical technology. In atomic physics and chemistry, for instance, the electric field is the attractive force holding the atomic nucleus and electrons together in atoms. It is also the force responsible for chemical bonding between atoms that result in molecules. The electric field is defined as a vector field that associates to each point in space the electrostatic ( Coulomb) for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separation Of Variables
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation. Ordinary differential equations (ODE) Suppose a differential equation can be written in the form :\frac f(x) = g(x)h(f(x)) which we can write more simply by letting y = f(x): :\frac=g(x)h(y). As long as ''h''(''y'') ≠ 0, we can rearrange terms to obtain: : = g(x) \, dx, so that the two variables ''x'' and ''y'' have been separated. ''dx'' (and ''dy'') can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. A formal definition of ''dx'' as a differential (infinitesimal) is somewhat advanced. Alternative notation Those who dislike Leibniz's notation may prefer to write this as :\frac \frac = g(x), but that ... [...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. 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 simplest exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
