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]   |
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Eigenvalue Problem
In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scalar multiplication, scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative number, negative or complex number, complex number). Euclidean vector, Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation Rotation (mathematics), rotates, Scaling (geometry), stretches, or Shear mapping, shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with nei ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle \theta, the sine and cosine functions are denoted as \sin(\theta) and \cos(\theta). The definitions of sine and cosine have been extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves, the posit ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Alfred Clebsch
Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Humboldt University of Berlin, Berlin. He subsequently taught in Berlin and University of Karlsruhe, Karlsruhe. His collaboration with Paul Gordan in University of Giessen, Giessen led to the introduction of Clebsch–Gordan coefficients for spherical harmonics, which are now widely used in quantum mechanics. Together with Carl Neumann at University of Göttingen, Göttingen, he founded the mathematical research journal ''Mathematische Annalen'' in 1868. In 1883, Adhémar Jean Claude Barré de Saint-Venant, Saint-Venant translated Clebsch's work on Elasticity (physics), elasticity into French and published it as ''Théorie de l'élasticité des Corps Solides''. Books by A. Clebsch Vorlesungen über Geometrie(Teubner, Leipzig, 1876-1891) edited b ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Gabriel Lamé
Gabriel Lamé (22 July 1795 – 1 May 1870) was a French mathematician who contributed to the theory of partial differential equations by the use of curvilinear coordinates, and the mathematical theory of elasticity (for which linear elasticity and finite strain theory elaborate the mathematical abstractions). Biography Lamé was born in Tours, in today's ''département'' of Indre-et-Loire. He became well known for his general theory of curvilinear coordinates and his notation and study of classes of ellipse-like curves, now known as Lamé curves or superellipses, and defined by the equation: : \left, \,\,\^n + \left, \,\,\^n = 1 where ''n'' is any positive real number. He is also known for his running time analysis of the Euclidean algorithm, marking the beginning of computational complexity theory. In 1844, using Fibonacci numbers, he proved that when finding the greatest common divisor of integers ''a'' and ''b'', the algorithm runs in no more than 5''k'' steps, where ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Siméon Denis Poisson
Baron Siméon Denis Poisson (, ; ; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electricity and magnetism, thermodynamics, elasticity, and fluid mechanics. Moreover, he predicted the Arago spot in his attempt to disprove the wave theory of Augustin-Jean Fresnel. Biography Poisson was born in Pithiviers, now in Loiret, France, the son of Siméon Poisson, an officer in the French Army. In 1798, he entered the École Polytechnique, in Paris, as first in his year, and immediately began to attract the notice of the professors of the school, who left him free to make his own decisions as to what he would study. In his final year of study, less than two years after his entry, he published two memoirs: one on Étienne Bézout's method of elimination, the other on the number of integrals of a finite difference equation. This was so ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics technology may be called an Acoustical engineering, acoustical engineer. The application of acoustics is present in almost all aspects of modern society with the most obvious being the audio and noise control industries. Hearing (sense), Hearing is one of the most crucial means of survival in the animal world and speech is one of the most distinctive characteristics of human development and culture. Accordingly, the science of acoustics spreads across many facets of human society—music, medicine, architecture, industrial production, warfare and more. Likewise, animal species such as songbirds and frogs use sound and hearing as a key element of mating rituals or for marking territories. Art, ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Seismology
Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes (or generally, quakes) and the generation and propagation of elastic waves through planetary bodies. It also includes studies of the environmental effects of earthquakes such as tsunamis; other seismic sources such as volcanoes, plate tectonics, glaciers, rivers, oceanic microseisms, and the atmosphere; and artificial processes such as explosions. Paleoseismology is a related field that uses geology to infer information regarding past earthquakes. A recording of Earth's motion as a function of time, created by a seismograph is called a seismogram. A seismologist is a scientist who works in basic or applied seismology. History Scholarly interest in earthquakes can be traced back to antiquity. Early speculations on the natural causes of earthquakes were included in the writings of Thales of Miletu ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Electromagnetic Radiation
In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength, ranging from radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. All forms of EMR travel at the speed of light in a vacuum and exhibit wave–particle duality, behaving both as waves and as discrete particles called photons. Electromagnetic radiation is produced by accelerating charged particles such as from the Sun and other celestial bodies or artificially generated for various applications. Its interaction with matter depends on wavelength, influencing its uses in communication, medicine, industry, and scientific research. Radio waves enable broadcasting and wireless communication, infrared is used in thermal imaging, visible light is essential for vision, and higher-energy radiation, such ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." It is one of the most fundamental scientific disciplines. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Hyperbolic Partial Differential Equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first n - 1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ... are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation. In one spatial dimension, this is \frac = c^2 \frac The equation has the property that, if and its first time derivative are arbitrarily specified initial data on the line (with sufficient smoothness properties), th ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Laplace Transform
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a function of a Complex number, complex variable s (in the complex-valued frequency domain, also known as ''s''-domain, or ''s''-plane). The transform is useful for converting derivative, differentiation and integral, integration in the time domain into much easier multiplication and Division (mathematics), division in the Laplace domain (analogous to how logarithms are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in science and engineering, mostly as a tool for solving linear differential equations and dynamical systems by simplifying ordinary differential equations and integral equations into algebraic equation, algebraic polynomial equations, and by simplifyin ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |