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Green's Function For The Three-variable Laplace Equation
In physics, the Green's function (or fundamental solution) for the Laplacian (or Laplace operator) in three variables is used to describe the response of a particular type of physical system to a point source. In particular, this Green's function arises in systems that can be described by Poisson's equation, a partial differential equation (PDE) of the form \nabla^2 u(\mathbf) = f(\mathbf) where \nabla^2 is the Laplace operator in \mathbb^3, f(\mathbf) is the source term of the system, and u(\mathbf) is the solution to the equation. Because \nabla^2 is a linear differential operator, the solution u(\mathbf) to a general system of this type can be written as an integral over a distribution of source given by f(\mathbf): u(\mathbf) = \int G(\mathbf,\mathbf)f(\mathbf)d\mathbf' where the Green's function for Laplacian in three variables G(\mathbf,\mathbf) describes the response of the system at the point \mathbf to a point source located at \mathbf: \nabla^2 G(\mathbf,\mathbf) = \de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Poisson Equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate the corresponding electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and physicist Siméon Denis Poisson who published it in 1823. Statement of the equation Poisson's equation is \Delta\varphi = f, where \Delta is the Laplace operator, and f and \varphi are real number, real or complex number, complex-valued function (mathematics), functions on a manifold. Usually, f is given, and \varphi is sought. When the manifold is Euclidean space, the Laplace operator is often denoted as , and so Poisson's equation is frequently written as \nabla^2 \varphi = f. In three- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Oblate Spheroidal Coordinates
Oblate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the non-focal axis of the ellipse, i.e., the symmetry axis that separates the foci. Thus, the two foci are transformed into a ring of radius a in the ''x''-''y'' plane. (Rotation about the other axis produces prolate spheroidal coordinates.) Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length. Oblate spheroidal coordinates are often useful in solving partial differential equations when the boundary conditions are defined on an oblate spheroid or a hyperboloid of revolution. For example, they played an important role in the calculation of the Perrin friction factors, which contributed to the awarding of the 1926 Nobel Prize in Physics to Jean Baptiste Perrin. These friction factors determine the rotational d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prolate Spheroidal Coordinates
Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length. Prolate spheroidal coordinates can be used to solve various partial differential equations in which the boundary conditions match its symmetry and shape, such as solving for a field produced by two centers, which are taken as the foci on the ''z''-axis. One example is solving for the wavefunction of an electron moving in the electromagnetic field of two positively charged nuclei, as in the hydrogen molecular ion, H2+. Another example is solving for the electric field generated by two small electrode ... [...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] |
Cylindrical Coordinates
A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite curvilinear surface in various modern branches of geometry and topology. The shift in the basic meaning—solid versus surface (as in a solid ball versus sphere surface)—has created some ambiguity with terminology. The two concepts may be distinguished by referring to solid cylinders and cylindrical surfaces. In the literature the unadorned term "cylinder" could refer to either of these or to an even more specialized object, the '' right circular cylinder''. Types The definitions and results in this section are taken from the 1913 text ''Plane and Solid Geometry'' by George A. Wentworth and David Eugene Smith . A ' is a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosine Transform
In mathematics, the Fourier sine and cosine transforms are integral equations that decompose arbitrary functions into a sum of sine waves representing the odd component of the function plus cosine waves representing the even component of the function. The modern Fourier transform concisely contains both the sine and cosine transforms. Since the sine and cosine transforms use sine and cosine waves instead of complex exponentials and don't require complex numbers or negative frequency, they more closely correspond to Joseph Fourier's original transform equations and are still preferred in some signal processing and statistics applications and may be better suited as an introduction to Fourier analysis. Definition The Fourier sine transform of f(t) is: If t means time, then \xi is frequency in cycles per unit time, but in the abstract, they can be any dual pair of variables (e.g. position and spatial frequency). The sine transform is necessarily an odd function of frequency, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Whipple Formulae
In the theory of special functions Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ..., Whipple's transformation for Legendre functions, named after Francis John Welsh Whipple, arise from a general expression, concerning associated Legendre functions. These formulae have been presented previously in terms of a viewpoint aimed at spherical harmonics, now that we view the equations in terms of toroidal coordinates, whole new symmetries of Legendre functions arise. For associated Legendre functions of the first and second kind, :P_^\biggl(\frac\biggr)= \frac and :Q_^\biggl(\frac\biggr)= -i(\pi/2)^\Gamma(-\nu-\mu)(z^2-1)^e^ P_\nu^\mu(z). These expressions are valid for all parameters \nu, \mu, and z. By shifting the complex degree and order in an appropriate fashion, we obtain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toroidal Coordinates
Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci F_1 and F_2 in bipolar coordinates become a ring of radius a in the xy plane of the toroidal coordinate system; the z-axis is the axis of rotation. The focal ring is also known as the reference circle. Definition The most common definition of toroidal coordinates (\tau, \sigma, \phi) is : x = a \ \frac \cos \phi : y = a \ \frac \sin \phi : z = a \ \frac together with \mathrm(\sigma)=\mathrm(z). The \sigma coordinate of a point P equals the angle F_ P F_ and the \tau coordinate equals the natural logarithm of the ratio of the distances d_ and d_ to opposite sides of the focal ring : \tau = \ln \frac. The coordinate ranges are -\pi<\sigma\le\pi, and Coordinate surfaces Surfaces of c ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Function
In physical science and mathematics, the Legendre functions , and associated Legendre functions , , and Legendre functions of the second kind, , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated Legendre polynomials are also solutions of the differential equation in special cases, which, by virtue of being polynomials, have a large number of additional properties, mathematical structure, and applications. For these polynomial solutions, see the separate Wikipedia articles. Legendre's differential equation The general Legendre equation reads \left(1 - x^2\right) y'' - 2xy' + \left[\lambda(\lambda+1) - \frac\right] y = 0, where the numbers and may be complex, and are called the degree and order of the relevant function, respectively. The polynomial solutions when is an integer (denoted ), and are the Legendre polynomials ; and when is an integer (denoted ), and is also an integer with are the associated Legendre polynomi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
