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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 con ...[...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 con ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Integrals
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function which can be expressed in the form f(x) = \int_^ R \left(t, \sqrt \right) \, dt, where is a rational function of its two arguments, is a polynomial of degree 3 or 4 with no repeated roots, and is a constant. In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of or if the integral is pseudo-elliptic. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legend ... [...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] |
Whipple Formulae
In the theory of special functions, 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 Whipple formulae for general complex index interchange of general associated Legendre functions of the first and second kind. These are given by : P_^\mu(z)=\frac\biggl \pi\sin\mu\pi P_^\nu\biggl(\frac\biggr)+\cos\pi(\nu+\mu)e^Q_^\nu\biggl(\frac\biggr) ... [...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] |
Electric Potential
The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in an electric field. More precisely, it is the energy per unit charge for a test charge that is so small that the disturbance of the field under consideration is negligible. Furthermore, the motion across the field is supposed to proceed with negligible acceleration, so as to avoid the test charge acquiring kinetic energy or producing radiation. By definition, the electric potential at the reference point is zero units. Typically, the reference point is earth or a point at infinity, although any point can be used. In classical electrostatics, the electrostatic field is a vector quantity expressed as the gradient of the electrostatic potential, which is a scalar quantity denoted by or occasionally , equal to the electric potential energy o ... [...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] |
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] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function which can be expressed in the form f(x) = \int_^ R \left(t, \sqrt \right) \, dt, where is a rational function of its two arguments, is a polynomial of degree 3 or 4 with no repeated roots, and is a constant. In general, integrals in this form cannot be expressed in terms of elementary functions. Exceptions to this general rule are when has repeated roots, or when contains no odd powers of or if the integral is pseudo-elliptic. However, with the appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions and the three Legend ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associated Legendre Functions
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\righty = 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 polynomials ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Coordinates
In mathematics, orthogonal coordinates are defined as a set of ''d'' coordinates q = (''q''1, ''q''2, ..., ''q''''d'') in which the coordinate hypersurfaces all meet at right angles (note: superscripts are indices, not exponents). A coordinate surface for a particular coordinate ''q''''k'' is the curve, surface, or hypersurface on which ''q''''k'' is a constant. For example, the three-dimensional Cartesian coordinates (''x'', ''y'', ''z'') is an orthogonal coordinate system, since its coordinate surfaces ''x'' = constant, ''y'' = constant, and ''z'' = 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 tho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
