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Bipolar Coordinates
Bipolar coordinates are a two-dimensional orthogonal coordinates, orthogonal coordinate system based on the Apollonian circles.Eric W. Weisstein, Concise Encyclopedia of Mathematics CD-ROM, ''Bipolar Coordinates'', CD-ROM edition 1.0, May 20, 1999 There is also a third system, based on two poles (biangular coordinates). The term "bipolar" is further used on occasion to describe other curves having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. However, the term ''bipolar coordinates'' is reserved for the coordinates described here, and never used for systems associated with those other curves, such as elliptic coordinates. Definition The system is based on two Focus (geometry), foci ''F''1 and ''F''2. Referring to the figure at right, the ''σ''-coordinate of a point ''P'' equals the angle ''F''1 ''P'' ''F''2, and the ''τ''-coordinate equals the natural logarithm of the ratio of the distances ''d''1 and ''d''2: : \tau = \ln \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipolar Tau Isosurfaces
Bipolar may refer to: Astronomy * Bipolar nebula, a distinctive nebular formation * Bipolar outflow, two continuous flows of gas from the poles of a star Mathematics * Bipolar coordinates, a two-dimensional orthogonal coordinate system * Bipolar set, a derivative of a polar set * Bipolar theorem, a theorem in convex analysis which provides necessary and sufficient conditions for a cone to be equal to its bipolar Medicine * Bipolar disorder, a mental disorder that causes periods of depression and periods of elevated mood ** Bipolar I disorder, a bipolar spectrum disorder characterized by the occurrence of at least one manic or mixed episode ** Bipolar II disorder, a bipolar spectrum disorder characterized by at least one episode of hypomania and at least one episode of major depression ** Bipolar disorder not otherwise specified, a diagnosis for bipolar disorder when it does not fall within the other established sub-types * Bipolar neuron, a type of neuron which has two extension ... [...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] |
Elliptic Coordinate System
In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are 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 coordinates (\mu, \nu) is :\begin x &= a \ \cosh \mu \ \cos \nu \\ y &= a \ \sinh \mu \ \sin \nu \end where \mu is a nonnegative real number and \nu \in , 2\pi On the complex plane, an equivalent relationship is :x + iy = a \ \cosh(\mu + i\nu) 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 In an orthogonal coordinate system the lengths of t ... [...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] |
Bispherical Coordinates
Bispherical coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that connects the two foci. Thus, the two foci F_ and F_ in bipolar coordinates remain points (on the z-axis, the axis of rotation) in the bispherical coordinate system. Definition The most common definition of bispherical coordinates (\tau, \sigma, \phi) is :\begin x &= a \ \frac \cos \phi, \\ y &= a \ \frac \sin \phi, \\ z &= a \ \frac, \end where 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 the foci : \tau = \ln \frac The coordinates ranges are −∞ < < ∞, 0 ≤ ≤ and 0 ≤ ≤ 2. Coordinate surfaces Surfaces of constant correspond to intersecting tori of different ra ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipolar Cylindrical Coordinates
Bipolar cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional bipolar coordinate system in the perpendicular z-direction. The two lines of foci F_ and F_ of the projected Apollonian circles are generally taken to be defined by x=-a and x=+a, respectively, (and by y=0) in the Cartesian coordinate system. The term "bipolar" is often used to describe other curves having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. However, the term ''bipolar coordinates'' is never used to describe coordinates associated with those curves, e.g., elliptic coordinates. Basic definition The most common definition of bipolar cylindrical coordinates (\sigma, \tau, z) is : x = a \ \frac : y = a \ \frac : z = \ z where 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 the focal lines : \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electric Field
An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) describes their capacity to exert attractive or repulsive forces on another charged object. Charged particles exert attractive forces on each other when the sign of their charges are opposite, one being positive while the other is negative, and repel each other when the signs of the charges are the same. Because these forces are exerted mutually, two charges must be present for the forces to take place. These forces are described by Coulomb's law, which says that the greater the magnitude of the charges, the greater the force, and the greater the distance between them, the weaker the force. Informally, the greater the charge of an object, the stronger its electric field. Similarly, an electric field is stronger nearer charged objects and weaker f ... [...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 differential equation, 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) A differential equation for the unknown f(x) is separable if it can be written in the form :\frac f(x) = g(x)h(f(x)) where g and h are given functions. This is perhaps more transparent when written using y = f(x) as: :\frac=g(x)h(y). So now as long as ''h''(''y'') ≠ 0, we can rearrange terms to obtain: : = g(x) \, dx, where the two variables ''x'' and ''y'' have been separated. Note ''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. Al ... [...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] |
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] |
Partial Differential Equations
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae 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 existence an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
