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Uniqueness Theorem For Poisson's Equation
The uniqueness theorem for Poisson's equation states that, for a large class of boundary conditions, the equation may have many solutions, but the gradient of every solution is the same. In the case of electrostatics, this means that there is a unique electric field derived from a potential function satisfying Poisson's equation under the boundary conditions. __TOC__ Proof The general expression for Poisson's equation in electrostatics is :\mathbf^2 \varphi = -\frac, where \varphi is the electric potential and \rho_f is the charge distribution over some region V with boundary surface S . The uniqueness of the solution can be proven for a large class of boundary conditions as follows. Suppose that we claim to have two solutions of Poisson's equation. Let us call these two solutions \varphi_1 and \varphi_2. Then :\mathbf^2 \varphi_1 = - \frac, and :\mathbf^2 \varphi_2 = - \frac. It follows that \varphi=\varphi_2-\varphi_1 is a solution of Laplace's equation, which is a sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson's 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 or complex-valued 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-dimensional Cartesian coordinates, it takes the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Calculus
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, \mathbb^3. The term ''vector calculus'' is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. It is used extensively in physics and engineering, especially in the description of electromagnetic fields, gravitational fields, and fluid flow. Vector calculus was developed from the theory of quaternions by J. Willard Gibbs and Oliver Heaviside near the end of the 19th century, and most of the notation and terminology was established by Gibbs and Edwin Bidwell Wilson in their 1901 book, '' Vector Analysis'', though earlier mathematicians such as Isaac Newton pioneered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electrostatics
Electrostatics is a branch of physics that studies slow-moving or stationary electric charges. Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles after triboelectric effect, rubbing. The Greek language, Greek word (), meaning 'amber', was thus the Root (linguistics), root of the word ''electricity''. Electrostatic phenomena arise from the forces that electric charges exert on each other. Such forces are described by Coulomb's law. There are many examples of electrostatic phenomena, from those as simple as the attraction of plastic wrap to one's hand after it is removed from a package, to the apparently spontaneous explosion of grain silos, the damage of electronic components during manufacturing, and photocopier and laser printing, laser printer operation. The electrostatic model accurately predicts electrical phenomena in "classical" cases where the velocities are low and the system is macroscopic so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American Multinational corporation, multinational Publishing, publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and produces books, Academic journal, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, Technology, technical, and engineering subject areas, abandoning its literary interests. Wiley's son Joh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Butterworth–Heinemann
Butterworth–Heinemann is a British publishing company specialised in professional information and learning materials for higher education and professional training, in printed and electronic forms. It was formed in 1990 by the merger of Heinemann Professional Publishing and Butterworths Scientific, both subsidiaries of Reed International. With its earlier constituent companies, the founding dates back to 1923. It has publishing units in Oxford (UK) and Waltham, Massachusetts (United States). As of 2006, it is an imprint of Elsevier Elsevier ( ) is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell (journal), Cell'', the ScienceDirect collection of electronic journals, .... See also * LexisNexis Butterworths References External links * Book publishing companies of the United Kingdom Elsevier imprints {{publish-corp-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, every function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a cen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniqueness Theorem
In mathematics, a uniqueness theorem, also called a unicity theorem, is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. Examples of uniqueness theorems include: * Cauchy's theorem (geometry), Cauchy's rigidity theorem and Alexandrov's uniqueness theorem for three-dimensional polyhedra. * Black hole uniqueness theorem * Cauchy–Kowalevski theorem is the main local existence theorem, existence and uniqueness theorem for analytic function, analytic partial differential equations associated with Cauchy problem, Cauchy initial value problems. * Cauchy–Kowalevski theorem#Cauchy–Kowalevski–Kashiwara theorem, Cauchy–Kowalevski–Kashiwara theorem is a wide generalization of the Cauchy–Kowalevski theorem for systems of linear partial differential equations with analytic coefficients. *Euclidean division#Statement of the theorem, Division theorem, the uniqueness of quotient and remainder ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Green's Function
In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if L is a linear differential operator, then * the Green's function G is the solution of the equation where \delta is Dirac's delta function; * the solution of the initial-value problem L y = f is the convolution Through the superposition principle, given a linear ordinary differential equation (ODE), one can first solve for each , and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of . Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead. Under many ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Method Of Images
The method of images (or method of mirror images) is a mathematical tool for solving differential equations, in which boundary value problem, boundary conditions are satisfied by combining a solution not restricted by the boundary conditions with its possibly weighted mirror image. Generally, original singularities are inside the domain of interest but the function is made to satisfy boundary conditions by placing additional singularities outside the domain of interest. Typically the locations of these additional singularities are determined as the virtual location of the original singularities as viewed in a mirror placed at the location of the boundary conditions. Most typically, the mirror is a hyperplane or hypersphere. The method of images can also be used in solving discrete problems with boundary conditions, such counting the number of restricted discrete random walks. Method of image charges The method of image charges is used in electrostatics to simply calculate or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coulomb's Law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric force is conventionally called the ''electrostatic force'' or Coulomb force. Although the law was known earlier, it was first published in 1785 by French physicist Charles-Augustin de Coulomb. Coulomb's law was essential to the development of the classical electromagnetism, theory of electromagnetism and maybe even its starting point, as it allowed meaningful discussions of the amount of electric charge in a particle. The law states that the magnitude, or absolute value, of the attractive or repulsive electrostatic force between two point Electric charge, charges is directly proportional to the product of the magnitudes of their charges and inversely proportional to the square of the distance between them. Coulomb discovered that bodies with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Boundary Condition
In mathematics, a mixed boundary condition for a partial differential equation defines a boundary value problem in which the solution of the given equation is required to satisfy different boundary conditions on disjoint parts of the boundary of the domain where the condition is stated. Precisely, in a mixed boundary value problem, the solution is required to satisfy a Dirichlet or a Neumann boundary condition in a mutually exclusive way on disjoint parts of the boundary. For example, given a solution to a partial differential equation on a domain with boundary , it is said to satisfy a mixed boundary condition if, consisting of two disjoint parts, and , such that , verifies the following equations: :\left. u \_ = u_0and\left. \frac\_ = g, where and are given functions defined on those portions of the boundary. The mixed boundary condition differs from the Robin boundary condition in that the latter requires a linear combination, possibly with pointwise variable coeffi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
