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Laplace 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 simple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...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] |
Stokes' Theorem
Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls, or simply the curl theorem, is a theorem in vector calculus on \R^3. Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical theorem of Stokes can be stated in one sentence: : The line integral of a vector field over a loop is equal to the surface integral of its '' curl'' over the enclosed surface. Stokes' theorem is a special case of the generalized Stokes theorem. In particular, a vector field on \R^3 can be considered as a 1-form in which case its curl is its exterior derivative, a 2-form. Theorem Let \Sigma be a smooth oriented surface in \R^3 with boundary \partial \Sigma \equiv \Gamma . If a vector field \mathbf(x,y,z) = (F_x(x, y, z), F_y(x, y, z), F_z(x, y, z)) is defined and has continuous first ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy–Riemann Equations
In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin-Louis Cauchy, Augustin Cauchy and Bernhard Riemann, consist of a system of differential equations, system of two partial differential equations which form a necessary and sufficient condition for a complex function of a complex variable to be complex differentiable. These equations are and where and are real differentiable function#Differentiability in higher dimensions, bivariate differentiable functions. Typically, and are respectively the real part, real and imaginary parts of a complex number, complex-valued function of a single complex variable where and are real variables; and are real differentiable functions of the real variables. Then is complex differentiable at a complex point if and only if the partial derivatives of and satisfy the Cauchy–Riemann equations at that point. A holomorphic function is a complex function that is differentiable at eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to the sum function given by its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable, that is, '' holomorphic functions''. The concept can be extended to functions of several complex variables. Complex analysis is contrasted with real analysis, which dea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superposition Principle
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input ''A'' produces response ''X'', and input ''B'' produces response ''Y'', then input (''A'' + ''B'') produces response (''X'' + ''Y''). A function F(x) that satisfies the superposition principle is called a linear function. Superposition can be defined by two simpler properties: additivity F(x_1 + x_2) = F(x_1) + F(x_2) and homogeneity F(ax) = a F(x) for scalar . This principle has many applications in physics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a linear system where the input stimulus is the load on the beam and the output response is the deflection of the beam. The importance of linear systems is that they are easier to analyze mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if for every x_0 in its domain, its Taylor series about x_0 converges to the function in some neighborhood of x_0 . This is stronger than merely being infinitely differentiable at x_0 , and therefore having a well-defined Taylor series; the Fabius function provides an example of a function that is infinitely differentiable but not analytic. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + \cdots in which the coefficients a_0, a_1, \dots a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Derivative
In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point. The directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a direction specified by v. The directional derivative of a scalar function ''f'' with respect to a vector v at a point (e.g., position) x may be denoted by any of the following: \begin \nabla_(\mathbf) &=f'_\mathbf(\mathbf)\\ &=D_\mathbff(\mathbf)\\ &=Df(\mathbf)(\mathbf)\\ &=\partial_\mathbff(\mathbf)\\ &=\mathbf\cdot\\ &=\mathbf\cdot \frac.\\ \end It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant. The directional derivative is a special case of the Gateaux derivative. Definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neumann Boundary Condition
In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann. When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative applied at the boundary of the domain. It is possible to describe the problem using other boundary conditions: a Dirichlet boundary condition specifies the values of the solution itself (as opposed to its derivative) on the boundary, whereas the Cauchy boundary condition, mixed boundary condition and Robin boundary condition are all different types of combinations of the Neumann and Dirichlet boundary conditions. Examples ODE For an ordinary differential equation, for instance, :y'' + y = 0, the Neumann boundary conditions on the interval take the form :y'(a)= \alpha, \quad y'(b) = \beta, where and are given numbers. PDE For a partial differential equation, for instance, :\nabla^2 y + y = 0, where denotes the Laplace operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Problem
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the problem can be stated as follows: :Given a function ''f'' that has values everywhere on the boundary of a region in \mathbb^n, is there a unique continuous function u twice continuously differentiable in the interior and continuous on the boundary, such that u is harmonic in the interior and u=f on the boundary? This requirement is called the Dirichlet boundary condition. The main issue is to prove the existence of a solution; uniqueness can be proven using the maximum principle. History The Dirichlet problem goes back to George Green, who studied the problem on general domains with general boundary conditions in his ''Essay on the Application o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christoffel Symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surface (topology), surfaces or other manifolds endowed with a metric tensor, metric, allowing distances to be measured on that surface. In differential geometry, an affine connection can be defined without reference to a metric, and many additional concepts follow: parallel transport, covariant derivatives, geodesics, etc. also do not require the concept of a metric. However, when a metric is available, these concepts can be directly tied to the "shape" of the manifold itself; that shape is determined by how the tangent space is attached to the cotangent space by the metric tensor. Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "vierbein, frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
