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Spherical Mean
In mathematics, the spherical mean of a function (mathematics), function around a point is the average of all values of that function on a sphere of given radius centered at that point. Definition Consider an open set ''U'' in the Euclidean space R''n'' and a continuous function ''u'' defined on ''U'' with real number, real or complex number, complex values. Let ''x'' be a point in ''U'' and ''r'' > 0 be such that the closed set, closed ball (mathematics), ball ''B''(''x'', ''r'') of center ''x'' and radius ''r'' is contained in ''U''. The spherical mean over the sphere of radius ''r'' centered at ''x'' is defined as : \frac\int\limits_ \! u(y) \, \mathrm S(y) where ∂''B''(''x'', ''r'') is the n-sphere, (''n'' − 1)-sphere forming the boundary (topology), boundary of ''B''(''x'', ''r''), d''S'' denotes integration with respect to spherical measure and ''ω''''n''−1(''r'') is the "surface area" of this (''n'' &minus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Mean
In mathematics, the spherical mean of a function (mathematics), function around a point is the average of all values of that function on a sphere of given radius centered at that point. Definition Consider an open set ''U'' in the Euclidean space R''n'' and a continuous function ''u'' defined on ''U'' with real number, real or complex number, complex values. Let ''x'' be a point in ''U'' and ''r'' > 0 be such that the closed set, closed ball (mathematics), ball ''B''(''x'', ''r'') of center ''x'' and radius ''r'' is contained in ''U''. The spherical mean over the sphere of radius ''r'' centered at ''x'' is defined as : \frac\int\limits_ \! u(y) \, \mathrm S(y) where ∂''B''(''x'', ''r'') is the n-sphere, (''n'' − 1)-sphere forming the boundary (topology), boundary of ''B''(''x'', ''r''), d''S'' denotes integration with respect to spherical measure and ''ω''''n''−1(''r'') is the "surface area" of this (''n'' &minus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boundary (topology)
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set include \operatorname(S), \operatorname(S), and \partial S. Some authors (for example Willard, in ''General Topology'') use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, ''Metric Spaces'' by E. T. Copson uses the term boundary to refer to Hausdorff's border, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term residue, which is defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations. In the simplest case, consider a function of two variables such that :\frac+\frac=0. The weak maximum principle, in this setting, says that for any open precompact subset of the domain of , the maximum of on the closure of is achieved on the boundary of . The strong maximum principle says that, unless is a constant function, the maximum cannot also be achieved anywhere on itself. Such statements give a striking qualitative picture of solutions of the given differential equation. Such a qualitative picture can be extended to many kinds of differential equations. In many situations, one can also use such maximum principles to draw precise quantitative conclusions about solutions of differential equations, such as control over the size ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \frac + \frac + \cdots + \frac = 0 everywhere on . This is usually written as : \nabla^2 f = 0 or :\Delta f = 0 Etymology of the term "harmonic" The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as ''harmonics''. Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit ''n''-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and over time "harmonic" ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all Order of derivation, orders in its Domain of a function, domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. Single mechanical or electromagnetic waves propagating in a pre-defined direction can also be described with the first-order one-way wave equation which is much easier to solve and also valid for inhomogenious media. Introduction The (two-way) wave equation is a second-order partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions of a time variable (a variable repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
D'Alembert's Formula
In mathematics, and specifically partial differential equations (PDEs), d'Alembert's formula is the general solution to the one-dimensional wave equation u_(x,t) = c^2 u_(x,t) (where subscript indices indicate partial differentiation, using the d'Alembert operator, the PDE becomes: \Box u = 0). The solution depends on the initial conditions at t = 0: u(x, 0) and u_t(x, 0). It consists of separate terms for the initial conditions u(x,0) and u_t(x,0): u(x,t) = \frac\left[u(x-ct, 0) + u(x+ct, 0)\right] + \frac \int_^ u_t(\xi, 0) \, d\xi. It is named after the mathematician Jean le Rond d'Alembert, who derived it in 1747 as a solution to the problem of a String vibration, vibrating string. Details The method of characteristics, characteristics of the PDE are x \pm ct = \mathrm (where \pm sign states the two solutions to quadratic equation), so we can use the change of variables \mu = x + ct (for the positive solution) and \eta = x-ct (for the negative solution) to transform the PD ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. Single mechanical or electromagnetic waves propagating in a pre-defined direction can also be described with the first-order one-way wave equation which is much easier to solve and also valid for inhomogenious media. Introduction The (two-way) wave equation is a second-order partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions of a time variable (a variable repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Of A Function
Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function ''f'' assigns an output ''f''(''x'') to every input ''x''. We say that the function has a limit ''L'' at an input ''p,'' if ''f''(''x'') gets closer and closer to ''L'' as ''x'' moves closer and closer to ''p''. More specifically, when ''f'' is applied to any input ''sufficiently'' close to ''p'', the output value is forced ''arbitrarily'' close to ''L''. On the other hand, if some inputs very close to ''p'' are taken to outputs that stay a fixed distance apart, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Measure
In mathematics — specifically, in geometric measure theory — spherical measure ''σ''''n'' is the "natural" Borel measure on the ''n''-sphere S''n''. Spherical measure is often normalized so that it is a probability measure on the sphere, i.e. so that ''σ''''n''(S''n'') = 1. Definition of spherical measure There are several ways to define spherical measure. One way is to use the usual "round" or " arclength" metric ''ρ''''n'' on S''n''; that is, for points ''x'' and ''y'' in S''n'', ''ρ''''n''(''x'', ''y'') is defined to be the (Euclidean) angle that they subtend at the centre of the sphere (the origin of R''n''+1). Now construct ''n''-dimensional Hausdorff measure ''H''''n'' on the metric space (S''n'', ''ρ''''n'') and define :\sigma^ = \frac H^. One could also have given S''n'' the metric that it inherits as a subspace of the Euclidean space R''n''+1; the same spherical measure results from this choice of metric. Ano ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-sphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, called the ''center''. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit -sphere or simply the -sphere for brevity. In terms of the standard norm, the -sphere is defined as : S^n = \left\ , and an -sphere of radius can be defined as : S^n(r) = \left\ . The dimension of -sphere is , and must not be confused with the dimension of the Euclidean space in which it is naturally embedded. An -sphere is the surface or boundary of an -dimensional ball. In particular: *the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere, *a circle, which i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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