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Integral Operator (other)
An integral operator is an operator that involves integration. Special instances are: * The operator of integration itself, denoted by the integral symbol * Integral linear operators, which are linear operators induced by bilinear forms involving integrals * Integral transforms, which are maps between two function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ve ...s, which involve integrals {{Authority control Integral calculus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator (mathematics)
In mathematics, an operator is generally a Map (mathematics), mapping or function (mathematics), function that acts on elements of a space (mathematics), space to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an ''operator'', but the term is often used in place of ''function'' when the domain of a function, domain is a set of functions or other structured objects. Also, the domain of an operator is often difficult to characterize explicitly (for example in the case of an integral operator), and may be extended so as to act on related objects (an operator that acts on functions may act also on differential equations whose solutions are functions that satisfy the equation). (see Operator (physics) for other examples) The most basic operators are linear maps, which act on vector spaces. Linear operators refer to linear maps whose domain and range are the same space, for example from \mathbb^n to \mathbb^n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. the other being Derivative, differentiation. Integration was initially used to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Usage of integration expanded to a wide variety of scientific fields thereafter. A definite integral computes the signed area of the region in the plane that is bounded by the Graph of a function, graph of a given Function (mathematics), function between two points in the real line. Conventionally, areas above the horizontal Coordinate axis, axis of the plane are positive while areas below are n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Symbol
The integral symbol (''see below'') is used to denote integrals and Antiderivative, antiderivatives in mathematics, especially in calculus. History The notation was introduced by the Germany, German mathematician Gottfried Wilhelm Leibniz in 1675 in his private writings; it first appeared publicly in the article "" (On a hidden geometry and analysis of indivisibles and infinites), published in ''Acta Eruditorum'' in June 1686. The symbol was based on the ſ (long s) character and was chosen because Leibniz thought of the integral as an infinite summation, sum of infinitesimal summands. Typography in Unicode and LaTeX Fundamental symbol The integral symbol is in Unicode and \int in LaTeX. In HTML, it is written as ∫ (hexadecimal), ∫ (decimal) and ∫ (List of XML and HTML character entity references#Character entity references in HTML, named entity). The original IBM PC code page 437 character set included a couple of characters ⌠,⎮ an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Linear Operator
In mathematical analysis, an integral linear operator is a linear operator ''T'' given by integration; i.e., :(Tf)(x) = \int f(y) K(x, y) \, dy where K(x, y) is called an integration kernel. More generally, an integral bilinear form is a bilinear functional that belongs to the continuous dual space of X \widehat_ Y, the injective tensor product of the locally convex topological vector spaces (TVSs) ''X'' and ''Y''. An integral linear operator is a continuous linear operator that arises in a canonical way from an integral bilinear form. These maps play an important role in the theory of nuclear spaces and nuclear maps. Definition - Integral forms as the dual of the injective tensor product Let ''X'' and ''Y'' be locally convex TVSs, let X \otimes_ Y denote the projective tensor product, X \widehat_ Y denote its completion, let X \otimes_ Y denote the injective tensor product, and X \widehat_ Y denote its completion. Suppose that \operatorname : X \otimes_ Y \to X \wid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a linear endomorphism. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear map' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilinear Form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linear in each argument separately: * and * and The dot product on \R^n is an example of a bilinear form which is also an inner product. An example of a bilinear form that is not an inner product would be the four-vector product. The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When is the field of complex numbers , one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument. Coordinate representation Let be an - dimensional vector space with basis . The matrix ''A'', defined by is called the ''matrix of the bilinear form'' on the basis . If the matrix represents a ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Transform
In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the ''inverse transform''. General form An integral transform is any transform ''T'' of the following form: :(Tf)(u) = \int_^ f(t)\, K(t, u)\, dt The input of this transform is a function ''f'', and the output is another function ''Tf''. An integral transform is a particular kind of mathematical operator. There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables, that is called the kernel or nucleus of the transform. Some kernels have an associated ''inverse kernel'' K^( u,t ) which (roughly speaking) yields an inverse transform: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Space
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication. In other scenarios, the function space might inherit a topological or metric structure, hence the name function ''space''. In linear algebra Let be a field and let be any set. The functions → can be given the structure of a vector space over where the operations are defined pointwise, that is, for any , : → , any in , and any in , define \begin (f+g)(x) &= f(x)+g(x) \\ (c\cdot f)(x) &= c\cdot f(x) \end When the domain has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure. For example, if and also itself are vector spaces over , the se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
