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Order Of Integration (calculus)
In calculus, interchange of the order of integration is a methodology that transforms iterated integrals (or multiple integrals through the use of Fubini's theorem) of functions into other, hopefully simpler, integrals by changing the order in which the integrations are performed. In some cases, the order of integration can be validly interchanged; in others it cannot. Problem statement The problem for examination is evaluation of an integral of the form : \iint_D \ f(x,y ) \ dx \,dy , where ''D'' is some two-dimensional area in the ''xy''–plane. For some functions ''f'' straightforward integration is feasible, but where that is not true, the integral can sometimes be reduced to simpler form by changing the order of integration. The difficulty with this interchange is determining the change in description of the domain ''D''. The method also is applicable to other multiple integrals. Sometimes, even though a full evaluation is difficult, or perhaps requires a numerical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing. Today, calculus has widespread uses in scienc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime (symbol)
The prime symbol , double prime symbol , triple prime symbol , and quadruple prime symbol are used to designate units and for other purposes in mathematics, science, linguistics and music. Although the characters differ little in appearance from those of the apostrophe and single and double quotation marks, the uses of the prime symbol are quite different. While an apostrophe is now often used in place of the prime, and a double quote in place of the double prime (due to the lack of prime symbols on everyday writing keyboards), such substitutions are not considered appropriate in formal materials or in typesetting. Designation of units The prime symbol is commonly used to represent feet (ft), and the double prime is used to represent inches (in). The triple prime as used in watchmaking represents a ( of a ''French'' inch or '' pouce'', about ). Primes are also used for angles. The prime symbol is used for arcminutes ( of a degree), and the double prime for arcsecon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Murray H
Murray may refer to: Businesses * Murray (bicycle company), an American manufacturer of low-cost bicycles * Murrays, an Australian bus company * Murray International Trust, a Scottish investment trust * D. & W. Murray Limited, an Australian wholesale drapery business * John Murray (publishing house), a British publishing house Fictional characters * Murray Monster, a muppet in ''Sesame Street'' *Little Murray Sparkles, a cat in ''Sesame Street'' * Murray (''Monkey Island''), a character in the video game series * Murray (''Sly Cooper''), a character in the video game series * Murray Slaughter, a regular character in ''The Mary Tyler Moore Show'' *Murray, the mascot of the band Dio *Murray, in the 2015 Netflix series '' Richie Rich'' *Murray, a ''Hotel Transylvania'' character *Murray the Cop, in ''Fat Pizza'' *Murray Smith, in '' Swift and Shift Couriers'' People *Murray (surname) * Murray (given name) Places Australia * Division of Murray, federal electoral district in V ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sokhotski–Plemelj Theorem
The Sokhotski–Plemelj theorem (Polish spelling is ''Sochocki'') is a theorem in complex analysis, which helps in evaluating certain integrals. The real-line version of it ( see below) is often used in physics, although rarely referred to by name. The theorem is named after Julian Sochocki, who proved it in 1868, and Josip Plemelj, who rediscovered it as a main ingredient of his solution of the Riemann–Hilbert problem in 1908. Statement of the theorem Let ''C'' be a smooth closed simple curve in the plane, and \varphi an analytic function on ''C''. Note that the Cauchy-type integral : \phi(z) = \frac \int_C\frac, cannot be evaluated for any ''z'' on the curve ''C''. However, on the interior and exterior of the curve, the integral produces analytic functions, which will be denoted \phi_i inside ''C'' and \phi_e outside. The Sokhotski–Plemelj formulas relate the limiting boundary values of these two analytic functions at a point ''z'' on ''C'' and the Cauchy princ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Fractions In Integration
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as a sum of a polynomial (possibly zero) and one or several fractions with a simpler denominator. The importance of the partial fraction decomposition lies in the fact that it provides algorithms for various computations with rational functions, including the explicit computation of antiderivatives, Taylor series expansions, inverse Z-transforms, and inverse Laplace transforms. The concept was discovered independently in 1702 by both Johann Bernoulli and Gottfried Leibniz. In symbols, the ''partial fraction decomposition'' of a rational fraction of the form \frac, where and are polynomials, is its expression as \frac=p(x) + \sum_j \frac where is a polynomial, and, for each , the denominator is a power of an irreducible polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Course Of Modern Analysis
''A Course of Modern Analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions'' (colloquially known as Whittaker and Watson) is a landmark textbook on mathematical analysis written by Edmund T. Whittaker and George N. Watson, first published by Cambridge University Press in 1902. The first edition was Whittaker's alone, but later editions were co-authored with Watson. History Its first, second, third, and the fourth edition were published in 1902, 1915, 1920, and 1927, respectively. Since then, it has continuously been reprinted and is still in print today. A revised, expanded and digitally reset fifth edition, edited by Victor H. Moll, was published in 2021. The book is notable for being the standard reference and textbook for a generation of Cambridge mathematicians including Littlewood and Godfrey H. Hardy. Mary L. Cartwright studied it as preparation for her final hono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Principal Value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. Formulation Depending on the type of singularity in the integrand , the Cauchy principal value is defined according to the following rules: In some cases it is necessary to deal simultaneously with singularities both at a finite number and at infinity. This is usually done by a limit of the form \lim_\, \lim_ \,\left ,\int_^ f(x)\,\mathrmx \,~ + ~ \int_^ f(x)\,\mathrmx \,\right In those cases where the integral may be split into two independent, finite limits, \lim_ \, \left, \,\int_a^ f(x)\,\mathrmx \,\\; < \;\infty and then the function is integrable in the ordinary sense. The result of the procedure for principal value is the same as the ordinary integral; since it no longer matches the definition, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange's Notation
In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians. The usefulness of each notation varies with the context, and it is sometimes advantageous to use more than one notation in a given context. The most common notations for differentiation (and its opposite operation, the antidifferentiation or indefinite integration) are listed below. Leibniz's notation The original notation employed by Gottfried Leibniz is used throughout mathematics. It is particularly common when the equation is regarded as a functional relationship between dependent and independent variables and . Leibniz's notation makes this relationship explicit by writing the derivative as :\frac. Furthermore, the derivative of at is therefore written :\frac(x)\text\frac\text\frac f(x). Higher derivatives are written as :\frac, \frac, \frac, \ldots, \frac. Thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integration By Parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation. The integration by parts formula states: \begin \int_a^b u(x) v'(x) \, dx & = \Big (x) v(x)\Biga^b - \int_a^b u'(x) v(x) \, dx\\ & = u(b) v(b) - u(a) v(a) - \int_a^b u'(x) v(x) \, dx. \end Or, letting u = u(x) and du = u'(x) \,dx while v = v(x) and dv = v'(x) \, dx, the formula can be written more compactly: \int u \, dv \ =\ uv - \int v \, du. Mathematician Brook Taylor discovered integration by parts, first publishing the idea in 1715. More general formulations of integration by parts ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterated Integral
In multivariable calculus, an iterated integral is the result of applying integrals to a function of more than one variable (for example f(x,y) or f(x,y,z)) in a way that each of the integrals considers some of the variables as given constants. For example, the function f(x,y), if y is considered a given parameter, can be integrated with respect to x, \int f(x,y)\,dx. The result is a function of y and therefore its integral can be considered. If this is done, the result is the iterated integral :\int\left(\int f(x,y)\,dx\right)\,dy. It is key for the notion of iterated integrals that this is different, in principle, from the multiple integral In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ... :\iint f(x,y)\,dx\,dy. In general, although these two can be different, Fubini's theor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integration Order
Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, performed by a specific class of recombinase enzymes ("integrases") Economics and law *Economic integration, trade unification between different states *Horizontal integration and vertical integration, in microeconomics and strategic management, styles of ownership and control *Regional integration, in which states cooperate through regional institutions and rules * Integration clause, a declaration that a contract is the final and complete understanding of the parties *A step in the process of money laundering *Integrated farming, a farm management system * Integration (tax), a feature of corporate and personal income tax in some countries Engineering *Data integration * Digital integration *Enterprise integration *Integrated archit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to ''quadrature'') is more or less a synonym for ''numerical integration'', especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take ''quadrature'' to include higher-dimensional integration. The basic problem in numerical integration is to compute an approximate solution to a definite integral :\int_a^b f(x) \, dx to a given degree of accuracy. If is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
