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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 :\iint f(x,y)\,dx\,dy. In general, although these two can be different, Fubini's theorem In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the ... ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariable Calculus
Multivariable calculus (also known as multivariate calculus) is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving several variables, rather than just one. Multivariable calculus may be thought of as an elementary part of advanced calculus. For advanced calculus, see calculus on Euclidean space. The special case of calculus in three dimensional space is often called vector calculus. Typical operations Limits and continuity A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions. For example, there are scalar functions of two variables with points in their domain which give different limits when approached along different paths. E.g., the function. :f(x,y) = \frac approaches zero whenever the point (0,0) is approached along lines through the origin (y=kx). However, when the origin is appr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In 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 ..., an integral assigns numbers to functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with Derivative, differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the function and the set is called the codomain of the function.Codomain ''Encyclopedia of Mathematics'Codomain. ''Encyclopedia of Mathematics''/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians Al-Biruni and Sharaf al-Din al-Tusi. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function Of Several Real Variables
In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or complex. However, the study of the complex-valued functions may be easily reduced to the study of the real-valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real-valued functions will be considered in this article. The domain of a function of variables is the subset of for which the function is defined. As usual, the domain of a function of several real variables is supposed to contain a nonempty open subset of . General definition A real-valued function of real variables is a function that take ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant (mathematics)
In mathematics, the word constant conveys multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other value); as a noun, it has two different meanings: * A fixed and well-defined number or other non-changing mathematical object. The terms '' mathematical constant'' or '' physical constant'' are sometimes used to distinguish this meaning. * A function whose value remains unchanged (i.e., a constant function). Such a constant is commonly represented by a variable which does not depend on the main variable(s) in question. For example, a general quadratic function is commonly written as: :a x^2 + b x + c\, , where , and are constants (or parameters), and a variable—a placeholder for the argument of the function being studied. A more explicit way to denote this function is :x\mapsto a x^2 + b x + c \, , which makes the function-argument status of (and by extension the constancy of , and ) clear. In this example , and are co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc. ''Parameter'' has more specific meanings within various disciplines, including mathematics, computer programming, engineering, statistics, logic, linguistics, and electronic musical composition. In addition to its technical uses, there are also extended uses, especially in non-scientific contexts, where it is used to mean defining characteristics or boundaries, as in the phrases 'test parameters' or 'game play parameters'. Modelization When a system is modeled by equations, the values that describe the system are called ''parameters''. For example, in mechanics, the masses, the dimensions and shapes (for solid bodies), the densities and the viscosities ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 plane) are called double integrals, and integrals of a function of three variables over a region in \mathbb^3 (real-number 3D space) are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration. Introduction Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the -axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function (on the three-dimensional Cartesian plane where ) and the plane which contains its domain. If there are more variables, a multiple integral will yield hypervolumes of multidimens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fubini's Theorem
In mathematical analysis Fubini's theorem is a result that gives conditions under which it is possible to compute a double integral by using an iterated integral, introduced by Guido Fubini in 1907. One may switch the order of integration if the double integral yields a finite answer when the integrand is replaced by its absolute value. \, \iint\limits_ f(x,y)\,\text(x,y) = \int_X\left(\int_Y f(x,y)\,\texty\right)\textx=\int_Y\left(\int_X f(x,y) \, \textx \right) \texty \qquad \text \qquad \iint\limits_ , f(x,y), \,\text(x,y) <+\infty. Fubini's theorem implies that two iterated integrals are equal to the corresponding double integral across its integrands. Tonelli's theorem, introduced by in 1909, is similar, but applies to a non-negative measurable function rather than one integrable over their domains. A related theorem is oft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Order Of Operations
In mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For example, in mathematics and most computer languages, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation. Thus, the expression is interpreted to have the value , and not . When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication, and could be placed only as a superscript to the right of their base. Thus and . These conventions exist to eliminate notational ambiguity, while allowing notation to be as brief as possible. Where it is desired to override the precedence conventions, or even simply to emphasize them, parentheses ( ) can be used. For example, forces addition to precede multipli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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