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 picture info Continuous Function * v * t * e In mathematics , a CONTINUOUS FUNCTION is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism . Continuity of functions is one of the core concepts of topology , which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers . A stronger form of continuity is uniform continuity . In addition, this article discusses the definition for the more general case of functions between two metric spaces . In order theory , especially in domain theory , one considers a notion of continuity known as Scott continuity . Other forms of continuity do exist but they are not discussed in this article [...More...] "Continuous Function" on: Wikipedia Google Yahoo Partial Fractions In Integration PARTIAL may refer to: Mathematics * Partial derivative Partial derivative * ∂ , the partial derivative symbol, often read as "partial" * Partial differential equation [...More...] "Partial Fractions In Integration" on: Wikipedia Google Yahoo Differential (mathematics) In mathematics , DIFFERENTIAL refers to infinitesimal differences or to the derivatives of functions. This article links to differentials in various branches of mathematics such as calculus , differential geometry , algebraic geometry and algebraic topology . CONTENTS * 1 Basic notions * 2 Differential geometry Differential geometry * 3 Algebraic geometry Algebraic geometry * 4 Other meanings * 5 References * 6 External links BASIC NOTIONS* In calculus , the differential represents a change in the linearization of a function . * The total differential is its generalization for functions of multiple variables. * In traditional approaches to calculus, the differentials (e.g. dx, dy, dt, etc.) are interpreted as infinitesimals [...More...] "Differential (mathematics)" on: Wikipedia Google Yahoo General Leibniz Rule In calculus , the GENERAL LEIBNIZ RULE, named after Gottfried Wilhelm Leibniz , generalizes the product rule (which is also known as "Leibniz's rule"). It states that if f {displaystyle f} and g {displaystyle g} are n {displaystyle n} -times differentiable functions , then the product f g {displaystyle fg} is also n {displaystyle n} -times differentiable and its n {displaystyle n} th derivative is given by ( f g ) ( n ) ( x ) = k = 0 n ( n k ) f ( n k ) ( x ) g ( k ) ( x ) {displaystyle (fg)^{(n)}(x)=sum _{k=0}^{n}{n choose k}f^{(n-k)}(x)g^{(k)}(x)} where ( n k ) = n ! k ! ( n k ) ! {displaystyle {n choose k}={n! over k!(n-k)!}} is the binomial coefficient and f ( 0 ) ( x ) = f ( x ) {displaystyle f^{(0)}(x)=f(x)} . This can be proved by using the product rule and mathematical induction (see proof below) [...More...] "General Leibniz Rule" on: Wikipedia Google Yahoo picture info Shell Integration SHELL INTEGRATION (the SHELL METHOD in integral calculus ) is a means of calculating the volume of a solid of revolution , when integrating along an axis perpendicular to the axis of revolution. This is in contrast to disk integration which integrates along the axis parallel to the axis of revolution. CONTENTS * 1 Definition * 2 Example * 3 See also * 4 References DEFINITIONThe shell method goes as follows: Consider a volume in three dimensions obtained by rotating a cross-section in the xy-plane around the y-axis. Suppose the cross-section is defined by the graph of the positive function f(x) on the interval [...More...] "Shell Integration" on: Wikipedia Google Yahoo Trigonometric Substitution In mathematics , TRIGONOMETRIC SUBSTITUTION is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities to simplify certain integrals containing radical expressions : SUBSTITUTION 1. If the integrand contains a2 − x2, let x = a sin {displaystyle x=asin theta } and use the identity 1 sin 2 = cos 2 . {displaystyle 1-sin ^{2}theta =cos ^{2}theta .} SUBSTITUTION 2. If the integrand contains a2 + x2, let x = a tan {displaystyle x=atan theta } and use the identity 1 + tan 2 = sec 2 . {displaystyle 1+tan ^{2}theta =sec ^{2}theta .} SUBSTITUTION 3. If the integrand contains x2 − a2, let x = a sec {displaystyle x=asec theta } and use the identity sec 2 1 = tan 2 [...More...] "Trigonometric Substitution" on: Wikipedia Google Yahoo picture info Order Of Integration (calculus) * v * t * e 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. CONTENTS * 1 Problem statement * 2 Relation to integration by parts * 3 Principal-value integrals * 4 Basic theorems * 5 See also * 6 References and notes * 7 External links PROBLEM STATEMENTThe problem for examination is evaluation of an integral of the form D f ( x , y ) d x d y , {displaystyle 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 [...More...] "Order Of Integration (calculus)" on: Wikipedia Google Yahoo picture info Disc Integration DISC INTEGRATION, also known in integral calculus as the DISC METHOD, is a means of calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution . This method models the resulting three-dimensional shape as a stack of an infinite number of discs of varying radius and infinitesimal thickness. It is also possible to use the same principles with rings instead of discs (the "WASHER METHOD") to obtain hollow solids of revolutions. This is in contrast to shell integration which integrates along the axis perpendicular to the axis of revolution [...More...] "Disc Integration" on: Wikipedia Google Yahoo picture info Integration By Parts In calculus , and more generally in mathematical analysis , INTEGRATION BY PARTS or PARTIAL INTEGRATION is a theorem that relates the integral of a product of functions to the integral 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 derived in one line simply by integrating the product rule of differentiation [...More...] "Integration By Parts" on: Wikipedia Google Yahoo Power Rule In calculus , the POWER RULE is used to differentiate functions of the form f ( x ) = x r {displaystyle f(x)=x^{r}} , whenever r {displaystyle r} is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated using this rule [...More...] "Power Rule" on: Wikipedia Google Yahoo Quotient Rule In calculus , the QUOTIENT RULE is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let f ( x ) = g ( x ) / h ( x ) , {displaystyle f(x)=g(x)/h(x),} where both g {displaystyle g} and h {displaystyle h} are differentiable and h ( x ) 0. {displaystyle h(x)not =0.} The quotient rule states that the derivative of f ( x ) {displaystyle f(x)} is f ( x ) = g ( x ) h ( x ) g ( x ) h ( x ) 2 [...More...] "Quotient Rule" on: Wikipedia Google Yahoo Lists Of Integrals Integration is the basic operation in integral calculus . While differentiation has easy rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives [...More...] "Lists Of Integrals" on: Wikipedia Google Yahoo picture info Integration By Reduction Formulae * v * t * e INTEGRATION BY REDUCTION FORMULA in integral calculus is a technique of integration, in the form of a recurrence relation . It is used when an expression containing an integer parameter , usually in the form of powers of elementary functions, or products of transcendental functions and polynomials of arbitrary degree , can't be integrated directly. But using other methods of integration a REDUCTION FORMULA can be set up to obtain the integral of the same or similar expression with a lower integer parameter, progressively simplifying the integral until it can be evaluated. This method of integration is one of the earliest used [...More...] "Integration By Reduction Formulae" on: Wikipedia Google Yahoo picture info Series (mathematics) In mathematics , a SERIES is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis . Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics ), through generating functions . In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics , computer science , statistics and finance . For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical by mathematicians and philosophers . This paradox was resolved using the concept of a limit during the 19th century [...More...] "Series (mathematics)" on: Wikipedia Google Yahoo picture info Power Series In mathematics , a POWER SERIES (in one variable) is an infinite series of the form n = 0 a n ( x c ) n = a 0 + a 1 ( x c ) 1 + a 2 ( x c ) 2 + {displaystyle sum _{n=0}^{infty }a_{n}left(x-cright)^{n}=a_{0}+a_{1}(x-c)^{1}+a_{2}(x-c)^{2}+ldots } where an represents the coefficient of the nth term and c is a constant. This series usually arises as the Taylor series of some known function . In many situations c (the center of the series) is equal to zero, for instance when considering a Maclaurin series . In such cases, the power series takes the simpler form n = 0 a n x n = a 0 + a 1 x + a 2 x 2 + [...More...] "Power Series" on: Wikipedia Google Yahoo Binomial Series In mathematics , the BINOMIAL SERIES is the Maclaurin series for the function f {displaystyle f} given by f ( x ) = ( 1 + x ) {displaystyle f(x)=(1+x)^{alpha }} , where C {displaystyle alpha in mathbb {C} } is an arbitrary complex number . Explicitly, ( 1 + x ) = k = 0 ( k ) x k ( 1 ) = 1 + x + ( 1 ) 2 ! x 2 + , {displaystyle {begin{aligned}(1+x)^{alpha }{alpha choose k};x^{k}qquad qquad qquad (1)\ width:41.704ex; height:13.176ex;" alt="begin{align} (1 + x)^alpha {alpha choose k} ; x^k qquadqquadqquad (1) \ "> ( k ) := ( 1 ) ( 2 ) ( k + 1 ) k ! [...More...] "Binomial Series" on: Wikipedia Google Yahoo
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