1.
Calculus
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Calculus 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 branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first. The formal study of calculus brought together Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, in other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series. He did not publish all these discoveries, and at this time infinitesimal methods were considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
2.
Fundamental theorem of calculus
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The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the functions integral. This part of the guarantees the existence of antiderivatives for continuous functions. This part of the theorem has key practical applications because it simplifies the computation of definite integrals. The fundamental theorem of calculus relates differentiation and integration, showing that two operations are essentially inverses of one another. Before the discovery of this theorem, it was not recognized that two operations were related. Ancient Greek mathematicians knew how to compute area via infinitesimals, an operation that we would now call integration, the first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character, was by James Gregory. Isaac Barrow proved a more generalized version of the theorem, while his student Isaac Newton completed the development of the mathematical theory. Gottfried Leibniz systematized the knowledge into a calculus for infinitesimal quantities, for a continuous function y = f whose graph is plotted as a curve, each value of x has a corresponding area function A, representing the area beneath the curve between 0 and x. The function A may not be known, but it is given that it represents the area under the curve. The area under the curve between x and x + h could be computed by finding the area between 0 and x + h, then subtracting the area between 0 and x, in other words, the area of this “sliver” would be A − A. There is another way to estimate the area of this same sliver, as shown in the accompanying figure, h is multiplied by f to find the area of a rectangle that is approximately the same size as this sliver. So, A − A ≈ f h In fact, this becomes a perfect equality if we add the red portion of the excess area shown in the diagram. So, A − A = f h + Rearranging terms, as h approaches 0 in the limit, the last fraction can be shown to go to zero. This is true because the area of the red portion of region is less than or equal to the area of the tiny black-bordered rectangle. More precisely, | f − A − A h | = | Red Excess | h ≤ h | f − f | h = | f − f |, by the continuity of f, the latter expression tends to zero as h does. Therefore, the left-hand side tends to zero as h does and that is, the derivative of the area function A exists and is the original function f, so, the area function is simply an antiderivative of the original function. Computing the derivative of a function and “finding the area” under its curve are opposite operations and this is the crux of the Fundamental Theorem of Calculus. Intuitively, the theorem states that the sum of infinitesimal changes in a quantity over time adds up to the net change in the quantity
3.
Limit of a function
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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 to every input x. We say the function has a limit L at an input 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 distance apart. The notion of a limit has many applications in modern calculus, in particular, the many definitions of continuity employ the limit, roughly, a function is continuous if all of its limits agree with the values of the function. It also appears in the definition of the derivative, in the calculus of one variable, however, his work was not known during his lifetime. Weierstrass first introduced the definition of limit in the form it is usually written today. He also introduced the notations lim and limx→x0, the modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908. Imagine a person walking over a landscape represented by the graph of y = f and her horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system. Her altitude is given by the coordinate y and she is walking towards the horizontal position given by x = p. As she gets closer and closer to it, she notices that her altitude approaches L, if asked about the altitude of x = p, she would then answer L. What, then, does it mean to say that her altitude approaches L. It means that her altitude gets nearer and nearer to L except for a small error in accuracy. For example, suppose we set a particular goal for our traveler. She reports back that indeed she can get within ten meters of L, since she notes that when she is within fifty horizontal meters of p, the accuracy goal is then changed, can she get within one vertical meter. If she is anywhere within seven meters of p, then her altitude always remains within one meter from the target L. This explicit statement is quite close to the definition of the limit of a function with values in a topological space. To say that lim x → p f = L, means that ƒ can be made as close as desired to L by making x close enough, the following definitions are the generally accepted ones for the limit of a function in various contexts. Suppose f, R → R is defined on the real line, the value of the limit does not depend on the value of f, nor even that p be in the domain of f
4.
Continuous function
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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 concepts of topology. The introductory portion of this focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the general case of functions between two metric spaces. In order theory, especially in 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, as an example, consider the function h, which describes the height of a growing flower at time t. By contrast, if M denotes the amount of money in an account at time t, then the function jumps at each point in time when money is deposited or withdrawn. A form of the definition of continuity was first given by Bernard Bolzano in 1817. Cauchy defined infinitely small quantities in terms of quantities. The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasnt published until the 1930s, all three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of continuity in 1872. This is not a definition of continuity since the function f =1 x is continuous on its whole domain of R ∖ A function is continuous at a point if it does not have a hole or jump. A “hole” or “jump” in the graph of a function if the value of the function at a point c differs from its limiting value along points that are nearby. Such a point is called a discontinuity, a function is then continuous if it has no holes or jumps, that is, if it is continuous at every point of its domain. Otherwise, a function is discontinuous, at the points where the value of the function differs from its limiting value, there are several ways to make this definition mathematically rigorous. These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a function is continuous or not. In the definitions below, f, I → R. is a function defined on a subset I of the set R of real numbers and this subset I is referred to as the domain of f
5.
Mean value theorem
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This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval. More precisely, if a function f is continuous on the closed interval and it is one of the most important results in real analysis. A special case of this theorem was first described by Parameshvara, from the Kerala school of astronomy and mathematics in India, in his commentaries on Govindasvāmi and Bhāskara II. A restricted form of the theorem was proved by Rolle in 1691, the result was what is now known as Rolles theorem, the mean value theorem in its modern form was stated and proved by Cauchy in 1823. Let f, → R be a function on the closed interval, and differentiable on the open interval. Then there exists c in such that f ′ = f − f b − a. The mean value theorem is a generalization of Rolles theorem, which assumes f = f, the mean value theorem is still valid in a slightly more general setting. One only needs to assume that f, → R is continuous on, If finite, that limit equals f ′. An example where this version of the theorem applies is given by the cube root function mapping x → x 13. Note that the theorem, as stated, is if a differentiable function is complex-valued instead of real-valued. For example, define f = e x i for all real x, then f − f =0 =0 while f ′ ≠0 for any real x. Thus the Mean value theorem says that given any chord of a smooth curve, the following proof illustrates this idea. Define g = f − r x, where r is a constant, since f is continuous on and differentiable on, the same is true for g. We now want to choose r so that g satisfies the conditions of Rolles theorem, Assume that f is a continuous, real-valued function, defined on an arbitrary interval I of the real line. If the derivative of f at every point of the interval I exists and is zero. Proof, Assume the derivative of f at every point of the interval I exists and is zero. Let be an open interval in I. By the mean value theorem, there exists a point c in such that 0 = f ′ = f − f b − a, thus, f is constant on the interior of I and thus is constant on I by continuity
6.
Rolle's theorem
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If a real-valued function f is continuous on a proper closed interval, differentiable on the open interval, and f = f, then there exists at least one c in the open interval such that f ′ =0. This version of Rolles theorem is used to prove the mean value theorem and it is also the basis for the proof of Taylors theorem. Indian mathematician Bhāskara II is credited with knowledge of Rolles theorem, although the theorem is named after Michel Rolle, Rolles 1691 proof covered only the case of polynomial functions. His proof did not use the methods of calculus, which at that point in his life he considered to be fallacious. The theorem was first proved by Cauchy in 1823 as a corollary of a proof of the mean value theorem, the name Rolles theorem was first used by Moritz Wilhelm Drobisch of Germany in 1834 and by Giusto Bellavitis of Italy in 1846. For a radius r >0, consider the function f = r 2 − x 2, x ∈ and its graph is the upper semicircle centered at the origin. This function is continuous on the interval and differentiable in the open interval. Since f = f, Rolles theorem applies, and indeed, note that the theorem applies even when the function cannot be differentiated at the endpoints because it only requires the function to be differentiable in the open interval. If differentiability fails at a point of the interval, the conclusion of Rolles theorem may not hold. Consider the absolute value function f = | x |, x ∈, then f = f, but there is no c between −1 and 1 for which the derivative is zero. This is because that function, although continuous, is not differentiable at x =0, note that the derivative of f changes its sign at x =0, but without attaining the value 0. The theorem cannot be applied to this function, clearly, because it does not satisfy the condition that the function must be differentiable for x in the open interval. However, when the differentiability requirement is dropped from Rolles theorem, f will still have a number in the open interval. The second example illustrates the generalization of Rolles theorem, Consider a real-valued. If the right- and left-hand limits agree for every x, then they agree in particular for c, if f is convex or concave, then the right- and left-hand derivatives exist at every inner point, hence the above limits exist and are real numbers. This generalized version of the theorem is sufficient to prove convexity when the derivatives are monotonically increasing. Since the proof for the version of Rolles theorem and the generalization are very similar. In particular, if the derivative exists, it must be zero at c, by assumption, f is continuous on, and by the extreme value theorem attains both its maximum and its minimum in
7.
Derivative
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The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a tool of calculus. For example, the derivative of the position of an object with respect to time is the objects velocity. The derivative of a function of a variable at a chosen input value. The tangent line is the best linear approximation of the function near that input value, for this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives may be generalized to functions of real variables. In this generalization, the derivative is reinterpreted as a transformation whose graph is the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables and it can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of variables, the Jacobian matrix reduces to the gradient vector. The process of finding a derivative is called differentiation, the reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration, differentiation and integration constitute the two fundamental operations in single-variable calculus. Differentiation is the action of computing a derivative, the derivative of a function y = f of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x, If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point. The simplest case, apart from the case of a constant function, is when y is a linear function of x. This formula is true because y + Δ y = f = m + b = m x + m Δ x + b = y + m Δ x. Thus, since y + Δ y = y + m Δ x and this gives an exact value for the slope of a line. If the function f is not linear, however, then the change in y divided by the change in x varies, differentiation is a method to find an exact value for this rate of change at any given value of x. The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the value of the ratio of the differences Δy / Δx as Δx becomes infinitely small
8.
Differential of a function
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In calculus, the differential represents the principal part of the change in a function y = f with respect to changes in the independent variable. The differential dy is defined by d y = f ′ d x, where f ′ is the derivative of f with respect to x, one also writes d f = f ′ d x. The precise meaning of the variables dy and dx depends on the context of the application, traditionally, the variables dx and dy are considered to be very small, and this interpretation is made rigorous in non-standard analysis. The quotient dy/dx is not infinitely small, rather it is a real number, the use of infinitesimals in this form was widely criticized, for instance by the famous pamphlet The Analyst by Bishop Berkeley. Augustin-Louis Cauchy defined the differential without appeal to the atomism of Leibnizs infinitesimals, in physical treatments, such as those applied to the theory of thermodynamics, the infinitesimal view still prevails. Courant & John reconcile the use of infinitesimal differentials with the mathematical impossibility of them as follows. The differentials represent finite non-zero values that are smaller than the degree of accuracy required for the purpose for which they are intended. Thus physical infinitesimals need not appeal to a corresponding mathematical infinitesimal in order to have a precise sense, following twentieth-century developments in mathematical analysis and differential geometry, it became clear that the notion of the differential of a function could be extended in a variety of ways. In real analysis, it is desirable to deal directly with the differential as the principal part of the increment of a function. This leads directly to the notion that the differential of a function at a point is a functional of an increment Δx. This approach allows the differential to be developed for a variety of more sophisticated spaces, in non-standard calculus, differentials are regarded as infinitesimals, which can themselves be put on a rigorous footing. The differential is defined in modern treatments of calculus as follows. The differential of a function f of a real variable x is the function df of two independent real variables x and Δx given by d f = d e f f ′ Δ x. One or both of the arguments may be suppressed, i. e. one may see df or simply df, if y = f, the differential may also be written as dy. The partial differential is therefore ∂ y ∂ x 1 d x 1 involving the partial derivative of y with respect to x1. The total differential is then defined as d y = ∂ y ∂ x 1 Δ x 1 + ⋯ + ∂ y ∂ x n Δ x n. Since, with this definition, d x i = Δ x i, in measurement, the total differential is used in estimating the error Δf of a function f based on the errors Δx, Δy. of the parameters x, y. As they are assumed to be independent, the analysis describes the worst-case scenario, the absolute values of the component errors are used, because after simple computation, the derivative may have a negative sign
9.
Total differential
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In calculus, the differential represents the principal part of the change in a function y = f with respect to changes in the independent variable. The differential dy is defined by d y = f ′ d x, where f ′ is the derivative of f with respect to x, one also writes d f = f ′ d x. The precise meaning of the variables dy and dx depends on the context of the application, traditionally, the variables dx and dy are considered to be very small, and this interpretation is made rigorous in non-standard analysis. The quotient dy/dx is not infinitely small, rather it is a real number, the use of infinitesimals in this form was widely criticized, for instance by the famous pamphlet The Analyst by Bishop Berkeley. Augustin-Louis Cauchy defined the differential without appeal to the atomism of Leibnizs infinitesimals, in physical treatments, such as those applied to the theory of thermodynamics, the infinitesimal view still prevails. Courant & John reconcile the use of infinitesimal differentials with the mathematical impossibility of them as follows. The differentials represent finite non-zero values that are smaller than the degree of accuracy required for the purpose for which they are intended. Thus physical infinitesimals need not appeal to a corresponding mathematical infinitesimal in order to have a precise sense, following twentieth-century developments in mathematical analysis and differential geometry, it became clear that the notion of the differential of a function could be extended in a variety of ways. In real analysis, it is desirable to deal directly with the differential as the principal part of the increment of a function. This leads directly to the notion that the differential of a function at a point is a functional of an increment Δx. This approach allows the differential to be developed for a variety of more sophisticated spaces, in non-standard calculus, differentials are regarded as infinitesimals, which can themselves be put on a rigorous footing. The differential is defined in modern treatments of calculus as follows. The differential of a function f of a real variable x is the function df of two independent real variables x and Δx given by d f = d e f f ′ Δ x. One or both of the arguments may be suppressed, i. e. one may see df or simply df, if y = f, the differential may also be written as dy. The partial differential is therefore ∂ y ∂ x 1 d x 1 involving the partial derivative of y with respect to x1. The total differential is then defined as d y = ∂ y ∂ x 1 Δ x 1 + ⋯ + ∂ y ∂ x n Δ x n. Since, with this definition, d x i = Δ x i, in measurement, the total differential is used in estimating the error Δf of a function f based on the errors Δx, Δy. of the parameters x, y. As they are assumed to be independent, the analysis describes the worst-case scenario, the absolute values of the component errors are used, because after simple computation, the derivative may have a negative sign
10.
Second derivative
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In calculus, the second derivative, or the second order derivative, of a function f is the derivative of the derivative of f. In Leibniz notation, a = d v d t = d 2 x d t 2, on the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a second derivative bows downward. The second derivative of a function f is denoted f ″. That is, f ″ = ′ When using Leibnizs notation for derivatives and this notation is derived from the following formula, d 2 y d x 2 = d d x. Given the function f = x 3, the derivative of f is the function f ′ =3 x 2, the second derivative of f is the derivative of f′, namely f ″ =6 x. The second derivative of a function f measures the concavity of the graph of f, a function whose second derivative is positive will be concave up, meaning that the tangent line will lie below the graph of the function. Similarly, a function whose derivative is negative will be concave down. If the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, a point where this occurs is called an inflection point. Assuming the second derivative is continuous, it must take a value of zero at any inflection point, the relation between the second derivative and the graph can be used to test whether a stationary point for a function is a local maximum or a local minimum. Specifically, If f ′ ′ <0 then f has a maximum at x. If f ′ ′ >0 then f has a minimum at x. If f ′ ′ =0, the second derivative test says nothing about the point x, the reason the second derivative produces these results can be seen by way of a real-world analogy. Consider a vehicle that at first is moving forward at a great velocity, the same is true for the minimum, with a vehicle that at first has a very negative velocity but positive acceleration. It is possible to write a single limit for the second derivative, the limit is called the second symmetric derivative. Note that the symmetric derivative may exist even when the second derivative does not. The expression on the right can be written as a quotient of difference quotients. This limit can be viewed as a version of the second difference for sequences
11.
Implicit function
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In mathematics, an implicit equation is a relation of the form R =0, where R is a function of several variables. For example, the equation of the unit circle is x 2 + y 2 −1 =0. An implicit function is a function that is defined implicitly by an implicit equation, thus, an implicit function for y in the context of the unit circle is defined implicitly by x 2 + f 2 −1 =0. This implicit equation defines f as a function of x only if −1 ≤ x ≤1, the implicit function theorem provides conditions under which a relation defines an implicit function. A common type of function is an inverse function. If f is a function of x, then the function of f. This solution is x = f −1, intuitively, an inverse function is obtained from f by interchanging the roles of the dependent and independent variables. Stated another way, the function gives the solution for x of the equation R = y − f =0. Example The product log is a function giving the solution for x of the equation y − x ex =0. An algebraic function is a function satisfies a polynomial equation whose coefficients are themselves polynomials. Algebraic functions play an important role in analysis and algebraic geometry. A simple example of a function is given by the left side of the unit circle equation. Solving for y gives a solution, y = ±1 − x 2. But even without specifying this explicit solution, it is possible to refer to the solution of the unit circle equation. Nevertheless, one can refer to the implicit solution y = g involving the multi-valued implicit function g. Not every equation R =0 implies a graph of a single-valued function, another example is an implicit function given by x − C =0 where C is a cubic polynomial having a hump in its graph. Thus, for a function to be a true function it might be necessary to use just part of the graph. An implicit function can sometimes be successfully defined as a function only after zooming in on some part of the x-axis
12.
Related rates
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In differential calculus, related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time, because science and engineering often relate quantities to each other, the methods of related rates have broad applications in these fields. Because problems involve several variables, differentiation with respect to time or one of the other variables requires application of the chain rule. Fundamentally, if a function F is defined such that F = f and we assume x is a function of t, i. e. x = g. Then F = f, so F ′ = f ′ ⋅ g ′ Written in Leibniz notation, the value of this is, if it is known how x changes with respect to t, then we can determine how F changes with respect to t and vice versa. We can extend this application of the rule with the sum, difference, product and quotient rules of calculus. If F = G + H then d F d x ⋅ d x d t = d G d y ⋅ d y d t + d H d z ⋅ d z d t. The most common way to approach related rates problems is the following, Identify the known variables, including rates of change, construct an equation relating the quantities whose rates of change are known to the quantity whose rate of change is to be found. Differentiate both sides of the equation with respect to time, often, the chain rule is employed at this step. Substitute the known rates of change and the known quantities into the equation, solve for the wanted rate of change. Errors in this procedure are often caused by plugging in the values for the variables before finding the derivative with respect to time. A 10-meter ladder is leaning against the wall of a building, how fast is the top of the ladder sliding down the wall when the base of the ladder is 6 meters from the wall. The distance between the base of the ladder and the wall, x, and the height of the ladder on the wall, y, represent the sides of a triangle with the ladder as the hypotenuse. The objective is to find dy/dt, the rate of change of y with respect to time, t, when h, x and dx/dt, the rate of change of x, are known. Step 1, x =6 h =10 d x d t =3 d h d t =0 d y d t =. Step 2, From the Pythagorean theorem, the equation x 2 + y 2 = h 2, step 4 &5, Using the variables from step 1 gives us, d y d t = h d h d t − x d x d t y. D y d t =10 ×0 −6 ×3 y = −18 y, in doing such, the top of the ladder is sliding down the wall at a rate of 9⁄4 meters per second. Because one physical quantity often depends on another, which, in turn depends on others, such as time and this section presents an example of related rates kinematics and electromagnetic induction
13.
Taylor's theorem
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In calculus, Taylors theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial. For analytic functions the Taylor polynomials at a point are finite order truncations of its Taylor series. The exact content of Taylors theorem is not universally agreed upon, indeed, there are several versions of it applicable in different situations, and some of them contain explicit estimates on the approximation error of the function by its Taylor polynomial. Taylors theorem is named after the mathematician Brook Taylor, who stated a version of it in 1712, yet an explicit expression of the error was not provided until much later on by Joseph-Louis Lagrange. An earlier version of the result was already mentioned in 1671 by James Gregory, Taylors theorem is taught in introductory level calculus courses and it is one of the central elementary tools in mathematical analysis. Within pure mathematics it is the point of more advanced asymptotic analysis. Taylors theorem also generalizes to multivariate and vector valued functions f, R n → R m on any dimensions n and m and this generalization of Taylors theorem is the basis for the definition of so-called jets which appear in differential geometry and partial differential equations. If a real-valued function f is differentiable at the point a then it has an approximation at the point a. This means that there exists a function h1 such that f = f + f ′ + h 1, here P1 = f + f ′ is the linear approximation of f at the point a. The graph of y = P1 is the tangent line to the graph of f at x = a, the error in the approximation is R1 = f − P1 = h 1. Note that this goes to zero a little bit faster than x − a as x tends to a, if we wanted a better approximation to f, we might instead try a quadratic polynomial instead of a linear function. Instead of just matching one derivative of f at a, we can match two derivatives, thus producing a polynomial that has the slope and concavity as f at a. The quadratic polynomial in question is P2 = f + f ′ + f ″22, Taylors theorem ensures that the quadratic approximation is, in a sufficiently small neighborhood of the point a, a better approximation than the linear approximation. Specifically, f = P2 + h 22, lim x → a h 2 =0. Here the error in the approximation is R2 = f − P2 = h 22 which, given the behavior of h 2. Similarly, we might get better approximations to f if we use polynomials of higher degree. In general, the error in approximating a function by a polynomial of degree k will go to zero a little bit faster than k as x tends to a. Find the smallest degree k for which the polynomial Pk approximates f to within an error on a given interval
14.
Product rule
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In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated as ′ = f ′ ⋅ g + f ⋅ g ′ or in the Leibniz notation d d x = u ⋅ d v d x + v ⋅ d u d x. In differentials notation, this can be written as d = u d v + v d u, discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. Here is Leibnizs argument, Let u and v be two functions of x. Then the differential of uv is d = ⋅ − u ⋅ v = u ⋅ d v + v ⋅ d u + d u ⋅ d v. Since the term du·dv is negligible, Leibniz concluded that d = v ⋅ d u + u ⋅ d v, suppose we want to differentiate ƒ = x2 sin. By using the rule, one gets the derivative ƒ = 2x sin + x2cos. This follows from the rule since the derivative of any constant is zero. This, combined with the sum rule for derivatives, shows that differentiation is linear, the rule for integration by parts is derived from the product rule, as is the quotient rule. Let h = f g, and suppose that f and g are each differentiable at x and we want to prove that h is differentiable at x and that its derivative h is given by f g + f g. To do this f g − f g is added to the numerator to permit its factoring, a rigorous proof of the product rule can be given using the definition of the derivative as a limit, and the basic properties of limits. Let h = f g, and suppose that f and g are each differentiable at x0 and we want to prove that h is differentiable at x0 and that its derivative h′ is given by f′ g + f g′. Let Δh = h − h, note that although x0 is fixed, Δh depends on the value of Δx, which is thought of as being small. The function h is differentiable at x0 if the limit lim Δ x →0 Δ h Δ x exists, as with Δh, let Δf = f − f and Δg = g − g which, like Δh, also depends on Δx. Then f = f + Δf and g = g + Δg, using the basic properties of limits and the definition of the derivative, we can tackle this term-by term. First, lim Δ x →0 = f ′ g, similarly, lim Δ x →0 = f g ′. The third term, corresponding to the small rectangle, winds up being negligible because Δf Δg vanishes to second order. Then, f g − f g = − f g = f ′ g h + f g ′ h + O Taking the limit for small h gives the result, Let f = uv and suppose u and v are positive functions of x
15.
Chain rule
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In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. This can be more explicitly in terms of the variable. Let F = f ∘ g, or equivalently, F = f for all x, then one can also write F ′ = f ′ g ′. The chain rule may be written, in Leibnizs notation, in the following way. If a variable z depends on the y, which itself depends on the variable x, so that y and z are therefore dependent variables, then z, via the intermediate variable of y. The chain rule states, d z d x = d z d y ⋅ d y d x. In integration, the counterpart to the rule is the substitution rule. The chain rule seems to have first been used by Leibniz and he used it to calculate the derivative of a + b z + c z 2 as the composite of the square root function and the function a + b z + c z 2. He first mentioned it in a 1676 memoir, the common notation of chain rule is due to Leibniz. LHôpital uses the chain rule implicitly in his Analyse des infiniment petits, the chain rule does not appear in any of Leonhard Eulers analysis books, even though they were written over a hundred years after Leibnizs discovery. Suppose that a skydiver jumps from an aircraft, assume that t seconds after his jump, his height above sea level in meters is given by g =4000 −4. 9t2. One model for the pressure at a height h is f =101325 e−0. 0001h. These two equations can be differentiated and combined in ways to produce the following data, g′ = −9. 8t is the velocity of the skydiver at time t. F′ = −10. 1325e−0. 0001h is the rate of change in pressure with respect to height at the height h and is proportional to the buoyant force on the skydiver at h meters above sea level. Is the atmospheric pressure the skydiver experiences t seconds after his jump, ′ is the rate of change in atmospheric pressure with respect to time at t seconds after the skydivers jump and is proportional to the buoyant force on the skydiver at t seconds after his jump. The chain rule gives a method for computing ′ in terms of f′, while it is always possible to directly apply the definition of the derivative to compute the derivative of a composite function, this is usually very difficult. The utility of the rule is that it turns a complicated derivative into several easy derivatives. The chain rule states that, under conditions, ′ = f ′ ⋅ g ′
16.
Inverse functions and differentiation
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In mathematics, the inverse of a function y = f is a function that, in some fashion, undoes the effect of f. The inverse of f is denoted f −1, the statements y = f and x = f −1 are equivalent. Their two derivatives, assuming they exist, are reciprocal, as the Leibniz notation suggests, that is, d x d y ⋅ d y d x =1. This is a consequence of the chain rule, since d x d y ⋅ d y d x = d x d x. This reflection operation turns the gradient of any line into its reciprocal, Y = x 2 has inverse x = y. D y d x =2 x, d x d y =12 y =12 x d y d x ⋅ d x d y =2 x ⋅12 x =1. At x =0, however, there is a problem and this is only useful if the integral exists. In particular we need f ′ to be non-zero across the range of integration and it follows that a function that has a continuous derivative has an inverse in a neighbourhood of every point where the derivative is non-zero. This need not be if the derivative is not continuous. The chain rule given above is obtained by differentiating the identity x = f −1 with respect to x, one can continue the same process for higher derivatives. Replacing the first derivative, using the identity obtained earlier, we get d 2 y d x 2 = − d 2 x d y 2 ⋅3 and these formulas can also be written using Lagranges notation. If f and g are inverses, then g ″ = − f ″3 y = e x has the inverse x = ln y, calculus Inverse functions Chain rule Inverse function theorem Implicit function theorem Integration of inverse functions
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Integral
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In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two operations of calculus, with its inverse, differentiation, being the other. The area above the x-axis adds to the total and that below the x-axis subtracts from the total, roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the integral may also refer to the related notion of the antiderivative. In this case, it is called an integral and is written. The integrals discussed in this article are those termed definite integrals, a rigorous mathematical definition of the integral was given by Bernhard Riemann. It is based on a procedure which approximates the area of a curvilinear region by breaking the region into thin vertical slabs. A line integral is defined for functions of two or three variables, and the interval of integration is replaced by a curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space and this method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle. A similar method was developed in China around the 3rd century AD by Liu Hui. This method was used in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi. The next significant advances in integral calculus did not begin to appear until the 17th century, further steps were made in the early 17th century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation. Barrow provided the first proof of the theorem of calculus. Wallis generalized Cavalieris method, computing integrals of x to a power, including negative powers. The major advance in integration came in the 17th century with the independent discovery of the theorem of calculus by Newton. The theorem demonstrates a connection between integration and differentiation and this connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the mathematical framework that both Newton and Leibniz developed
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Antiderivative
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In calculus, an antiderivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F ′ = f, the process of solving for antiderivatives is called antidifferentiation and its opposite operation is called differentiation, which is the process of finding a derivative. The discrete equivalent of the notion of antiderivative is antidifference, the function F = x3/3 is an antiderivative of f = x2, as the derivative of x3/3 is x2. As the derivative of a constant is zero, x2 will have a number of antiderivatives, such as x3/3, x3/3 +1, x3/3 -2. Thus, all the antiderivatives of x2 can be obtained by changing the value of C in F = x3/3 + C, essentially, the graphs of antiderivatives of a given function are vertical translations of each other, each graphs vertical location depending upon the value of C. In physics, the integration of acceleration yields velocity plus a constant, the constant is the initial velocity term that would be lost upon taking the derivative of velocity because the derivative of a constant term is zero. This same pattern applies to further integrations and derivatives of motion, C is called the arbitrary constant of integration. If the domain of F is a disjoint union of two or more intervals, then a different constant of integration may be chosen for each of the intervals. For instance F = { −1 x + C1 x <0 −1 x + C2 x >0 is the most general antiderivative of f =1 / x 2 on its natural domain ∪. Every continuous function f has an antiderivative, and one antiderivative F is given by the integral of f with variable upper boundary. Varying the lower boundary produces other antiderivatives and this is another formulation of the fundamental theorem of calculus. There are many functions whose antiderivatives, even though they exist, cannot be expressed in terms of elementary functions. Examples of these are ∫ e − x 2 d x, ∫ sin x 2 d x, ∫ sin x x d x, ∫1 ln x d x, ∫ x x d x. From left to right, the first four are the function, the Fresnel function, the trigonometric integral. See also Differential Galois theory for a detailed discussion. Finding antiderivatives of elementary functions is often harder than finding their derivatives. For some elementary functions, it is impossible to find an antiderivative in terms of elementary functions. See the articles on elementary functions and nonelementary integral for further information, integrals which have already been derived can be looked up in a table of integrals
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Improper integral
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Such an integral is often written symbolically just like a standard definite integral, in some cases with infinity as a limit of integration. By abuse of notation, improper integrals are often written symbolically just like standard definite integrals, when the definite integral exists, this ambiguity is resolved as both the proper and improper integral will coincide in value. The original definition of the Riemann integral does not apply to a such as 1 / x 2 on the interval [1, ∞). The narrow definition of the Riemann integral also does not cover the function 1 / x on the interval, the problem here is that the integrand is unbounded in the domain of integration. However, the integral does exist if understood as the limit ∫011 x d x = lim a →0 + ∫ a 11 x d x = lim a →0 + =2. Sometimes integrals may have two singularities where they are improper, consider, for example, the function 1/ integrated from 0 to ∞. At the lower bound, as x goes to 0 the function goes to ∞, thus this is a doubly improper integral. Integrated, say, from 1 to 3, an ordinary Riemann sum suffices to produce a result of π/6, to integrate from 1 to ∞, a Riemann sum is not possible. However, any upper bound, say t, gives a well-defined result,2 arctan − π/2. This has a limit as t goes to infinity, namely π/2. Similarly, the integral from 1/3 to 1 allows a Riemann sum as well, replacing 1/3 by an arbitrary positive value s is equally safe, giving π/2 −2 arctan. This, too, has a limit as s goes to zero. This process does not guarantee success, a limit might fail to exist, for example, over the bounded interval from 0 to 1 the integral of 1/x does not converge, and over the unbounded interval from 1 to ∞ the integral of 1/√x does not converge. It might also happen that an integrand is unbounded near an interior point, for the integral as a whole to converge, the limit integrals on both sides must exist and must be bounded. But the similar integral ∫ −11 d x x cannot be assigned a value in this way, as the integrals above, an improper integral converges if the limit defining it exists. It is also possible for an integral to diverge to infinity. In that case, one may assign the value of ∞ to the integral, for instance lim b → ∞ ∫1 b 1 x d x = ∞. However, other improper integrals may simply diverge in no particular direction, such as lim b → ∞ ∫1 b x sin d x and this is called divergence by oscillation
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Riemann integral
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In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. For many functions and practical applications, the Riemann integral can be evaluated by the theorem of calculus or approximated by numerical integration. The Riemann integral is unsuitable for many theoretical purposes, some of the technical deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral. Let f be a nonnegative real-valued function on the interval, and let S = be the region of the plane under the graph of the function f and we are interested in measuring the area of S. Once we have measured it, we denote the area by. The basic idea of the Riemann integral is to use very simple approximations for the area of S, by taking better and better approximations, we can say that in the limit we get exactly the area of S under the curve. A partition of an interval is a sequence of numbers of the form a = x 0 < x 1 < x 2 < ⋯ < x n = b Each is called a subinterval of the partition. The mesh or norm of a partition is defined to be the length of the longest subinterval, a tagged partition P of an interval is a partition together with a finite sequence of numbers t0. Tn −1 subject to the conditions that for each i, in other words, it is a partition together with a distinguished point of every subinterval. The mesh of a partition is the same as that of an ordinary partition. Suppose that two partitions P and Q are both partitions of the interval. We say that Q is a refinement of P if for each i, with i ∈, there exists an integer r such that xi = yr and such that ti = sj for some j with j ∈ [r. Said more simply, a refinement of a tagged partition breaks up some of the subintervals and adds tags to the partition where necessary, thus it refines the accuracy of the partition. We can define a partial order on the set of all tagged partitions by saying that one tagged partition is greater or equal to if the former is a refinement of the latter. Let f be a function defined on the interval. The Riemann sum of f with respect to the tagged partition x0, tn −1 is ∑ i =0 n −1 f. Each term in the sum is the product of the value of the function at a given point, consequently, each term represents the area of a rectangle with height f and width xi +1 − xi. The Riemann sum is the area of all the rectangles, a closely related concept are the lower and upper Darboux sums
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Lebesgue integration
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In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral extends the integral to a class of functions. It also extends the domains on which these functions can be defined, however, as the need to consider more irregular functions arose—e. g. Also, one wish to integrate on spaces more general than the real line. The Lebesgue integral provides the right abstractions needed to do this important job, the Lebesgue integral plays an important role in probability theory, in the branch of mathematics called real analysis and in many other fields in the mathematical sciences. It is named after Henri Lebesgue, who introduced the integral and it is also a pivotal part of the axiomatic theory of probability. The integral of a function f between limits a and b can be interpreted as the area under the graph of f and this is easy to understand for familiar functions such as polynomials, but what does it mean for more exotic functions. In general, for class of functions does area under the curve make sense. The answer to this question has great theoretical and practical importance, as part of a general movement toward rigor in mathematics in the nineteenth century, mathematicians attempted to put integral calculus on a firm foundation. The Riemann integral—proposed by Bernhard Riemann —is a broadly successful attempt to provide such a foundation, riemanns definition starts with the construction of a sequence of easily calculated areas that converge to the integral of a given function. This definition is successful in the sense that it gives the answer for many already-solved problems. However, Riemann integration does not interact well with taking limits of sequences of functions and this is important, for instance, in the study of Fourier series, Fourier transforms, and other topics. The Lebesgue integral is able to describe how and when it is possible to take limits under the integral sign. For this reason, the Lebesgue definition makes it possible to calculate integrals for a class of functions. For example, the Dirichlet function, which is 0 where its argument is irrational and 1 otherwise, has a Lebesgue integral, but does not have a Riemann integral. Lebesgue summarized his approach to integration in a letter to Paul Montel, I have to pay a certain sum, I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. After I have taken all the out of my pocket I order the bills and coins according to identical values. The insight is that one should be able to rearrange the values of a function freely and this process of rearrangement can convert a very pathological function into one that is nice from the point of view of integration, and thus let such pathological functions be integrated
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Methods of contour integration
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In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is related to the calculus of residues, a method of complex analysis. One use for contour integrals is the evaluation of integrals along the line that are not readily found by using only real variable methods. In complex analysis a contour is a type of curve in the complex plane, in contour integration, contours provide a precise definition of the curves on which an integral may be suitably defined. A curve in the plane is defined as a continuous function from a closed interval of the real line to the complex plane. This definition of a curve coincides with the notion of a curve. This more precise definition allows us to consider what properties a curve must have for it to be useful for integration, moreover, we will restrict the pieces from crossing over themselves, and we require that each piece have a finite continuous derivative. Contours are often defined in terms of directed smooth curves and these provide a precise definition of a piece of a smooth curve, of which a contour is made. A smooth curve is a curve z, → C with a non-vanishing, continuous derivative such that each point is traversed only once, with the possible exception of a curve such that the endpoints match. In the case where the match the curve is called closed, and the function is required to be one-to-one everywhere else. A smooth curve that is not closed is often referred to as a smooth arc, the parametrization of a curve provides a natural ordering of points on the curve, z comes before z if x < y. This leads to the notion of a smooth curve. It is most useful to consider curves independent of the specific parametrization and this can be done by considering equivalence classes of smooth curves with the same direction. A directed smooth curve can then be defined as a set of points in the complex plane that is the image of some smooth curve in their natural order. Note that not all orderings of the points are the natural ordering of a smooth curve, in fact, a given smooth curve has only two such orderings. Also, a closed curve can have any point as its endpoint. Contours are the class of curves on which we define contour integration, a contour is a directed curve which is made up of a finite sequence of directed smooth curves whose endpoints are matched to give a single direction. This requires that the sequence of curves γ1, …, γn be such that the point of γi coincides with the initial point of γi+1
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Integration by parts
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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 general formulations of integration by parts exist for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals. The discrete analogue for sequences is called summation by parts, the theorem can be derived as follows. Suppose u and v are two differentiable functions. The product rule states, d d x = v d d x + u d d x and it is not actually necessary for u and v to be continuously differentiable. Integration by parts works if u is continuous and the function designated v is Lebesgue integrable. This is only if we choose v = − exp . One can also come up with similar examples in which u and v are not continuously differentiable. This visualisation also explains why integration by parts may help find the integral of an inverse function f−1 when the integral of the f is known. Indeed, the x and y are inverses, and the integral ∫x dy may be calculated as above from knowing the integral ∫y dx. The following form is useful in illustrating the best strategy to take, as a simple example, consider, ∫ ln x 2 d x. Since the derivative of ln is 1/x, one makes part u, since the antiderivative of 1/x2 is -1/x, the formula now yields, ∫ ln x 2 d x = − ln x − ∫ d x. The antiderivative of −1/x2 can be found with the rule and is 1/x. Alternatively, one may choose u and v such that the product u simplifies due to cancellation, for example, suppose one wishes to integrate, ∫ sec 2 ⋅ ln d x. The integrand simplifies to 1, so the antiderivative is x, finding a simplifying combination frequently involves experimentation. Some other special techniques are demonstrated in the examples below, exponentials and trigonometric functions An example commonly used to examine the workings of integration by parts is I = ∫ e x cos d x. Here, integration by parts is performed twice, then, ∫ e x sin d x = e x sin − ∫ e x cos d x. Putting these together, ∫ e x cos d x = e x cos + e x sin − ∫ e x cos d x, the same integral shows up on both sides of this equation
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Shell integration
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Shell integration 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 parallel to the axis of revolution. The 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 on the interval. Consider the volume, depicted below, whose cross section on the interval is defined by, because the volume is hollow in the middle we will find two functions, one that defines the inner solid and one that defines the outer solid. After integrating these two functions with the method we subtract them to yield the desired volume. With the shell method all we need is the formula,2 π ∫12 x 22 d x By expanding the polynomial the integral becomes very simple. In the end we find the volume is π/10 cubic units, solid of revolution Disk integration Weisstein, Eric W. Method of Shells
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Order of integration (calculus)
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In some cases, the order of integration can be validly interchanged, in others it cannot. The problem for examination is evaluation of an integral of the form ∬ D f d x d y, 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 integration, reduction to a single integration makes a numerical evaluation much easier and more efficient. Consider the iterated integral ∫ a z ∫ a x h d y d x and this forms a three dimensional slice dx wide along the x-axis, from y=a to y=x along the y axis, and in the z direction z=f. Notice that if the thickness dx is infinitesimal, x varies only infinitesimally on the slice and we can assume that x is constant. This integration is as shown in the panel of Figure 1. The integral can be reduced to an integration by reversing the order of integration as shown in the right panel of the figure. For application to principal-value integrals, see Whittaker and Watson, Gakhov, Lu, see also the discussion of the Poincaré-Bertrand transformation in Obolashvili. The second form is evaluated using a partial fraction expansion and an evaluation using the Sokhotski–Plemelj formula, the notation ∫ L ∗ indicates a Cauchy principal value. A good discussion of the basis for reversing the order of integration is found in the book Fourier Analysis by T. W. Körner. He introduces his discussion with an example where interchange of integration leads to two different answers because the conditions of Theorem II below are not satisfied. Here is the example, ∫1 ∞ x 2 − y 22 d y =1 ∞ = −11 + x 2, ron Miechs UCLA Calculus Problems More complex examples of changing the order of integration Duane Nykamps University of Minnesota website A general introduction
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Integration by reduction formulae
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Integration by reduction formula in integral calculus is a technique of integration, in the form of a recurrence relation. This method of integration is one of the earliest used and this makes the reduction formula a type of recurrence relation. In other words, the formula expresses the integral I n = ∫ f d x, in terms of I k = ∫ f d x. To compute the integral, we set n to its value, then we back-substitute the previous results until we have computed In. Below are examples of the procedure, cosine integral Typically, integrals like ∫ cos n x d x, can be evaluated by a reduction formula. Start by setting, I n = ∫ cos n x d x. Now re-write as, I n = ∫ cos n −1 x cos x d x, Integrating by this substitution, cos x d x = d, I n = ∫ cos n −1 x d. To supplement the example, the above can be used to evaluate the integral for n =5, I5 = ∫ cos 5 x d x, exponential integral Another typical example is, ∫ x n e a x d x. Start by setting, I n = ∫ x n e a x d x
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Series (mathematics)
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In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a sequence has defined first and last terms. To emphasize that there are a number of terms, a series is often called an infinite series. In order to make the notion of an infinite sum mathematically rigorous, given an infinite sequence, the associated series is the expression obtained by adding all those terms together, a 1 + a 2 + a 3 + ⋯. These can be written compactly as ∑ i =1 ∞ a i, by using the summation symbol ∑. The sequence can be composed of any kind of object for which addition is defined. A series is evaluated by examining the finite sums of the first n terms of a sequence, called the nth partial sum of the sequence, and taking the limit as n approaches infinity. If this limit does not exist, the infinite sum cannot be assigned a value, and, in this case, the series is said to be divergent. On the other hand, if the partial sums tend to a limit when the number of terms increases indefinitely, then the series is said to be convergent, and the limit is called the sum of the series. An example is the series from Zenos dichotomy and its mathematical representation, ∑ n =1 ∞12 n =12 +14 +18 + ⋯. The study of series is a part of mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, 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 any sequence of numbers, real numbers, complex numbers, functions thereof. By definition the series ∑ n =0 ∞ a n converges to a limit L if and this definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k. When the index set is the natural numbers I = N, a series indexed on the natural numbers is an ordered formal sum and so we rewrite ∑ n ∈ N as ∑ n =0 ∞ in order to emphasize the ordering induced by the natural numbers. Thus, we obtain the common notation for a series indexed by the natural numbers ∑ n =0 ∞ a n = a 0 + a 1 + a 2 + ⋯. When the semigroup G is also a space, then the series ∑ n =0 ∞ a n converges to an element L ∈ G if. This definition is usually written as L = ∑ n =0 ∞ a n ⇔ L = lim k → ∞ s k, a series ∑an is said to converge or to be convergent when the sequence SN of partial sums has a finite limit
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Geometric series
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In mathematics, a geometric series is a series with a constant ratio between successive terms. For example, the series 12 +14 +18 +116 + ⋯ is geometric, Geometric series are among the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, geometric series played an important role in the development of calculus. Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, the terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. This relationship allows for the representation of a series using only two terms, r and a. The term r is the ratio, and a is the first term of the series. In the case above, where r is one half, the series has the sum one, if r is greater than one or less than minus one the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, if r is equal to one, all of the terms of the series are the same. If r is one the terms take two values alternately. The sum of the oscillates between two values. This is a different type of divergence and again the series has no sum, see for example Grandis series,1 −1 +1 −1 + ···. The sum can be computed using the self-similarity of the series, consider the sum of the following geometric series, s =1 +23 +49 +827 + ⋯. This series has common ratio 2/3, if we multiply through by this common ratio, then the initial 1 becomes a 2/3, the 2/3 becomes a 4/9, and so on,23 s =23 +49 +827 +1681 + ⋯. This new series is the same as the original, except that the first term is missing, subtracting the new series s from the original series s cancels every term in the original but the first, s −23 s =1, so s =3. A similar technique can be used to evaluate any self-similar expression, as n goes to infinity, the absolute value of r must be less than one for the series to converge. When a =1, this can be simplified to 1 + r + r 2 + r 3 + ⋯ =11 − r, the formula also holds for complex r, with the corresponding restriction, the modulus of r is strictly less than one. Since = 1−rn+1 and rn+1 →0 for | r | <1, convergence of geometric series can also be demonstrated by rewriting the series as an equivalent telescoping series. Consider the function, g = r K1 − r