1.
Summation
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In mathematics, summation is the addition of a sequence of numbers, the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a sum, prefix sum. The numbers to be summed may be integers, rational numbers, real numbers, besides numbers, other types of values can be added as well, vectors, matrices, polynomials and, in general, elements of any additive group. For finite sequences of elements, summation always produces a well-defined sum. The summation of a sequence of values is called a series. A value of such a series may often be defined by means of a limit, another notion involving limits of finite sums is integration. The summation of the sequence is an expression whose value is the sum of each of the members of the sequence, in the example,1 +2 +4 +2 =9. Addition is also commutative, so permuting the terms of a sequence does not change its sum. There is no notation for the summation of such explicit sequences. If, however, the terms of the sequence are given by a pattern, possibly of variable length. For the summation of the sequence of integers from 1 to 100. In this case, the reader can guess the pattern. However, for more complicated patterns, one needs to be precise about the used to find successive terms. Using this sigma notation the above summation is written as, ∑ i =1100 i, the value of this summation is 5050. It can be found without performing 99 additions, since it can be shown that ∑ i =1 n i = n 2 for all natural numbers n, more generally, formulae exist for many summations of terms following a regular pattern. By contrast, summation as discussed in this article is called definite summation, when it is necessary to clarify that numbers are added with their signs, the term algebraic sum is used. Mathematical notation uses a symbol that compactly represents summation of many terms, the summation symbol, ∑. The i = m under the symbol means that the index i starts out equal to m
2.
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
3.
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
4.
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
5.
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
6.
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
7.
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
8.
Differential calculus
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In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two divisions of calculus, the other being integral calculus. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, the derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation, geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point. Differential calculus and integral calculus are connected by the theorem of calculus. Differentiation has applications to nearly all quantitative disciplines, for example, in physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. The derivative of the momentum of a body equals the applied to the body. The reaction rate of a reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials, derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena, derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory, and abstract algebra. Suppose that x and y are real numbers and that y is a function of x and this relationship can be written as y = f. If f is the equation for a line, then there are two real numbers m and b such that y = mx + b. In this slope-intercept form, the m is called the slope and can be determined from the formula, m = change in y change in x = Δ y Δ x. It follows that Δy = m Δx, a general function is not a line, so it does not have a slope. Geometrically, the derivative of f at the point x = a is the slope of the tangent line to the function f at the point a and this is often denoted f ′ in Lagranges notation or dy/dx|x = a in Leibnizs notation. Since the derivative is the slope of the approximation to f at the point a. If every point a in the domain of f has a derivative, for example, if f = x2, then the derivative function f ′ = dy/dx = 2x
9.
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
10.
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
11.
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
12.
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
13.
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
14.
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
15.
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
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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
17.
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 ′
18.
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
19.
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
20.
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
21.
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
22.
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
23.
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
24.
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
25.
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
26.
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
27.
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
28.
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
29.
Taylor series
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In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the functions derivatives at a single point. The concept of a Taylor series was formulated by the Scottish mathematician James Gregory, a function can be approximated by using a finite number of terms of its Taylor series. Taylors theorem gives quantitative estimates on the error introduced by the use of such an approximation, the polynomial formed by taking some initial terms of the Taylor series is called a Taylor polynomial. The Taylor series of a function is the limit of that functions Taylor polynomials as the degree increases, a function may not be equal to its Taylor series, even if its Taylor series converges at every point. A function that is equal to its Taylor series in an interval is known as an analytic function in that interval. The Taylor series of a real or complex-valued function f that is differentiable at a real or complex number a is the power series f + f ′1. Which can be written in the more compact sigma notation as ∑ n =0 ∞ f n, N where n. denotes the factorial of n and f denotes the nth derivative of f evaluated at the point a. The derivative of order zero of f is defined to be f itself and 0 and 0. are both defined to be 1, when a =0, the series is also called a Maclaurin series. The Maclaurin series for any polynomial is the polynomial itself. The Maclaurin series for 1/1 − x is the geometric series 1 + x + x 2 + x 3 + ⋯ so the Taylor series for 1/x at a =1 is 1 − +2 −3 + ⋯. The Taylor series for the exponential function ex at a =0 is x 00, + ⋯ =1 + x + x 22 + x 36 + x 424 + x 5120 + ⋯ = ∑ n =0 ∞ x n n. The above expansion holds because the derivative of ex with respect to x is also ex and this leaves the terms n in the numerator and n. in the denominator for each term in the infinite sum. The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a result, but rejected it as an impossibility. It was through Archimedess method of exhaustion that a number of progressive subdivisions could be performed to achieve a finite result. Liu Hui independently employed a similar method a few centuries later, in the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama. The Kerala school of astronomy and mathematics further expanded his works with various series expansions, in the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a method for constructing these series for all functions for which they exist was finally provided by Brook Taylor. The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, if f is given by a convergent power series in an open disc centered at b in the complex plane, it is said to be analytic in this disc
30.
Ratio test
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In mathematics, the ratio test is a test for the convergence of a series ∑ n =1 ∞ a n, where each term is a real or complex number and an is nonzero when n is large. The test was first published by Jean le Rond dAlembert and is known as dAlemberts ratio test or as the Cauchy ratio test. The sum of the first m terms is given by,1 −12 m, as m increases, this converges to 1, so the sum of the series is 1. On the other hand given this geometric series, ∑ n =1 ∞2 n =2 +4 +8 + ⋯ The quotient a n +1 a n of any two adjacent terms is 2. The sum of the first m terms is given by 2 m +1 −2, more generally, the sum of the first m terms of the geometric series is given by, ∑ n =1 m r n = r r m −1 r −1. Whether this converges or diverges as m increases depends on whether r, however, as n increases, the ratio still tends in the limit towards the same constant 1/2. The ratio test generalizes the simple test for geometric series to more complex series like this one where the quotient of two terms is not fixed, but in the limit tends towards a fixed value. The rules are similar, if the quotient approaches a value less than one, the series converges, whereas if it approaches a value greater than one, the series diverges. It is possible to make the ratio test applicable to cases where the limit L fails to exist, if limit superior. The test criteria can also be refined so that the test is sometimes even when L =1. More specifically, let R = lim sup | a n +1 a n | r = lim inf | a n +1 a n |, if the limit L in exists, we must have L = R = r. So the original ratio test is a version of the refined one. As every term is positive, the series converges, consider the series ∑ n =1 ∞ e n n. Putting this into the ratio test, L = lim n → ∞ | a n +1 a n | = lim n → ∞ | e n +1 n +1 e n n | = e >1. Consider the three series ∑ n =1 ∞1, ∑ n =1 ∞1 n 2, ∑ n =1 ∞ n 1 n, the first series diverges, the second one converges absolutely and the third one converges conditionally. However, the term-by-term magnitude ratios | a n +1 a n | of the three series are respectively 1, n 22 and n n +1. So, in all three cases, we have lim n → ∞ | a n +1 a n | =1 and this illustrates that when L=1, the series may converge or diverge and hence the original ratio test is inconclusive. Below is a proof of the validity of the ratio test
31.
Integral test for convergence
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In mathematics, the integral test for convergence is a method used to test infinite series of non-negative terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is known as the Maclaurin–Cauchy test. Consider an integer N and a non-negative, continuous function f defined on the unbounded interval [N, ∞), then the infinite series ∑ n = N ∞ f converges to a real number if and only if the improper integral ∫ N ∞ f d x is finite. In other words, if the integral diverges, then the series diverges as well, if the improper integral is finite, then the proof also gives the lower and upper bounds for the infinite series. The proof basically uses the comparison test, comparing the term f with the integral of f over the intervals [n −1, n) and [n, n + 1), respectively. Since f is a decreasing function, we know that f ≤ f for all x ∈ [ n, ∞ ) and f ≤ f for all x ∈. Combining these two estimates yields ∫ N M +1 f d x ≤ ∑ n = N M f ≤ f + ∫ N M f d x, letting M tend to infinity, the bounds in and the result follow. From we get the upper estimate ζ = ∑ x =1 ∞1 x 1 + ε ≤1 + ε ε, once such a sequence is found, a similar question can be asked with f taking the role of 1/n, and so on. In this way it is possible to investigate the borderline between divergence and convergence of infinite series, using the integral test for convergence, one can show that, for every natural number k, the series still diverges but converges for every ε >0. Here lnk denotes the composition of the natural logarithm defined recursively by ln k = { ln for k =1. Knopp, Konrad, Infinite Sequences and Series, Dover Publications, ISBN 0-486-60153-6 Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, fourth edition, Cambridge University Press,1963, ISBN 0-521-58807-3 Ferreira, Jaime Campos, Ed Calouste Gulbenkian,1987, ISBN 972-31-0179-3
32.
Cauchy condensation test
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In mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a non-negative, non-increasing sequence f of real numbers, the series ∑ n =1 ∞ f converges if, moreover, if they converge, the sum of the condensed series is no more than twice as large as the sum of the original. The essential thrust of a proof follows, following the line of Oresmes proof of the divergence of the harmonic series. Pursuing this idea, the integral test for convergence gives us, in the case of monotone f, that ∑ n =1 ∞ f converges if and only if ∫1 ∞ f d x converges. The substitution x →2 x yields the integral log 2 ∫0 ∞2 x f d x, the test can be useful for series where n appears as in a denominator in f. For the most basic example of this sort, the harmonic series ∑ n =1 ∞1 / n is transformed into the series ∑1, as a more complex example, take f, = n − a − b − c. Here the series converges for a >1, and diverges for a <1. When a =1, the condensation transformation gives the series ∑ n − b − c, the logarithms shift to the left. So when a =1, we have convergence for b >1, when b =1 the value of c enters. Here f ∘ m denotes the mth compositional iterate of a function f, the lower limit of the sum, N, was chosen so that all terms of the series are positive. Notably, these series provide examples of infinite sums that converge or diverge arbitrarily slowly, for instance, in the case of k =2 and α =1, the partial sum exceeds 10 only after 1010100 terms yet the series diverges nevertheless. Let u be an increasing sequence of positive integers such that the ratio of successive differences is bounded. Then the convergence of the series ∑ n =1 ∞ f is equivalent to the convergence of, taking u =2 n so that Δ u =2 n, the Cauchy condensation test emerges as a special case
33.
Gradient
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In mathematics, the gradient is a multi-variable generalization of the derivative. While a derivative can be defined on functions of a variable, for functions of several variables. The gradient is a function, as opposed to a derivative. If f is a differentiable, real-valued function of several variables, like the derivative, the gradient represents the slope of the tangent of the graph of the function. More precisely, the gradient points in the direction of the greatest rate of increase of the function, the components of the gradient in coordinates are the coefficients of the variables in the equation of the tangent space to the graph. The Jacobian is the generalization of the gradient for vector-valued functions of several variables, a further generalization for a function between Banach spaces is the Fréchet derivative. Consider a room in which the temperature is given by a field, T. At each point in the room, the gradient of T at that point will show the direction in which the temperature rises most quickly, the magnitude of the gradient will determine how fast the temperature rises in that direction. Consider a surface whose height above sea level at point is H, the gradient of H at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at point is given by the magnitude of the gradient vector. The gradient can also be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, suppose that the steepest slope on a hill is 40%. If a road goes directly up the hill, then the steepest slope on the road will also be 40%, if, instead, the road goes around the hill at an angle, then it will have a shallower slope. This observation can be stated as follows. If the hill height function H is differentiable, then the gradient of H dotted with a unit vector gives the slope of the hill in the direction of the vector. More precisely, when H is differentiable, the dot product of the gradient of H with a unit vector is equal to the directional derivative of H in the direction of that unit vector. The gradient of a function f is denoted ∇f or ∇→f where ∇ denotes the vector differential operator. The notation grad f is commonly used for the gradient. The gradient of f is defined as the vector field whose dot product with any vector v at each point x is the directional derivative of f along v. That is
34.
Curl (mathematics)
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In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a 3-dimensional vector field. At every point in the field, the curl of that point is represented by a vector, the attributes of this vector characterize the rotation at that point. The direction of the curl is the axis of rotation, as determined by the rule. If the vector represents the flow velocity of a moving fluid. A vector field whose curl is zero is called irrotational, the curl is a form of differentiation for vector fields. The alternative terminology rotor or rotational and alternative notations rot F and ∇ × F are often used for curl F and this is a similar phenomenon as in the 3 dimensional cross product, and the connection is reflected in the notation ∇ × for the curl. The name curl was first suggested by James Clerk Maxwell in 1871, the curl of a vector field F, denoted by curl F, or ∇ × F, or rot F, at a point is defined in terms of its projection onto various lines through the point. As such, the curl operator maps continuously differentiable functions f, ℝ3 → ℝ3 to continuous functions g, in fact, it maps Ck functions in ℝ3 to Ck −1 functions in ℝ3. Implicitly, curl is defined by, ⋅ n ^ = d e f lim A →0 where ∮C F · dr is a line integral along the boundary of the area in question, and | A | is the magnitude of the area. Note that the equation for each component, k can be obtained by exchanging each occurrence of a subscript 1,2,3 in cyclic permutation, 1→2, 2→3, and 3→1. If are the Cartesian coordinates and are the coordinates, then h i =2 +2 +2 is the length of the coordinate vector corresponding to ui. The remaining two components of curl result from cyclic permutation of indices,3,1,2 →1,2,3 →2,3,1. Suppose the vector field describes the velocity field of a fluid flow, if the ball has a rough surface, the fluid flowing past it will make it rotate. The rotation axis points in the direction of the curl of the field at the centre of the ball, and the angular speed of the rotation is half the magnitude of the curl at this point. The notation ∇ × F has its origins in the similarities to the 3-dimensional cross product, such notation involving operators is common in physics and algebra. However, in coordinate systems, such as polar-toroidal coordinates. This expands as follows, i + j + k Although expressed in terms of coordinates, equivalently, = e k ε k l m ∇ l F m where ek are the coordinate vector fields. Equivalently, using the derivative, the curl can be expressed as, ∇ × F = ♯ Here ♭ and ♯ are the musical isomorphisms
35.
Directional derivative
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It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant. The directional derivative is a case of the Gâteaux derivative. The directional derivative of a function f = f along a vector v = is the function defined by the limit ∇ v f = lim h →0 f − f h. In the context of a function on a Euclidean space, some texts restrict the vector v to being a unit vector, without the restriction, this definition is valid in a broad range of contexts, for example where the norm of a vector is undefined. Intuitively, the derivative of f at a point x represents the rate of change of f with respect to time when moving past x at velocity v. Some authors define the derivative to be with respect to an arbitrary nonzero vector v after normalization, thus being independent of its magnitude. This definition gives the rate of increase of f per unit of distance moved in the given direction. In this case, one has ∇ v f = lim h →0 f − f h | v |, many of the familiar properties of the ordinary derivative hold for the directional derivative. These include, for any functions f and g defined in a neighborhood of, suppose that f is a function defined in a neighborhood of p, and differentiable at p. If v is a tangent vector to M at p, then the derivative of f along v, denoted variously as df, ∇ v f, L v f, or v p. Let γ, → M be a curve with γ = p. We translate a covector S along δ then δ′ and then subtract the translation along δ′, instead of building the directional derivative using partial derivatives, we use the covariant derivative. The rotation operator for an angle θ, i. e. See for example Neumann boundary condition, if the normal direction is denoted by n, then the directional derivative of a function f is sometimes denoted as ∂ f ∂ n. The directional directive provides a way of finding these derivatives. The definitions of directional derivatives for various situations are given below and it is assumed that the functions are sufficiently smooth that derivatives can be taken. Let f be a real valued function of the vector v, mathematical methods for physics and engineering
36.
Vector calculus identities
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The gradient of a tensor field, A, of order n, is generally written as grad = ∇ A and is a tensor field of order n +1. In particular, if the field has order 0, ψ. The divergence of a field, A, of non-zero order n, is generally written as div = ∇ ⋅ A and is a contraction to a tensor field of order n −1. Specifically, the divergence of a vector is a scalar, specifically, for the outer product of two vectors, ∇ ⋅ = b + b. For a 3-dimensional vector field v, curl is also a 3-dimensional vector field, generally written as, ∇ × v or in Einstein notation as, ε i j k ∂ v k ∂ x j where ε is the Levi-Civita symbol. In Cartesian coordinates, the Laplacian of a function f is Δ f = ∇2 f = f = ∂2 f ∂ x 2 + ∂2 f ∂ y 2 + ∂2 f ∂ z 2. For a tensor field, A, the laplacian is generally written as, in Feynman subscript notation, ∇ B = A × + B where the notation ∇B means the subscripted gradient operates on only the factor B. A less general but similar idea is used in geometric algebra where the so-called Hestenes overdot notation is employed, the above identity is then expressed as, ∇ ˙ = A × + B where overdots define the scope of the vector derivative. The dotted vector, in this case B, is differentiated, for the remainder of this article, Feynman subscript notation will be used where appropriate. Where JA denotes the Jacobian of A. Alternatively, using Feynman subscript notation, as a special case, when A = B,12 ∇ = J A T A = A + A ×. ∇ × = ∇ − ∇2 A Here, ∇2 is the vector Laplacian operating on the vector field A, exterior derivative Vector calculus Del in cylindrical and spherical coordinates Comparison of vector algebra and geometric algebra
37.
Divergence theorem
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More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that the sum of all sources gives the net out of a region. The divergence theorem is an important result for the mathematics of physics and engineering, in particular in electrostatics, in physics and engineering, the divergence theorem is usually applied in three dimensions. However, it generalizes to any number of dimensions, in one dimension, it is equivalent to the fundamental theorem of calculus. In two dimensions, it is equivalent to Greens theorem, the theorem is a special case of the more general Stokes theorem. If a fluid is flowing in some area, then the rate at which fluid flows out of a region within that area can be calculated by adding up the sources inside the region. The fluid flow is represented by a field, and the vector fields divergence at a given point describes the strength of the source or sink there. So, integrating the fields divergence over the interior of the region should equal the integral of the field over the regions boundary. The divergence theorem says that this is true, suppose V is a subset of R n which is compact and has a piecewise smooth boundary S. If F is a continuously differentiable vector field defined on a neighborhood of V, then we have, the left side is a volume integral over the volume V, the right side is the surface integral over the boundary of the volume V. The closed manifold ∂V is quite generally the boundary of V oriented by outward-pointing normals, the symbol within the two integrals stresses once more that ∂V is a closed surface. By replacing F in the divergence theorem with specific forms, other useful identities can be derived, with F → F g for a scalar function g and a vector field F, ∭ V d V = S g F ⋅ n d S. A special case of this is F = ∇ f , in case the theorem is the basis for Greens identities. With F → F × G for two vector fields F and G, ∭ V d V = S ⋅ d S. With F → f c for a scalar function f and vector c, ∭ V c ⋅ ∇ f d V = S ⋅ d S − ∭ V f d V. The last term on the right vanishes for constant c or any divergence free vector field, with F → c × F for vector field F and constant vector c, ∭ V c ⋅ d V = S ⋅ d S. Suppose we wish to evaluate S F ⋅ n d S, where S is the sphere defined by S =. Since the function y is positive in one hemisphere of W and negative in the other, in an equal and opposite way, the same is true for z, ∭ W y d V = ∭ W z d V =0
38.
Green's theorem
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In mathematics, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. It is named after George Green and is the special case of the more general Kelvin–Stokes theorem. Let C be an oriented, piecewise smooth, simple closed curve in a plane. If L and M are functions of defined on a region containing D and have continuous partial derivatives there. In physics, Greens theorem is used to solve two-dimensional flow integrals. In plane geometry, and in particular, area surveying, Greens theorem can be used to determine the area, the following is a proof of half of the theorem for the simplified area D, a type I region where C1 and C3 are curves connected by vertical lines. A similar proof exists for the half of the theorem when D is a type II region where C2. Putting these two together, the theorem is thus proven for regions of type III. The general case can then be deduced from this case by decomposing D into a set of type III regions. If it can be shown that if ∮ C L d x = ∬ D d A and ∮ C M d y = ∬ D d A are true and we can prove easily for regions of type I, and for regions of type II. Greens theorem then follows for regions of type III, assume region D is a type I region and can thus be characterized, as pictured on the right, by D = where g1 and g2 are continuous functions on. Compute the double integral in, ∬ D ∂ L ∂ y d A = ∫ a b ∫ g 1 g 2 ∂ L ∂ y d y d x = ∫ a b d x, now compute the line integral in. C can be rewritten as the union of four curves, C1, C2, C3, with C1, use the parametric equations, x = x, y = g1, a ≤ x ≤ b. Then ∫ C1 L d x = ∫ a b L d x, with C3, use the parametric equations, x = x, y = g2, a ≤ x ≤ b. Then ∫ C3 L d x = − ∫ − C3 L d x = − ∫ a b L d x, the integral over C3 is negated because it goes in the negative direction from b to a, as C is oriented positively. On C2 and C4, x remains constant, meaning ∫ C4 L d x = ∫ C2 L d x =0, combining with, we get for regions of type I. A similar treatment yields for regions of type II, putting the two together, we get the result for regions of type III. Write F for the vector-valued function F =, start with the left side of Greens theorem, ∮ C = ∮ C ⋅ = ∮ C F ⋅ d r
39.
Multivariable calculus
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A study of limits and continuity in multivariable calculus yields many counter-intuitive results not demonstrated by single-variable functions. Indeed, the function f = x 2 y x 4 + y 2 approaches zero along any line through the origin, however, when the origin is approached along a parabola y = x 2, it has a limit of 0.5. Since taking different paths toward the same point yields different values for the limit, continuity in each argument is not sufficient for multivariate continuity. For instance, in the case of a function with two real-valued parameters, f, continuity of f in x for fixed y and continuity of f in y for fixed x does not imply continuity of f. Consider f = { y x − y if 1 ≥ x > y ≥0 x y − x if 1 ≥ y > x ≥01 − x if x = y >00 else. It is easy to verify that all real-valued functions that are given by f y, = f are continuous in x, similarly, all f x are continuous as f is symmetric with regards to x and y. However, f itself is not continuous as can be seen by considering the sequence f which should converge to f =0 if f was continuous, however, lim n → ∞ f =1. Thus, function is not continuous at, the partial derivative generalizes the notion of the derivative to higher dimensions. A partial derivative of a function is a derivative with respect to one variable with all other variables held constant. Partial derivatives may be combined in interesting ways to more complicated expressions of the derivative. In vector calculus, the del operator is used to define the concepts of gradient, divergence, a matrix of partial derivatives, the Jacobian matrix, may be used to represent the derivative of a function between two spaces of arbitrary dimension. The derivative can thus be understood as a transformation which directly varies from point to point in the domain of the function. Differential equations containing partial derivatives are called differential equations or PDEs. These equations are more difficult to solve than ordinary differential equations. The multiple integral expands the concept of the integral to functions of any number of variables, double and triple integrals may be used to calculate areas and volumes of regions in the plane and in space. Fubinis theorem guarantees that an integral may be evaluated as a repeated integral or iterated integral as long as the integrand is continuous throughout the domain of integration. The surface integral and the integral are used to integrate over curved manifolds such as surfaces and curves. In single-variable calculus, the theorem of calculus establishes a link between the derivative and the integral
40.
Geometric calculus
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In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to other mathematical theories including differential geometry. With a geometric algebra given, let a and b be vectors, the directional derivative of F along b is defined as ∇ b F = lim ϵ →0 F − F ϵ provided that the limit exists, where the limit is taken for scalar ε. This is similar to the definition of a directional derivative. The standard order of operations for the derivative is that it acts only on the function closest to its immediate right. Given two functions F and G, then for example we have ∇ F G = G, although the partial derivative exhibits a product rule, the geometric derivative only partially inherits this property. A solution is to adopt the overdot notation, in which the scope of a derivative with an overdot is the multivector-valued function sharing the same overdot. In this case, if we define ∇ ˙ F G ˙ = e i F, neither the interior derivative operator nor the exterior derivative operator is invertible. Let be a set of vectors that span an n-dimensional vector space. From geometric algebra, we interpret the pseudoscalar e 1 ∧ e 2 ∧ ⋯ ∧ e n to be the volume of the n-parallelotope subtended by these basis vectors. If the basis vectors are orthonormal, then this is the unit pseudoscalar and we denote these selected basis vectors by. A general k-volume of the k-parallelotope subtended by these vectors is the grade k multivector e i 1 ∧ e i 2 ∧ ⋯ ∧ e i k. Even more generally, we may consider a new set of vectors proportional to the k basis vectors and we are free to choose components as infinitesimally small as we wish as long as they remain nonzero. Compare the Riemannian volume form in the theory of differential forms and we may divide this volume into a sum of simplices. Let be the coordinates of the vertices, at each vertex we assign a measure Δ U i as the average measure of the simplices sharing the vertex. The reason for defining the derivative and integral as above is that they allow a strong generalization of Stokes theorem. Let L be a function of r-grade input A and general position x. A sufficiently smooth k-surface in a space is deemed a manifold
41.
Partial derivative
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In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. Partial derivatives are used in calculus and differential geometry. The partial derivative of an f with respect to the variable x is variously denoted by f x ′, f x, ∂ x f, D x f, D1 f, ∂ ∂ x f. One of the first known uses of the symbol in mathematics is by Marquis de Condorcet from 1770, the modern partial derivative notation is by Adrien-Marie Legendre, though he later abandoned it, Carl Gustav Jacob Jacobi re-introduced the symbol in 1841. Suppose that ƒ is a function of more than one variable, for instance, z = f = x 2 + x y + y 2. The graph of this function defines a surface in Euclidean space, to every point on this surface, there is an infinite number of tangent lines. Partial differentiation is the act of choosing one of these lines, the graph and this plane are shown on the right. On the graph below it, we see the way the function looks on the plane y =1, therefore ∂ z ∂ x =3 at the point. That is, the derivative of z with respect to x at is 3. The function f can be reinterpreted as a family of functions of one variable indexed by the other variables, in other words, every value of y defines a function, denoted fy, which is a function of one variable x. That is, f y = x 2 + x y + y 2. Once a value of y is chosen, say a, then f determines a function fa which traces a curve x2 + ax + a2 on the xz plane, f a = x 2 + a x + a 2. In this expression, a is a constant, not a variable, so fa is a function of one real variable. Consequently, the definition of the derivative for a function of one variable applies, the above procedure can be performed for any choice of a. Assembling the derivatives together into a function gives a function describes the variation of f in the x direction. This is the derivative of f with respect to x. Here ∂ is a rounded d called the partial derivative symbol, to distinguish it from the letter d, ∂ is sometimes pronounced tho or partial. In general, the derivative of an n-ary function f in the direction xi at the point is defined to be
42.
Multiple integral
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The multiple integral is a collective term for the definite integral of functions of more than one real variable, for example, f or f. Integrals of a function of two variables over a region in R2 are called double integrals, and integrals of a function of three variables over a region of R3 are called triple integrals, if there are more variables, a multiple integral will yield hypervolumes of multidimensional functions. For n >1, consider a so-called half-open n-dimensional hyperrectangular domain T, defined as, partition each interval [aj, bj) into a finite family Ij of non-overlapping subintervals ijα, with each subinterval closed at the left end, and open at the right end. Then the finite family of subrectangles C given by C = I1 × I2 × ⋯ × I n is a partition of T, that is, let f, T → R be a function defined on T. The diameter of a subrectangle Ck is the largest of the lengths of the intervals whose Cartesian product is Ck, the diameter of a given partition of T is defined as the largest of the diameters of the subrectangles in the partition. Intuitively, as the diameter of the partition C is restricted smaller and smaller, the number of subrectangles m gets larger, and the measure m of each subrectangle grows smaller. Then the integral of the function over the original domain is defined to be the integral of the extended function over its rectangular domain. In what follows the Riemann integral in n dimensions will be called the multiple integral, multiple integrals have many properties common to those of integrals of functions of one variable. One important property of multiple integrals is that the value of an integral is independent of the order of integrands under certain conditions and this property is popularly known as Fubinis theorem. For continuous functions, this is justified by Fubinis theorem, sometimes, it is possible to obtain the result of the integration by direct examination without any calculations. The following are some methods of integration, When the integrand is a constant function c, the integral is equal to the product of c. If c =1 and the domain is a subregion of R2, the integral gives the area of the region, while if the domain is a subregion of R3, the integral gives the volume of the region. When the integrand is even with respect to this variable, the integral is equal to twice the integral over one half of the domain, as the integrals over the two halves of the domain are equal. Consider the function f =2 sin − 3y3 +5 integrated over the domain T =, a disc with radius 1 centered at the origin with the boundary included. Similarly, the function 3y3 is an odd function of y, and T is symmetric with respect to the x-axis, therefore the original integral is equal to the area of the disk times 5, or 5π. Consider the function f = x exp and as integration region the sphere with radius 2 centered at the origin, T =. The ball is symmetric about all three axes, but it is sufficient to integrate with respect to x-axis to show that the integral is 0, such a domain will be here called a normal domain. Elsewhere in the literature, normal domains are called type I or type II domains
43.
Surface integral
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In mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral, given a surface, one may integrate over its scalar fields, and vector fields. Surface integrals have applications in physics, particularly with the theories of classical electromagnetism, let such a parameterization be x, where varies in some region T in the plane. The surface integral can also be expressed in the equivalent form ∬ S f d Σ = ∬ T f g d s d t where g is the determinant of the first fundamental form of the mapping x. So that ∂ r ∂ x =, and ∂ r ∂ y =, one can recognize the vector in the second line above as the normal vector to the surface. Note that because of the presence of the product, the above formulas only work for surfaces embedded in three-dimensional space. This can be seen as integrating a Riemannian volume form on the parameterized surface, consider a vector field v on S, that is, for each x in S, v is a vector. The surface integral can be defined according to the definition of the surface integral of a scalar field. This applies for example in the expression of the field at some fixed point due to an electrically charged surface. Alternatively, if we integrate the normal component of the vector field, imagine that we have a fluid flowing through S, such that v determines the velocity of the fluid at x. The flux is defined as the quantity of flowing through S per unit time. This illustration implies that if the field is tangent to S at each point, then the flux is zero, because the fluid just flows in parallel to S. This also implies that if v does not just flow along S and we find the formula ∬ S v ⋅ d Σ = ∬ S d Σ = ∬ T ∥ ∥ d s d t = ∬ T v ⋅ d s d t. The cross product on the side of this expression is a surface normal determined by the parametrization. This formula defines the integral on the left and we may also interpret this as a special case of integrating 2-forms, where we identify the vector field with a 1-form, and then integrate its Hodge dual over the surface. The transformation of the forms are similar. Then, the integral of f on S is given by ∬ D d s d t where ∂ x ∂ s × ∂ x ∂ t = is the surface element normal to S. Let us note that the integral of this 2-form is the same as the surface integral of the vector field which has as components f x, f y and f z
44.
Volume integral
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In mathematics—in particular, in multivariable calculus—a volume integral refers to an integral over a 3-dimensional domain, that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example and it can also mean a triple integral within a region D in R3 of a function f, and is usually written as, ∭ D f d x d y d z. A volume integral in cylindrical coordinates is ∭ D f ρ d ρ d φ d z, and this is rather trivial however, and a volume integral is far more powerful. Multiple integral, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Weisstein, Eric W
45.
Jacobian matrix and determinant
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In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. When the matrix is a matrix, both the matrix and its determinant are referred to as the Jacobian in literature. Suppose f, ℝn → ℝm is a function takes as input the vector x ∈ ℝn. Then the Jacobian matrix J of f is an m×n matrix, usually defined and arranged as follows, J = = or, component-wise and this matrix, whose entries are functions of x, is also denoted by Df, Jf, and ∂/∂. This linear map is thus the generalization of the notion of derivative. If m = n, the Jacobian matrix is a matrix, and its determinant. It carries important information about the behavior of f. In particular, the f has locally in the neighborhood of a point x an inverse function that is differentiable if. The Jacobian determinant also appears when changing the variables in multiple integrals, if m =1, f is a scalar field and the Jacobian matrix is reduced to a row vector of partial derivatives of f—i. e. the gradient of f. These concepts are named after the mathematician Carl Gustav Jacob Jacobi, the Jacobian generalizes the gradient of a scalar-valued function of multiple variables, which itself generalizes the derivative of a scalar-valued function of a single variable. In other words, the Jacobian for a multivariate function is the gradient. The Jacobian can also be thought of as describing the amount of stretching, rotating or transforming that a transformation imposes locally, for example, if = f is used to transform an image, the Jacobian Jf, describes how the image in the neighborhood of is transformed. If p is a point in ℝn and f is differentiable at p, compare this to a Taylor series for a scalar function of a scalar argument, truncated to first order, f = f + f ′ + o. The Jacobian of the gradient of a function of several variables has a special name, the Hessian matrix. If m=n, then f is a function from ℝn to itself and we can then form its determinant, known as the Jacobian determinant. The Jacobian determinant is occasionally referred to as the Jacobian, the Jacobian determinant at a given point gives important information about the behavior of f near that point. For instance, the differentiable function f is invertible near a point p ∈ ℝn if the Jacobian determinant at p is non-zero. This is the inverse function theorem, furthermore, if the Jacobian determinant at p is positive, then f preserves orientation near p, if it is negative, f reverses orientation
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Fractional calculus
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In this context, the term powers refers to iterative application of a linear operator on a function, in some analogy to function composition acting on a variable, e. g. f 2 = f. Continuous semigroups are prevalent in mathematics, and have an interesting theory, Fractional differential equations are a generalization of differential equations through the application of fractional calculus. The derivative of a function f at a point x is a property only when a is an integer. In other words, it is not correct to say that the derivative at x of a function f depends only on values of f very near x. Therefore it is expected that the theory involves some sort of boundary conditions, to use a metaphor, the fractional derivative requires some peripheral vision. As far as the existence of such a theory is concerned, the fractional derivative of a function to order a is often now defined by means of the Fourier or Mellin integral transforms. A fairly natural question to ask is there exists a linear operator H, or half-derivative. Let f be a function defined for x >0, form the definite integral from 0 to x. Repeating this process gives = ∫0 x d t = ∫0 x d t, the Cauchy formula for repeated integration, namely =1. ∫0 x n −1 f d t, leads in a way to a generalization for real n. Using the gamma function to remove the discrete nature of the function gives us a natural candidate for fractional applications of the integral operator. =1 Γ ∫0 x α −1 f d t This is in fact a well-defined operator. It is straightforward to show that the J operator satisfies = = =1 Γ ∫0 x α + β −1 f d t This relationship is called the property of fractional differintegral operators. Unfortunately the comparable process for the derivative operator D is significantly more complex, let us assume that f is a monomial of the form f = x k. The first derivative is as usual f ′ = d d x f = k x k −1, repeating this gives the more general result that d a d x a x k = k. For example, the th derivative of the th derivative yields the 2nd derivative, also notice that setting negative values for a yields integrals. For example, D3 /2 f = D1 /2 D1 f = D1 /2 d d x f We can also come at the question via the Laplace transform. Noting that L = L =1 s and L =1 s =1 s 2 etc. we assert J α f = L −1, for example J α = L −1 = Γ Γ t α + k as expected
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Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
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Mathematical analysis
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Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. These theories are studied in the context of real and complex numbers. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis, analysis may be distinguished from geometry, however, it can be applied to any space of mathematical objects that has a definition of nearness or specific distances between objects. Mathematical analysis formally developed in the 17th century during the Scientific Revolution, early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, a geometric sum is implicit in Zenos paradox of the dichotomy. The explicit use of infinitesimals appears in Archimedes The Method of Mechanical Theorems, in Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle. Zu Chongzhi established a method that would later be called Cavalieris principle to find the volume of a sphere in the 5th century, the Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolles theorem in the 12th century. In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and his followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century. The modern foundations of analysis were established in 17th century Europe. During this period, calculus techniques were applied to approximate discrete problems by continuous ones, in the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the definition of continuity in 1816. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required a change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations, the contributions of these mathematicians and others, such as Weierstrass, developed the -definition of limit approach, thus founding the modern field of mathematical analysis. In the middle of the 19th century Riemann introduced his theory of integration, the last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the epsilon-delta definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of numbers without proof. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the size of the set of discontinuities of real functions, also, monsters began to be investigated
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Combinatorics
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Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general methods were developed. One of the oldest and most accessible parts of combinatorics is graph theory, Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist or a combinatorist, basic combinatorial concepts and enumerative results appeared throughout the ancient world. Greek historian Plutarch discusses an argument between Chrysippus and Hipparchus of a rather delicate enumerative problem, which was shown to be related to Schröder–Hipparchus numbers. In the Ostomachion, Archimedes considers a tiling puzzle, in the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. The Indian mathematician Mahāvīra provided formulae for the number of permutations and combinations, later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J. J. Sylvester and Percy MacMahon helped lay the foundation for enumerative, graph theory also enjoyed an explosion of interest at the same time, especially in connection with the four color problem. In the second half of the 20th century, combinatorics enjoyed a rapid growth, in part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical science, but at the same time led to a partial fragmentation of the field. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, fibonacci numbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a framework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis, in contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Partition theory studies various enumeration and asymptotic problems related to integer partitions, originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field. It incorporates the bijective approach and various tools in analysis and analytic number theory, graphs are basic objects in combinatorics