History of the creation of mathematical analysis. Mathematical analysis

The founders of modern science - Copernicus, Kepler, Galileo and Newton - approached the study of nature as mathematics. By studying motion, mathematicians developed such a fundamental concept as function, or the relationship between variables, for example d = kt 2 where d is the distance traveled by a freely falling body, and t- the number of seconds that the body is in free fall. The concept of function immediately became central in determining the speed at a given moment in time and the acceleration of a moving body. The mathematical difficulty of this problem was that at any moment the body travels zero distance in zero time. Therefore, determining the value of the speed at an instant of time by dividing the path by the time, we arrive at the mathematically meaningless expression 0/0.

The problem of determining and calculating instantaneous rates of change of various quantities attracted the attention of almost all mathematicians of the 17th century, including Barrow, Fermat, Descartes and Wallis. The disparate ideas and methods they proposed were combined into a systematic, universally applicable formal method by Newton and G. Leibniz (1646-1716), the creators of differential calculus. There were heated debates between them on the issue of priority in the development of this calculus, with Newton accusing Leibniz of plagiarism. However, as research by historians of science has shown, Leibniz created mathematical analysis independently of Newton. As a result of the conflict, the exchange of ideas between mathematicians in continental Europe and England was interrupted for many years, to the detriment of the English side. English mathematicians continued to develop the ideas of analysis in a geometric direction, while mathematicians of continental Europe, including I. Bernoulli (1667-1748), Euler and Lagrange achieved incomparably greater success following the algebraic, or analytical, approach.

The basis of all mathematical analysis is the concept of limit. Speed at an instant is defined as the limit to which the average speed tends d/t when the value t getting closer to zero. Differential calculus provides a computationally convenient general method for finding the rate of change of a function f (x) for any value X. This speed is called derivative. From the generality of the record f (x) it is clear that the concept of derivative is applicable not only in problems related to the need to find speed or acceleration, but also in relation to any functional dependence, for example, to some relationship from economic theory. One of the main applications of differential calculus is the so-called. maximum and minimum tasks; Another important range of problems is finding the tangent to a given curve.

It turned out that with the help of a derivative, specially invented for working with motion problems, it is also possible to find areas and volumes limited by curves and surfaces, respectively. The methods of Euclidean geometry did not have the necessary generality and did not allow obtaining the required quantitative results. Through the efforts of mathematicians of the 17th century. Numerous private methods were created that made it possible to find the areas of figures bounded by curves of one type or another, and in some cases the connection between these problems and problems of finding the rate of change of functions was noted. But, as in the case of differential calculus, it was Newton and Leibniz who realized the generality of the method and thereby laid the foundations of integral calculus.

The Newton-Leibniz method begins by replacing the curve that limits the area to be determined with a sequence of broken lines that approximates it, similar to what was done in the exhaustion method invented by the Greeks. The exact area is equal to the limit of the sum of areas n rectangles when n turns to infinity. Newton showed that this limit could be found by reversing the process of finding the rate of change of a function. The inverse operation of differentiation is called integration. The statement that summation can be accomplished by reversing differentiation is called the fundamental theorem of calculus. Just as differentiation is applicable to a much broader class of problems than finding velocities and accelerations, integration is applicable to any problem involving summation, such as physics problems involving the addition of forces.

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definition - Mathematical_analysis

In the educational process, analysis includes:

At the same time, elements of functional analysis and the theory of the Lebesgue integral are given optionally, and TFKP, calculus of variations, and the theory of differential equations are taught in separate courses. The rigor of the presentation follows late 19th century patterns and in particular makes use of naive set theory.

The program of the analysis course taught at universities in the Russian Federation roughly corresponds to the program of the Anglo-American course “Calculus”.

Story

The predecessors of mathematical analysis were the ancient method of exhaustion and the method of indivisibles. All three directions, including analysis, are related by a common initial idea: decomposition into infinitesimal elements, the nature of which, however, seemed rather vague to the authors of the idea. Algebraic approach ( infinitesimal calculus) begins to appear in Wallis, James Gregory and Barrow. The new calculus as a system was created in full by Newton, who, however, did not publish his discoveries for a long time.

The official date of birth of differential calculus can be considered May, when Leibniz published his first article "A New Method of Highs and Lows...". This article, in a concise and inaccessible form, set out the principles of a new method called differential calculus.

Leibniz and his students

These definitions are explained geometrically, while in Fig. infinitesimal increments are depicted as finite. The consideration is based on two requirements (axioms). First:

It is required that two quantities that differ from each other only by an infinitesimal amount can be taken [when simplifying expressions?] indifferently one instead of the other.

The continuation of each such line is called a tangent to the curve. Investigating the tangent passing through the point, L'Hopital attaches great importance to the quantity

reaching extreme values at the inflection points of the curve, while the relation to is not given any special significance.

It is noteworthy to find extremum points. If, with a continuous increase in diameter, the ordinate first increases and then decreases, then the differential is first positive compared to , and then negative.

But any continuously increasing or decreasing value cannot turn from positive to negative without passing through infinity or zero... It follows that the differential of the largest and smallest value must be equal to zero or infinity.

This formulation is probably not flawless, if we remember the first requirement: let, say, , then by virtue of the first requirement

;

at zero, the right hand side is zero and the left hand side is not. Apparently it should have been said that it can be transformed in accordance with the first requirement so that at the maximum point . . In the examples, everything is self-explanatory, and only in the theory of inflection points does L'Hopital write that it is equal to zero at the maximum point, being divided by .

Further, with the help of differentials alone, extremum conditions are formulated and a large number of complex problems related mainly to differential geometry on the plane are considered. At the end of the book, in chap. 10, sets out what is now called L'Hopital's rule, although in an unusual form. Let the ordinate of the curve be expressed as a fraction, the numerator and denominator of which vanish at . Then the point of the curve c has an ordinate equal to the ratio of the differential of the numerator to the differential of the denominator taken at .

According to L'Hôpital's plan, what he wrote constituted the first part of Analysis, while the second was supposed to contain integral calculus, that is, a method of finding the connection between variables based on the known connection of their differentials. Its first presentation was given by Johann Bernoulli in his Mathematical lectures on the integral method. Here a method is given for taking most elementary integrals and methods for solving many first-order differential equations are indicated.

Pointing to the practical usefulness and simplicity of the new method, Leibniz wrote:

What a person versed in this calculus can obtain directly in three lines, other learned men were forced to look for by following complex detours.

Euler

The changes that took place over the next half century are reflected in Euler's extensive treatise. The presentation of the analysis opens with a two-volume “Introduction”, which contains research on various representations of elementary functions. The term “function” first appears only in Leibniz, but it was Euler who put it in the first place. The original interpretation of the concept of a function was that a function is an expression for counting (German. Rechnungsausdrϋck) or analytical expression.

A variable quantity function is an analytical expression composed in some way from this variable quantity and numbers or constant quantities.

Emphasizing that “the main difference between functions lies in the way they are composed of variable and constant,” Euler lists the actions “through which quantities can be combined and mixed with each other; these actions are: addition and subtraction, multiplication and division, exponentiation and extraction of roots; This should also include the solution of [algebraic] equations. In addition to these operations, called algebraic, there are many others, transcendental, such as: exponential, logarithmic and countless others, delivered by integral calculus.” This interpretation made it possible to easily handle multi-valued functions and did not require an explanation of which field the function was being considered over: the counting expression was defined for complex values of variables even when this was not necessary for the problem under consideration.

Operations in the expression were allowed only in finite numbers, and the transcendental penetrated with the help of an infinitely large number. In expressions, this number is used along with natural numbers. For example, such an expression for the exponent is considered acceptable

in which only later authors saw the ultimate transition. Various transformations were made with analytical expressions, which allowed Euler to find representations for elementary functions in the form of series, infinite products, etc. Euler transforms expressions for counting as they do in algebra, without paying attention to the possibility of calculating the value of a function at a point for each from written formulas.

Unlike L'Hopital, Euler examines in detail transcendental functions and in particular their two most studied classes - exponential and trigonometric. He discovers that all elementary functions can be expressed using arithmetic operations and two operations - taking the logarithm and the exponent.

The proof itself perfectly demonstrates the technique of using the infinitely large. Having defined sine and cosine using the trigonometric circle, Euler derived the following from the addition formulas:

Assuming and , he gets

discarding infinitesimal quantities of higher order. Using this and a similar expression, Euler obtained his famous formula

Having indicated various expressions for functions that are now called elementary, Euler moves on to consider curves on a plane drawn by free movement of the hand. In his opinion, it is not possible to find a single analytical expression for every such curve (see also the String Dispute). In the 19th century, at the instigation of Casorati, this statement was considered erroneous: according to Weierstrass’s theorem, any continuous curve in the modern sense can be approximately described by polynomials. In fact, Euler was hardly convinced by this, because he still needed to rewrite the passage to the limit using the symbol.

Euler begins his presentation of differential calculus with the theory of finite differences, followed in the third chapter by a philosophical explanation that “an infinitesimal quantity is exactly zero,” which most of all did not suit Euler’s contemporaries. Then, differentials are formed from finite differences at an infinitesimal increment, and from Newton's interpolation formula - Taylor's formula. This method essentially goes back to the work of Taylor (1715). In this case, Euler has a stable relation , which, however, is considered as a relation of two infinitesimals. The last chapters are devoted to approximate calculation using series.

In the three-volume integral calculus, Euler interprets and introduces the concept of integral as follows:

The function whose differential is called its integral and is denoted by the sign placed in front.

In general, this part of Euler’s treatise is devoted to a more general, from a modern point of view, problem of the integration of differential equations. At the same time, Euler finds a number of integrals and differential equations that lead to new functions, for example, -functions, elliptic functions, etc. A rigorous proof of their non-elementary nature was given in the 1830s by Jacobi for elliptic functions and by Liouville (see elementary functions).

Lagrange

The next major work that played a significant role in the development of the concept of analysis was Theory of analytic functions Lagrange and Lacroix's extensive retelling of Lagrange's work in a somewhat eclectic manner.

Wanting to get rid of the infinitesimal altogether, Lagrange reversed the connection between derivatives and the Taylor series. By analytic function Lagrange understood an arbitrary function studied by analytical methods. He designated the function itself as , giving a graphical way to write the dependence - earlier Euler made do with only variables. To apply analysis methods, according to Lagrange, it is necessary that the function be expanded into a series

whose coefficients will be new functions. It remains to call it a derivative (differential coefficient) and denote it as . Thus, the concept of derivative is introduced on the second page of the treatise and without the help of infinitesimals. It remains to be noted that

therefore the coefficient is twice the derivative of the derivative, that is

etc.

This approach to the interpretation of the concept of derivative is used in modern algebra and served as the basis for the creation of Weierstrass's theory of analytic functions.

Lagrange operated with such series as formal ones and obtained a number of remarkable theorems. In particular, for the first time and quite rigorously he proved the solvability of the initial problem for ordinary differential equations in formal power series.

The question of assessing the accuracy of approximations provided by partial sums of the Taylor series was first posed by Lagrange: in the end Theories of analytic functions he derived what is now called Taylor's formula with a remainder term in Lagrange form. However, in contrast to modern authors, Lagrange did not see the need to use this result to justify the convergence of the Taylor series.

The question of whether the functions used in analysis can really be expanded into a power series subsequently became the subject of debate. Of course, Lagrange knew that at some points elementary functions may not be expanded into a power series, but at these points they are not differentiable in any sense. Cauchy in his Algebraic analysis cited the function as a counterexample

extended by zero at zero. This function is smooth everywhere on the real axis and at zero it has a zero Maclaurin series, which, therefore, does not converge to the value . Against this example, Poisson objected that Lagrange defined the function as a single analytical expression, while in Cauchy’s example the function is defined differently at zero and at . Only at the end of the 19th century did Pringsheim prove that there is an infinitely differentiable function, given by a single expression, for which the Maclaurin series diverges. An example of such a function is the expression

Further development

In the last third of the 19th century, Weierstrass arithmetized the analysis, considering the geometric justification to be insufficient, and proposed a classical definition of the limit through the ε-δ language. He also created the first rigorous theory of the set of real numbers. At the same time, attempts to improve the Riemann integrability theorem led to the creation of a classification of discontinuity of real functions. “Pathological” examples were also discovered (continuous functions that are nowhere differentiable, space-filling curves). In this regard, Jordan developed measure theory, and Cantor developed set theory, and at the beginning of the 20th century, mathematical analysis was formalized with their help. Another important development of the 20th century was the development of non-standard analysis as an alternative approach to justifying analysis.

Sections of mathematical analysis

Bibliography

Encyclopedic articles

Educational literature

Standard textbooks

For many years, the following textbooks have been popular in Russia:

Some universities have their own analysis guides:

Mathematics at a technical university Collection of textbooks in 21 volumes.

Bogdanov Yu. S. Lectures on mathematical analysis (in two parts). - Minsk: BSU, 1974. - 357 p.

Advanced textbooks

Textbooks:

Rudin U. Fundamentals of mathematical analysis. M., 1976 - a small book, written very clearly and concisely.

Problems of increased difficulty:

G. Polia, G. Szege, Problems and theorems from analysis.

Philosophy is considered the focus of all sciences, since it included the first sprouts of literature, astronomy, literature, natural science, mathematics and other areas. Over time, each field developed independently, mathematics was no exception. The first “hint” of analysis is considered to be the theory of decomposition into infinitesimal quantities, which many minds tried to approach, but it was vague and had no basis. This is due to an attachment to the old school of science, which was strict in its formulations. Isaac Newton came very close to forming the foundations, but was too late. As a result, mathematical analysis owes its emergence as a separate system to the philosopher Gottfried Leibniz. It was he who introduced to the scientific world such concepts as minimum and maximum, inflection points and convexity of the graph of a function, and formulated the foundations of differential calculus. From this moment on, mathematics is officially divided into elementary and higher.

Mathematical analysis. Our days

Any specialty, be it technical or humanitarian, includes analysis in the course of study. The depth of study varies, but the essence remains the same. Despite all the “abstractness,” it is one of the pillars on which natural science in its modern understanding rests. With his help, physics and economics developed, he is able to describe and predict the activities of the stock exchange, and help in building an optimal portfolio of shares. Introduction to mathematical analysis is based on elementary concepts:

multitudes;
basic operations on sets;
properties of operations on sets;
functions (otherwise known as mappings);
types of functions;
sequences;
number lines;
sequence limit;
properties of limits;
continuity of function.

It is worth highlighting separately such concepts as set, point, straight line, plane. All of them have no definitions, since they are the basic concepts on which all mathematics is built. All that can be done in the process is to explain what exactly they mean in individual cases.

Limit as a continuation

The fundamentals of mathematical analysis include limit. In practice, it represents the value to which a sequence or function strives, comes as close as desired, but does not reach it. It is denoted as lim; consider a special case of the limit of the function: lim (x-1)= 0 for x→1. From this simplest example it is clear that as x→1 the entire function tends to 0, since if we substitute the limit into the function itself, we get (1-1)=0. More detailed information, from elementary to complex special cases, is presented in a kind of “Bible” of analysis - the works of Fichtenholtz. It examines mathematical analysis, limits, their derivation and further application. For example, derivation of the number e (Euler's constant) would be impossible without the theory of limits. Despite the dynamic abstractness of the theory, limits are actively used in practice in economics and sociology. For example, you cannot do without them when calculating interest on a bank deposit.

During the ancient period, some ideas appeared that later led to integral calculus, but in that era these ideas were not developed in a rigorous, systematic manner. Calculations of volumes and areas, one of the purposes of integral calculus, can be found in the Moscow mathematical papyrus from Egypt (c. 1820 BC), but the formulas are more like instructions, without any indication of the method, and some are simply erroneous. In the era of Greek mathematics, Eudoxus (c. 408-355 BC) used the exhaustion method to calculate areas and volumes, which anticipates the concept of limit, and later this idea was further developed by Archimedes (c. 287-212 BC) , inventing heuristics that resemble methods of integral calculus. The exhaustion method was later invented in China by Liu Hui in the 3rd century AD, which he used to calculate the area of a circle. In the 5th AD, Zu Chongzhi developed a method for calculating the volume of a sphere, which would later be called Cavalieri's principle.

Middle Ages

In the 14th century, Indian mathematician Madhava Sangamagrama and the Kerala School of Astronomy and Mathematics introduced many components of calculus, such as Taylor series, approximation of infinite series, integral test of convergence, early forms of differentiation, term-by-term integration, iterative methods for solving nonlinear equations, and determining what area under the curve is its integral. Some consider Yuktibhāṣā to be the first work on mathematical analysis.

Modern era

In Europe, the seminal work was the treatise of Bonaventura Cavalieri, in which he argued that volumes and areas can be calculated as the sum of the volumes and areas of an infinitely thin section. The ideas were similar to what Archimedes outlined in his Method, but this treatise of Archimedes was lost until the first half of the 20th century. Cavalieri's work was not recognized because his methods could lead to erroneous results, and he gave infinitesimals a dubious reputation.

Formal research into infinitesimal calculus, which Cavalieri combined with finite difference calculus, was taking place in Europe around this time. Pierre Fermat, claiming that he borrowed it from Diophantus, introduced the concept of "quasi-equality" (English: adequality), which was equality up to an infinitesimal error. John Wallis, Isaac Barrow and James Gregory also made major contributions. The last two, around 1675, proved the second fundamental theorem of calculus.

Reasons

In mathematics, foundations refer to a strict definition of a subject, starting from precise axioms and definitions. At the initial stage of the development of calculus, the use of infinitesimal quantities was considered to be lax, and was severely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley. Berkeley excellently described the infinitesimals as "ghosts of dead quantities" in his book The Analyst in 1734. Developing a rigorous foundation for calculus occupied mathematicians for more than a century after Newton and Leibniz, and is still to some extent an active area of research today.

Several mathematicians, including Maclaurin, tried to prove the validity of the use of infinitesimals, but this was only done 150 years later with the work of Cauchy and Weierstrass, who finally found a way to evade the simple “little things” of infinitesimals, and the beginnings were made differential and integral calculus. In Cauchy's writings we find a universal range of fundamental approaches, including the definition of continuity in terms of infinitesimals and the (somewhat imprecise) prototype of the (ε, δ)-definition of limit in the definition of differentiation. In his work, Weierstrass formalizes the concept of limit and eliminates infinitesimal quantities. After this work of Weierstrass, the general basis of calculus became limits, and not infinitesimal quantities. Bernhard Riemann used these ideas to give a precise definition of the integral. Additionally, during this period, the ideas of calculus were generalized to Euclidean space and to the complex plane.

In modern mathematics, the fundamentals of calculus are included in the branch of real analysis, which contains complete definitions and proofs of the theorems of calculus. The scope of calculus research has become much broader. Henri Lebesgue developed the theory of set measures and used it to determine integrals of all but the most exotic functions. Laurent Schwartz introduced generalized functions, which can be used to calculate the derivatives of any function in general.

The introduction of limits determined not the only strict approach to the basis of calculus. An alternative would be, for example, Abraham Robinson's non-standard analysis. Robinson's approach, developed in the 1960s, uses technical tools from mathematical logic to extend the system of real numbers to infinitesimal and infinitely large numbers, as in the original Newton-Leibniz concept. These numbers, called hyperreals, can be used in the ordinary rules of calculus, much as Leibniz did.

Importance

Although some ideas of calculus had previously been developed in Egypt, Greece, China, India, Iraq, Persia, and Japan, the modern use of calculus began in Europe in the 17th century, when Isaac Newton and Gottfried Wilhelm Leibniz built on the work of earlier mathematicians to build on its basic principles. The development of calculus was based on earlier concepts of instantaneous motion and area under a curve.

Differential calculus is used in calculations related to speed and acceleration, curve slope, and optimization. Applications of integral calculus include calculations involving areas, volumes, arc lengths, centers of mass, work and pressure. More complex applications include calculations of power series and Fourier series.

Calculus [ ] is also used to gain a more accurate understanding of the nature of space, time and motion. For centuries, mathematicians and philosophers have wrestled with the paradoxes associated with dividing by zero or finding the sum of an infinite series of numbers. These questions arise when studying motion and calculating areas. The ancient Greek philosopher Zeno of Elea gave several famous examples of such paradoxes. Calculus provides tools for resolving these paradoxes, in particular limits and infinite series.

Limits and infinitesimals

Notes

Morris Kline, Mathematical thought from ancient to modern times, Vol. I
Archimedes, Method, in The Works of Archimedes ISBN 978-0-521-66160-7
Dun, Liu; Fan, Dainian; Cohen, Son Robertne. A comparison of Archimdes" and Liu Hui"s studies of circles (English): journal. - Springer, 1966. - Vol. 130. - P. 279. - ISBN 0-792-33463-9., Chapter, p. 279
Zill, Dennis G. Calculus: Early Transcendentals / Dennis G. Zill, Scott Wright, Warren S. Wright. - 3. - Jones & Bartlett Learning, 2009. - P. xxvii. - ISBN 0-763-75995-3.,Extract of page 27
Indian mathematics
von Neumann, J., "The Mathematician", in Heywood, R. B., ed., The Works of the Mind, University of Chicago Press, 1947, pp. 180-196. Reprinted in Bródy, F., Vámos, T., eds., The Neumann Compedium, World Scientific Publishing Co. Pte. Ltd., 1995, ISBN 9810222017, pp. 618-626.
André Weil: Number theory. An approach through history. From Hammurapi to Legendre. Birkhauser Boston, Inc., Boston, MA, 1984, ISBN 0-8176-4565-9, p. 28.
Leibniz, Gottfried Wilhelm. The Early Mathematical Manuscripts of Leibniz. Cosimo, Inc., 2008. Page 228. Copy
Unlu, Elif Maria Gaetana Agnesi (undefined) . Agnes Scott College (April 1995). Archived from the original on September 5, 2012.

Links

Ron Larson, Bruce H. Edwards (2010). "Calculus", 9th ed., Brooks Cole Cengage Learning. ISBN 978-0-547-16702-2
McQuarrie, Donald A. (2003). Mathematical Methods for Scientists and Engineers, University Science Books.