Antiquity

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 infinitesimal quantities 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. On initial stage During the development of calculus, the use of infinitesimal quantities was considered lax, and was subjected to severe criticism 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 basics 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 small numbers. large numbers, as it was 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 the 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

1. Morris Kline, Mathematical thought from ancient to modern times, Vol. I
2. Archimedes, Method, in The Works of Archimedes ISBN 978-0-521-66160-7
3. 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
4. Zill, Dennis G.; Wright, Scott; Wright, Warren S. Calculus: Early Transcendentals (undefined). - 3. - Jones & Bartlett Learning (English)Russian, 2009. - P. xxvii. - ISBN 0-763-75995-3.,Extract of page 27
5. Indian mathematics
6. 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.
7. 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.
8. Leibniz, Gottfried Wilhelm. The Early Mathematical Manuscripts of Leibniz. Cosimo, Inc., 2008. Page 228. Copy
9. 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. ISBN 978-1-891389-24-5
• James Stewart (2008). Calculus: Early Transcendentals, 6th ed., Brooks Cole Cengage Learning.

1. The period of creation of mathematics of variable quantities. Creation of analytical geometry, differential and integral calculus

In the 17th century A new period in the history of mathematics begins - the period of mathematics of variable quantities. Its emergence is associated primarily with the successes of astronomy and mechanics.

Kepler in 1609-1619 discovered and mathematically formulated the laws of planetary motion. By 1638, Galileo had created the mechanics of free movement of bodies, founded the theory of elasticity, and applied mathematical methods to study motion, to find patterns between the path of motion, its speed and acceleration. Newton formulated the law of universal gravitation by 1686.

The first decisive step in the creation of the mathematics of variable quantities was the appearance of Descartes’ book “Geometry”. Descartes' main services to mathematics are his introduction of variable quantities and the creation of analytical geometry. First of all, he was interested in the geometry of motion, and, applying algebraic methods to the study of objects, he became the creator of analytical geometry.

Analytical geometry began with the introduction of a coordinate system. In honor of the creator, a rectangular coordinate system consisting of two axes intersecting at right angles, measurement scales entered on them and a reference point - the point of intersection of these axes - is called a coordinate system on a plane. Together with the third axis, it is a rectangular Cartesian coordinate system in space.

By the 60s of the 17th century. Numerous methods have been developed to calculate the areas enclosed by various curved lines. Only one push was needed to create a single integral calculus from disparate methods.

Differential methods solved the main problem: knowing a curved line, find its tangents. Many practice problems led to the formulation of an inverse problem. In the process of solving the problem, it became clear that integration methods were applicable to it. Thus, a deep connection was established between differential and integral methods, which created the basis for a unified calculus. The earliest form of differential and integral calculus is the theory of fluxions, developed by Newton.

Mathematicians of the 18th century worked simultaneously in the fields of natural science and technology. Lagrange created the foundations of analytical mechanics. His work showed how many results can be obtained in mechanics thanks to powerful methods of mathematical analysis. Laplace's monumental work “Celestial Mechanics” summed up all previous work in this area.

XVIII century gave mathematics a powerful apparatus - the analysis of infinitesimals. During this period, Euler introduced the symbol f(x) for a function into mathematics and showed that functional dependence was the main object of study in mathematical analysis. Methods were developed for calculating partial derivatives, multiple and curvilinear integrals, and differentials of functions of many variables.

In the 18th century a number of important mathematical disciplines emerged from mathematical analysis: theory differential equations, calculus of variations. At this time, the development of probability theory began.

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Using derivative and integral to solve equations and inequalities

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Applying derivatives to problem solving

The 19th century is the beginning of a new, fourth period in the history of mathematics - the period of modern mathematics.

We already know that one of the main directions in the development of mathematics in the fourth period is the strengthening of the rigor of proofs in all mathematics, especially the restructuring of mathematical analysis on a logical basis. In the second half of the 18th century. numerous attempts were made to rebuild mathematical analysis: the introduction of the definition of a limit (D'Alembert et al.), the definition of the derivative as the limit of a ratio (Euler et al.), the results of Lagrange and Carnot, etc., but these works lacked a system, and sometimes they were unsuccessful. However, they prepared the ground on which perestroika in the 19th century. could be implemented. In the 19th century This direction of development of mathematical analysis became the leading one. It was taken up by O. Cauchy, B. Bolzano, K. Weierstrass and others.

1. Augustin Louis Cauchy (1789−1857) graduated from the Ecole Polytechnique and the Institute of Communications in Paris. Since 1816, member of the Paris Academy and professor at the Ecole Polytechnique. In 1830−1838 During the years of the republic, he was in exile because of his monarchist beliefs. Since 1848, Cauchy became a professor at the Sorbonne - University of Paris. He published more than 800 papers on mathematical analysis, differential equations, theory of functions of a complex variable, algebra, number theory, geometry, mechanics, optics, etc. The main areas of his scientific interests were mathematical analysis and theory of functions of a complex variable.

Cauchy published his lectures on analysis, given at the Ecole Polytechnique, in three works: “Course of Analysis” (1821), “Summary of Lectures on Infinitesimal Calculus” (1823), “Lecture on Applications of Analysis to Geometry”, 2 volumes (1826, 1828). In these books, for the first time, mathematical analysis is built on the basis of the theory of limits. they marked the beginning of a radical restructuring of mathematical analysis.

Cauchy gives the following definition of the limit of a variable: “If the values ​​successively assigned to the same variable approach a fixed value indefinitely, so that in the end they differ from it as little as possible, then the latter is called the limit of all others.” The essence of the matter is expressed well here, but the words “as little as desired” themselves need definition, and in addition, the definition of the limit of a variable, and not the limit of a function, is formulated here. Next, the author proves various properties of limits.

Then Cauchy gives the following definition of the continuity of a function: a function is called continuous (at a point) if an infinitesimal increment in the argument generates an infinitesimal increment in the function, i.e., in modern language

Then he has various properties of continuous functions.

The first book also examines the theory of series: it gives the definition of the sum of a number series as the limit of its partial sum, introduces a number of sufficient criteria for the convergence of number series, as well as power series and the region of their convergence - all this in both the real and complex domains.

He presents differential and integral calculus in his second book.

Cauchy defines the derivative of a function as the limit of the ratio of the increment of the function to the increment of the argument, when the increment of the argument tends to zero, and the differential as the limit of the ratio It follows from this that. The usual derivative formulas are discussed next; in this case, the author often uses Lagrange's mean value theorem.

In integral calculus, Cauchy first puts forward the definite integral as a basic concept. He also introduces it for the first time as the limit of integral sums. Here we prove an important theorem on the integrability of a continuous function. His indefinite integral is defined as a function of the argument that. In addition, expansions of functions in Taylor and Maclaurin series are considered here.

In the second half of the 19th century. a number of scientists: B. Riemann, G. Darboux and others found new conditions for the integrability of a function and even changed the very definition of a definite integral so that it could be applied to the integration of some discontinuous functions.

In the theory of differential equations, Cauchy was mainly concerned with proofs of fundamentally important existence theorems: the existence of a solution to an ordinary differential equation, first of the first and then of the th order; existence of a solution for a linear system of partial differential equations.

In the theory of functions of a complex variable, Cauchy is the founder; Many of his articles are devoted to it. In the 18th century Euler and d'Alembert laid only the beginning of this theory. In the university course on the theory of functions of a complex variable, we constantly come across the name of Cauchy: the Cauchy - Riemann conditions for the existence of a derivative, the Cauchy integral, the Cauchy integral formula, etc.; many theorems on residues of a function are also due to Cauchy. B. Riemann, K. Weierstrass, P. Laurent and others also obtained very important results in this area.

Let's return to the basic concepts of mathematical analysis. In the second half of the century, it became clear that the Czech scientist Bernard Bolzano (1781 - 1848) had done a lot in the field of substantiating analysis before Cauchy and Weierschtrass. Before Cauchy, he gave definitions of the limit, continuity of a function and the convergence of a number series, proved a criterion for the convergence of a number sequence, and also, long before it appeared in Weierstrass, the theorem: if a number set is bounded above (below), then it has an exact upper ( exact bottom edge. He considered a number of properties of continuous functions; Let us remember that in the university course of mathematical analysis there are the Bolzano–Cauchy and Bolzano–Weierstrass theorems on functions continuous on an interval. Bolzano also investigated some issues of mathematical analysis, for example, he constructed the first example of a function that is continuous on a segment, but does not have a derivative at any point on the segment. During his lifetime, Bolzano was able to publish only five small works, so his results became known too late.

2. In mathematical analysis, the lack of a clear definition of a function was felt more and more clearly. A significant contribution to resolving the dispute about what is meant by function was made by the French scientist Jean Fourier. He studied the mathematical theory of thermal conductivity in solids and, in connection with this, used trigonometric series (Fourier series)

these series later became widely used in mathematical physics, a science that deals with mathematical methods for studying partial differential equations encountered in physics. Fourier proved that any continuous curve, regardless of what dissimilar parts it is composed of, can be defined by a single analytical expression - a trigonometric series, and that this can also be done for some curves with discontinuities. Fourier's study of such series once again raised the question of what is meant by a function. Can such a curve be considered to define a function? (This is a renewal of the old 18th century debate about the relationship between function and formula at a new level.)

In 1837, the German mathematician P. Direchle first gave a modern definition of a function: “is a function of a variable (on an interval if each value (on this interval) corresponds to a completely specific value, and it does not matter how this correspondence is established - by an analytical formula, a graph, a table, or even just words." Noteworthy is the addition: "it does not matter how this correspondence is established." Direchle's definition received general recognition quite quickly. However, it is now customary to call the correspondence itself a function.

3. The modern standard of rigor in mathematical analysis first appeared in the works of Weierstrass (1815−1897). He worked for a long time as a mathematics teacher in gymnasiums, and in 1856 became a professor at the University of Berlin. The listeners of his lectures gradually published them in the form of separate books, thanks to which the content of Weierstrass's lectures became well known in Europe. It was Weierstrass who began to systematically use language in mathematical analysis. He gave a definition of the limit of a sequence, a definition of the limit of a function in language (which is often incorrectly called the Cauchy definition), rigorously proved theorems on limits and the so-called Weierstrass theorem on the limit of a monotone sequence: an increasing (decreasing) sequence, bounded from above (from below), has a finite limit. He began to use the concepts of the exact upper and exact lower bounds of a numerical set, the concept of a limit point of a set, proved the theorem (which has another author - Bolzano): a bounded numerical set has a limit point, and examined some properties of continuous functions. Weierstrass devoted many works to the theory of functions of a complex variable, substantiating it with the help of power series. He also studied the calculus of variations, differential geometry and linear algebra.

4. Let us dwell on the theory of infinite sets. Its creator was the German mathematician Cantor. Georg Kantor (1845-1918) worked for many years as a professor at the University of Halle. He published works on set theory starting in 1870. He proved the uncountability of the set of real numbers, thus establishing the existence of nonequivalent infinite sets, introduced general concept powers of a set, found out the principles of comparing powers. Cantor built a theory of transfinite, “improper” numbers, attributing the lowest, smallest transfinite number to the power of a countable set (in particular, the set of natural numbers), to the power of the set of real numbers - a higher, larger transfinite number, etc.; this gave him the opportunity to construct an arithmetic of transfinite numbers, similar to the ordinary arithmetic of natural numbers. Cantor systematically applied actual infinity, for example, the possibility of completely “exhausting” the natural series of numbers, while before him in mathematics of the 19th century. only potential infinity was used.

Cantor's set theory aroused objections from many mathematicians when it appeared, but recognition gradually came when its enormous importance for the justification of topology and the theory of functions of a real variable became clear. But logical gaps remained in the theory itself; in particular, paradoxes of set theory were discovered. Here is one of the most famous paradoxes. Let us denote by the set all such sets that are not elements of themselves. Does the inclusion also hold and is not an element since, by condition, only such sets are included as elements that are not elements of themselves; if the condition holds, inclusion is a contradiction in both cases.

These paradoxes were associated with the internal inconsistency of some sets. It became clear that not just any sets can be used in mathematics. The existence of paradoxes was overcome by the creation already at the beginning of the 20th century. axiomatic set theory (E. Zermelo, A. Frenkel, D. Neumann, etc.), which, in particular, answered the question: what sets can be used in mathematics? It turns out that you can use the empty set, the union of given sets, the set of all subsets of a given set, etc.

History of mathematical analysis

The 18th century is often called the century of the scientific revolution, which determined the development of society up to the present day. This revolution was based on the remarkable mathematical discoveries made in the 17th century and built on in the following century. “There is not a single object in the material world and not a single thought in the realm of the spirit that would not be affected by the influence of the scientific revolution of the 18th century. Not a single element of modern civilization could exist without the principles of mechanics, without analytical geometry and differential calculus. There is not a single branch of human activity that has not been strongly influenced by the genius of Galileo, Descartes, Newton and Leibniz.” These words of the French mathematician E. Borel (1871 - 1956), spoken by him in 1914, remain relevant in our time. Many great scientists contributed to the development of mathematical analysis: I. Kepler (1571 -1630), R. Descartes (1596 -1650), P. Fermat (1601 -1665), B. Pascal (1623 -1662), H. Huygens (1629 -1695), I. Barrow (1630 -1677), brothers J. Bernoulli (1654 -1705) and I. Bernoulli (1667 -1748) and others.

The innovation of these celebrities in understanding and describing the world around us:

movement, change and variability (life has entered with its dynamics and development);

statistical casts and one-time photographs of her conditions.

The mathematical discoveries of the 17th and 17th centuries were defined using concepts such as variable and function, coordinates, graph, vector, derivative, integral, series and differential equation.

Pascal, Descartes and Leibniz were not so much mathematicians as philosophers. It is the universal human and philosophical meaning of their mathematical discoveries that now constitutes the main value and is a necessary element of general culture.

Both serious philosophy and serious mathematics cannot be understood without mastering the corresponding language. Newton, in a letter to Leibniz about solving differential equations, sets out his method as follows: 5accdae10effh 12i…rrrssssttuu.

Founders 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 to the definition of speed in this moment time and 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.