Brief history of the development of mathematical analysis. Mathematical analysis

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MATHEMATICS HISTORY. The most ancient mathematical activities there was a bill. An account was necessary to keep track of livestock and conduct trade. Some primitive tribes counted the number of objects, correlating them with various parts body, mainly the fingers and toes. A rock painting that has survived to this day from the Stone Age depicts the number 35 as a series of 35 finger sticks lined up in a row. The first significant advances in arithmetic were the conceptualization of number and the invention of the four basic operations: addition, subtraction, multiplication and division. The first achievements of geometry are associated with such simple concepts as straight lines and circles. Further development mathematics began around 3000 BC. thanks to the Babylonians and Egyptians.

BABYLONIA AND EGYPT

Babylonia.

The source of our knowledge about the Babylonian civilization are well-preserved clay tablets covered with the so-called. cuneiform texts that date from 2000 BC. and up to 300 AD The mathematics on the cuneiform tablets was mainly related to farming. Arithmetic and simple algebra were used in exchanging money and paying for goods, calculating simple and compound interest, taxes and the share of the harvest handed over to the state, temple or landowner. Numerous arithmetic and geometric problems arose in connection with the construction of canals, granaries and other public works. A very important task of mathematics was the calculation of the calendar, since the calendar was used to determine the dates of agricultural work and religious holidays. The division of a circle into 360, and degrees and minutes into 60 parts, originates in Babylonian astronomy.

The Babylonians also created a number system that used base 10 for numbers from 1 to 59. The symbol for one was repeated the required number of times for numbers from 1 to 9. To represent numbers from 11 to 59, the Babylonians used a combination of the symbol for the number 10 and the symbol for one. To denote numbers starting from 60 and above, the Babylonians introduced a positional number system with a base of 60. A significant advance was the positional principle, according to which the same number sign (symbol) has different meanings depending on the place where it is located. An example is the meaning of six in the (modern) notation of the number 606. However, there was no zero in the ancient Babylonian number system, which is why the same set of symbols could mean both the number 65 (60 + 5) and the number 3605 (60 2 + 0 + 5). Ambiguities also arose in the interpretation of fractions. For example, the same symbols could mean the number 21, the fraction 21/60 and (20/60 + 1/60 2). Ambiguities were resolved depending on the specific context.

The Babylonians compiled tables of reciprocal numbers (which were used in division), tables of squares and square roots, as well as tables of cubes and cube roots. They knew a good approximation of the number . Cuneiform texts dealing with solving algebraic and geometric problems indicate that they used the quadratic formula to solve quadratic equations and could solve some special types of problems involving up to ten equations in ten unknowns, as well as certain varieties of cubic and quartic equations. Only the tasks and the main steps of the procedures for solving them are depicted on clay tablets. Since geometric terminology was used to designate unknown quantities, the solution methods mainly consisted of geometric operations with lines and areas. As for algebraic problems, they were formulated and solved in verbal notation.

Around 700 BC The Babylonians began to use mathematics to study the movements of the Moon and planets. This allowed them to predict the positions of the planets, which was important for both astrology and astronomy.

In geometry, the Babylonians knew about such relationships, for example, as the proportionality of the corresponding sides of similar triangles. They knew the Pythagorean theorem and the fact that an angle inscribed in a semicircle is a right angle. They also had rules for calculating the areas of simple plane figures, including regular polygons, and the volumes of simple bodies. Number p The Babylonians considered it equal to 3.

Egypt.

Our knowledge of ancient Egyptian mathematics is based mainly on two papyri dating from about 1700 BC. The mathematical information presented in these papyri dates back to an even earlier period - c. 3500 BC The Egyptians used mathematics to calculate the weight of bodies, the area of crops and the volume of granaries, the size of taxes and the number of stones required for the construction of certain structures. In the papyri one can also find problems related to determining the amount of grain needed to prepare a given number of glasses of beer, as well as more complex problems related to differences in types of grain; For these cases, conversion factors were calculated.

But the main area of application of mathematics was astronomy, or rather calculations related to the calendar. The calendar was used to determine the dates of religious holidays and to predict the annual flooding of the Nile. However, the level of development of astronomy in Ancient Egypt was much lower than the level of its development in Babylon.

Ancient Egyptian writing was based on hieroglyphs. The number system of that period was also inferior to the Babylonian one. The Egyptians used a non-positional decimal system, in which the numbers from 1 to 9 were indicated by the corresponding number of vertical bars, and individual symbols were introduced for successive powers of the number 10. By sequentially combining these symbols, any number could be written. With the advent of papyrus, the so-called hieratic cursive writing arose, which, in turn, contributed to the emergence of a new numerical system. For each of the numbers 1 through 9 and for each of the first nine multiples of 10, 100, etc. a special one was used identification symbol. Fractions were written as a sum of fractions with a numerator equal to one. With such fractions, the Egyptians performed all four arithmetic operations, but the procedure for such calculations remained very cumbersome.

Geometry among the Egyptians came down to calculating the areas of rectangles, triangles, trapezoids, circles, as well as formulas for calculating the volumes of certain bodies. It must be said that the mathematics that the Egyptians used to build the pyramids was simple and primitive.

The tasks and solutions given in the papyri are formulated purely by prescription, without any explanation. The Egyptians dealt only with the simplest types of quadratic equations and arithmetic and geometric progression, and therefore those general rules, which they were able to deduce were also of the simplest type. Neither Babylonian nor Egyptian mathematicians had general methods; the entire body of mathematical knowledge was a collection of empirical formulas and rules.

Although the Mayans of Central America did not influence the development of mathematics, their achievements dating back to around the 4th century are noteworthy. The Mayans were apparently the first to use a special symbol to represent zero in their 20-digit system. They had two number systems: one used hieroglyphs, and the other, more common, used a dot for one, a horizontal line for the number 5, and a symbol for zero. Positional designations began with the number 20, and numbers were written vertically from top to bottom.

GREEK MATHEMATICS

Classical Greece.

From a 20th century point of view. The founders of mathematics were the Greeks of the classical period (6th–4th centuries BC). Mathematics, as it existed in the earlier period, was a set of empirical conclusions. On the contrary, in deductive reasoning a new statement is derived from accepted premises in a way that excludes the possibility of its rejection.

The Greeks' insistence on deductive proof was an extraordinary step. No other civilization has reached the idea of arriving at conclusions solely on the basis of deductive reasoning, starting from explicitly stated axioms. We find one explanation for the Greeks' adherence to deductive methods in the structure of Greek society of the classical period. Mathematicians and philosophers (often these were the same people) belonged to the highest strata of society, where any Practical activities was considered an unworthy occupation. Mathematicians preferred abstract reasoning about numbers and spatial relationships to solving practical problems. Mathematics was divided into arithmetic - the theoretical aspect and logistics - the computational aspect. Logistics was left to the freeborn of the lower classes and slaves.

The deductive character of Greek mathematics was fully formed by the time of Plato and Aristotle. The invention of deductive mathematics is usually attributed to Thales of Miletus (c. 640–546 BC), who, like many ancient Greek mathematicians of the classical period, was also a philosopher. It has been suggested that Thales used deduction to prove some results in geometry, although this is doubtful.

Another great Greek whose name is associated with the development of mathematics was Pythagoras (c. 585–500 BC). It is believed that he could have become acquainted with Babylonian and Egyptian mathematics during his long wanderings. Pythagoras founded a movement that flourished in ca. 550–300 BC The Pythagoreans created pure mathematics in the form of number theory and geometry. They represented whole numbers in the form of configurations of dots or pebbles, classifying these numbers in accordance with the shape of the resulting figures (“curly numbers”). The word "calculation" (calculation, calculation) originates from the Greek word meaning "pebble". Numbers 3, 6, 10, etc. The Pythagoreans called it triangular, since the corresponding number of pebbles can be arranged in the form of a triangle, the numbers 4, 9, 16, etc. – square, since the corresponding number of pebbles can be arranged in the form of a square, etc.

From simple geometric configurations some properties of integers arose. For example, the Pythagoreans discovered that the sum of two consecutive triangular numbers is always equal to some square number. They discovered that if (in modern notation) n 2 is a square number, then n 2 + 2n +1 = (n+ 1) 2 . A number equal to the sum of all its own divisors, except this number itself, was called perfect by the Pythagoreans. Examples of perfect numbers are integers such as 6, 28 and 496. The Pythagoreans called two numbers friendly if each number is equal to the sum of the divisors of the other; for example, 220 and 284 are friendly numbers (and here the number itself is excluded from its own divisors).

For the Pythagoreans, any number represented something more than a quantitative value. For example, the number 2, according to their view, meant difference and was therefore identified with opinion. Four represented justice, as it was the first number equal to the product of two equal factors.

The Pythagoreans also discovered that the sum of certain pairs of square numbers is again a square number. For example, the sum of 9 and 16 is 25, and the sum of 25 and 144 is 169. Triples of numbers such as 3, 4 and 5 or 5, 12 and 13 are called Pythagorean numbers. They have a geometric interpretation if two numbers from three are equated to the lengths of the legs right triangle, then the third number will be equal to the length of its hypotenuse. This interpretation apparently led the Pythagoreans to realize a more general fact, now known as the Pythagorean theorem, according to which in any right triangle the square of the length of the hypotenuse equal to the sum squares of leg lengths.

Considering a right triangle with unit legs, the Pythagoreans discovered that the length of its hypotenuse was equal to , and this plunged them into confusion, for they tried in vain to represent a number as a ratio of two integers, which was extremely important for their philosophy. The Pythagoreans called quantities that cannot be represented as ratios of integers incommensurable; the modern term is “irrational numbers”. Around 300 BC Euclid proved that number is incommensurable. The Pythagoreans dealt with irrational numbers, representing all quantities in geometric images. If 1 is considered to be the length of some segments, then the difference between rational and irrational numbers is smoothed out. The product of numbers is the area of a rectangle with sides of length and. Even today we sometimes talk about the number 25 as the square of 5, and the number 27 as the cube of 3.

The ancient Greeks solved equations with unknowns using geometric constructions. Special constructions were developed to perform addition, subtraction, multiplication and division of segments, extracting square roots from the lengths of segments; now this method is called geometric algebra.

Bringing tasks to geometric view had a number of important consequences. In particular, numbers began to be considered separately from geometry, since it was possible to work with incommensurable relations only using geometric methods. Geometry became the basis of almost all rigorous mathematics at least until 1600. And even in the 18th century, when algebra and mathematical analysis were already sufficiently developed, rigorous mathematics was interpreted as geometry, and the word “geometer” was equivalent to the word “mathematician.”

It is to the Pythagoreans that we owe much of the mathematics that was then systematically presented and proven in Beginnings Euclid. There is reason to believe that it was they who discovered what is now known as theorems about triangles, parallel lines, polygons, circles, spheres and regular polyhedra.

One of the most prominent Pythagoreans was Plato (c. 427–347 BC). Plato was convinced that the physical world can only be understood through mathematics. It is believed that he is credited with inventing the analytical method of proof. (The analytical method begins with a statement that needs to be proven, and then consequences are successively deduced from it until some known fact; the proof is obtained using the reverse procedure.) It is generally accepted that the followers of Plato invented a method of proof called “proof by contradiction.” Aristotle, a student of Plato, occupies a prominent place in the history of mathematics. Aristotle laid the foundations of the science of logic and expressed a number of ideas regarding definitions, axioms, infinity and the possibility of geometric constructions.

The greatest of the Greek mathematicians of the classical period, second only to Archimedes in the importance of his results, was Eudoxus (c. 408–355 BC). It was he who introduced the concept of magnitude for such objects as line segments and angles. Having the concept of magnitude, Eudoxus logically and strictly substantiated the Pythagorean method of dealing with irrational numbers.

The work of Eudoxus made it possible to establish the deductive structure of mathematics on the basis of explicitly formulated axioms. He also took the first step in the creation of mathematical analysis, since it was he who invented the method of calculating areas and volumes, called the “exhaustion method.” This method consists of constructing inscribed and described flat figures or spatial bodies that fill (“exhaust”) the area or volume of the figure or body that is the subject of research. Eudoxus also owns the first astronomical theory that explains the observed movement of the planets. The theory proposed by Eudoxus was purely mathematical; it showed how combinations of rotating spheres with different radii and axes of rotation could explain the seemingly irregular movements of the Sun, Moon and planets.

Around 300 BC the results of many Greek mathematicians were combined into a single whole by Euclid, who wrote a mathematical masterpiece Beginnings. From a few shrewdly selected axioms, Euclid derived about 500 theorems, covering all the most important results of the classical period. Euclid began his work by defining such terms as straight line, angle and circle. He then stated ten self-evident truths, such as “the whole is greater than any of the parts.” And from these ten axioms, Euclid was able to derive all the theorems. Text for mathematicians Began Euclid served as a model of rigor for a long time, until in the 19th century. it was not found to have serious deficiencies, such as the unconscious use of assumptions that were not explicitly stated.

Apollonius (c. 262–200 BC) lived during the Alexandrian period, but his main work is in the spirit of the classical tradition. His proposed analysis of conic sections - circle, ellipse, parabola and hyperbola - was the culmination of the development of Greek geometry. Apollonius also became the founder of quantitative mathematical astronomy.

Alexandrian period.

During this period, which began around 300 BC, the nature of Greek mathematics changed. Alexandrian mathematics arose from the fusion of classical Greek mathematics with the mathematics of Babylonia and Egypt. In general, mathematicians of the Alexandrian period were more inclined to solve purely technical problems than to philosophy. The great Alexandrian mathematicians - Eratosthenes, Archimedes, Hipparchus, Ptolemy, Diophantus and Pappus - demonstrated the power of the Greek genius in theoretical abstraction, but were equally willing to apply their talent to solution practical problems and purely quantitative problems.

Eratosthenes (c. 275–194 BC) found a simple method for accurately calculating the circumference of the Earth, and he also created a calendar in which every fourth year has one more day than the others. The astronomer Aristarchus (c. 310–230 BC) wrote an essay About the sizes and distances of the Sun and Moon, which contained one of the first attempts to determine these sizes and distances; Aristarchus' work was geometric in nature.

The greatest mathematician of antiquity was Archimedes (c. 287–212 BC). He is the author of the formulations of many theorems about the areas and volumes of complex figures and bodies, which he quite strictly proved by the method of exhaustion. Archimedes always sought to obtain exact solutions and found upper and lower bounds for ir rational numbers. For example, working with the regular 96-gon, he flawlessly proved that the exact value of the number p is between 3 1/7 and 3 10/71. Archimedes also proved several theorems that contained new results in geometric algebra. He was responsible for the formulation of the problem of dissecting a ball by a plane so that the volumes of the segments are in a given ratio to each other. Archimedes solved this problem by finding the intersection of a parabola and an equilateral hyperbola.

Archimedes was the greatest mathematical physicist of antiquity. He used geometric considerations to prove theorems of mechanics. His essay About floating bodies laid the foundations of hydrostatics. According to legend, Archimedes discovered the law that bears his name, according to which a body immersed in water is subject to a buoyant force equal to the weight of the liquid displaced by it. While bathing, while in the bathroom, and unable to cope with the joy of discovery that gripped him, he ran out naked into the street shouting: “Eureka!” (“Opened!”)

In the time of Archimedes, they were no longer limited to geometric constructions that could only be done with a compass and a ruler. Archimedes used a spiral in his constructions, and Diocles (late 2nd century BC) solved the problem of doubling a cube using a curve he introduced, called the cissoid.

During the Alexandrian period, arithmetic and algebra were treated independently of geometry. The Greeks of the classical period had a logically substantiated theory of integers, but the Alexandrian Greeks, having adopted Babylonian and Egyptian arithmetic and algebra, largely lost their already developed ideas about mathematical rigor. Lived between 100 BC and 100 AD Heron of Alexandria transformed much of the geometric algebra of the Greeks into frankly lax computational procedures. However, in proving new theorems of Euclidean geometry, he was still guided by the standards of logical rigor of the classical period.

The first fairly voluminous book in which arithmetic was presented independently of geometry was Introduction to Arithmetic Nicomacheus (c. 100 AD). In the history of arithmetic, its role is comparable to that of Began Euclid in the history of geometry. For more than 1,000 years, it served as the standard textbook for its clear, concise, and comprehensive presentation of the teachings of whole numbers (prime, composite, coprime, and proportions). Repeating many Pythagorean statements, Introduction At the same time, Nicomachus moved on, since Nicomachus saw more general relations, although he cited them without proof.

A significant milestone in the algebra of the Alexandrian Greeks was the work of Diophantus (c. 250). One of his main achievements is associated with the introduction of symbolism into algebra. In his works, Diophantus did not propose general methods; he dealt with specific positive rational numbers, and not with their letter designations. He laid the foundations of the so-called. Diophantine analysis – study of uncertain equations.

The highest achievement of Alexandrian mathematicians was the creation of quantitative astronomy. We owe the invention of trigonometry to Hipparchus (c. 161–126 BC). His method was based on a theorem stating that in similar triangles the ratio of the lengths of any two sides of one of them is equal to the ratio of the lengths of two corresponding sides of the other. In particular, the ratio of the length of the leg lying opposite the acute angle A in a right triangle, to the length of the hypotenuse must be the same for all right triangles having the same sharp corner A. This ratio is known as the sine of the angle A. The ratios of the lengths of the other sides of a right triangle are called cosine and tangent of the angle A. Hipparchus invented a method for calculating such ratios and compiled their tables. With these tables and easily measurable distances on the surface of the Earth, he was able to calculate the length of its great circle and the distance to the Moon. According to his calculations, the radius of the Moon was one third of the Earth's radius; According to modern data, the ratio of the radii of the Moon and the Earth is 27/1000. Hipparchus determined the length of the solar year with an error of only 6 1/2 minutes; It is believed that it was he who introduced latitude and longitude.

Greek trigonometry and its applications to astronomy reached its peak in Almagest Egyptian Claudius Ptolemy (died 168 AD). IN Almagest the theory of motion was presented celestial bodies, which dominated until the 16th century, when it was replaced by the Copernican theory. Ptolemy sought to build the simplest mathematical model, realizing that his theory was just a convenient mathematical description of astronomical phenomena consistent with observations. Copernicus's theory prevailed precisely because it was simpler as a model.

Decline of Greece.

After the conquest of Egypt by the Romans in 31 BC. the great Greek Alexandrian civilization fell into decay. Cicero proudly argued that, unlike the Greeks, the Romans were not dreamers, and therefore applied their mathematical knowledge in practice, deriving real benefit from it. However, the contribution of the Romans to the development of mathematics itself was insignificant. The Roman number system was based on cumbersome notations for numbers. Its main feature was the additive principle. Even the subtractive principle, for example writing the number 9 as IX, came into widespread use only after the invention of typesetting in the 15th century. Roman number notation was used in some European schools until about 1600, and in accounting a century later.

INDIA AND ARAB

The successors of the Greeks in the history of mathematics were the Indians. Indian mathematicians did not engage in proofs, but they introduced original concepts and a number of effective methods. It was they who first introduced zero both as a cardinal number and as a symbol of the absence of units in the corresponding digit. Mahavira (850 AD) established rules for operations with zero, believing, however, that dividing a number by zero leaves the number unchanged. The correct answer for the case of dividing a number by zero was given by Bhaskara (b. 1114), and he also owned the rules for operating with irrational numbers. The Indians introduced the concept of negative numbers (to represent debts). We find their earliest use in Brahmagupta (c. 630). Aryabhata (p. 476) went further than Diophantus in the use of continued fractions in solving indefinite equations.

Our modern number system, based on the positional principle of writing numbers and zero as a cardinal number and the use of empty place notation, is called Indo-Arabic. On the wall of a temple built in India ca. 250 BC, several figures were discovered that resemble our modern figures in their outlines.

Around 800 Indian mathematics reached Baghdad. The term "algebra" comes from the beginning of the book's title Al-jabr wa-l-muqabala (Replenishment and opposition), written in 830 by the astronomer and mathematician al-Khwarizmi. In his essay he paid tribute to the merits of Indian mathematics. Al-Khwarizmi's algebra was based on the works of Brahmagupta, but Babylonian and Greek influences are clearly discernible. Another prominent Arab mathematician, Ibn al-Haytham (c. 965–1039), developed a method for obtaining algebraic solutions to quadratic and cubic equations. Arab mathematicians, including Omar Khayyam, were able to solve some cubic equations using geometric methods using conic sections. Arab astronomers introduced the concept of tangent and cotangent into trigonometry. Nasireddin Tusi (1201–1274) in Treatise on the Complete Quadrangle systematically outlined plane and spherical geometry and was the first to consider trigonometry separately from astronomy.

Yet the most important contribution of the Arabs to mathematics was their translations and commentaries on the great works of the Greeks. Europe became acquainted with these works after the Arab conquest North Africa and Spain, and later the works of the Greeks were translated into Latin.

MIDDLE AGES AND RENAISSANCE

Medieval Europe.

Roman civilization did not leave a noticeable mark on mathematics because it was too concerned with solving practical problems. The civilization that developed in early Middle Ages Europe (c. 400–1100) was not productive for exactly the opposite reason: intellectual life focused almost exclusively on theology and the afterlife. The level of mathematical knowledge did not rise above arithmetic and simple sections from Began Euclid. Astrology was considered the most important branch of mathematics in the Middle Ages; astrologers were called mathematicians. And since medical practice was based primarily on astrological indications or contraindications, doctors had no choice but to become mathematicians.

Around 1100, Western European mathematics began an almost three-century period of mastering the heritage preserved by the Arabs and Byzantine Greeks Ancient world and East. Since the Arabs owned almost all the works of the ancient Greeks, Europe received an extensive mathematical literature. The translation of these works into Latin contributed to the rise of mathematical research. All the great scientists of the time admitted that they drew inspiration from the works of the Greeks.

The first European mathematician worth mentioning was Leonardo of Pisa (Fibonacci). In his essay Book of abacus(1202) he introduced the Europeans to Indo-Arabic numerals and methods of calculation, as well as Arabic algebra. Over the next few centuries, mathematical activity in Europe waned. The body of mathematical knowledge of the era, compiled by Luca Pacioli in 1494, did not contain any algebraic innovations that Leonardo did not have.

Revival.

Among the best geometers of the Renaissance were artists who developed the idea of perspective, which required a geometry with converging parallel lines. The artist Leon Battista Alberti (1404–1472) introduced the concepts of projection and section. Straight rays of light from the observer's eye to various points in the depicted scene form a projection; the section is obtained by passing the plane through the projection. In order for the painted picture to look realistic, it had to be such a cross-section. The concepts of projection and section gave rise to purely mathematical questions. For example, what common geometric properties do the section and the original scene have, and what are the properties of two different sections of the same projection formed by two different planes intersecting the projection at different angles? From such questions projective geometry arose. Its founder, J. Desargues (1593–1662), using evidence based on projection and section, unified the approach to various types conic sections, which the great Greek geometer Apollonius considered separately.

THE BEGINNING OF MODERN MATHEMATICS

Advance of the 16th century. in Western Europe was marked important achievements in algebra and arithmetic. Were put into circulation decimals and rules for arithmetic operations with them. A real triumph was the invention of logarithms in 1614 by J. Napier. By the end of the 17th century. the understanding of logarithms as exponents with any positive number other than one as the base has finally emerged. From the beginning of the 16th century. Irrational numbers began to be used more widely. B. Pascal (1623–1662) and I. Barrow (1630–1677), I. Newton’s teacher at Cambridge University, argued that a number such as , can only be interpreted as a geometric quantity. However, in those same years, R. Descartes (1596–1650) and J. Wallis (1616–1703) believed that irrational numbers are acceptable on their own, without reference to geometry. In the 16th century Controversy continued over the legality of introducing negative numbers. Complex numbers that arose when solving quadratic equations, such as those called “imaginary” by Descartes, were considered even less acceptable. These numbers were under suspicion even in the 18th century, although L. Euler (1707–1783) used them with success. Complex numbers were finally recognized only at the beginning of the 19th century, when mathematicians became familiar with their geometric representation.

Advances in algebra.

In the 16th century Italian mathematicians N. Tartaglia (1499–1577), S. Dal Ferro (1465–1526), L. Ferrari (1522–1565) and D. Cardano (1501–1576) found general solutions to equations of the third and fourth degrees. To make algebraic reasoning and notation more precise, many symbols were introduced, including +, –, ґ, =, > and<.>b 2 – 4 ac] quadratic equation, namely, that the equation ax 2 + bx + c= 0 has equal real, different real, or complex conjugate roots, depending on whether the discriminant b 2 – 4ac equal to zero, greater than or less than zero. In 1799, K. Friedrich Gauss (1777–1855) proved the so-called. fundamental theorem of algebra: every polynomial n-th degree has exactly n roots

The main task of algebra is to find a general solution algebraic equations- continued to occupy mathematicians at the beginning of the 19th century. When talking about the general solution of a second degree equation ax 2 + bx + c= 0, mean that each of its two roots can be expressed using a finite number of addition, subtraction, multiplication, division and rooting operations performed on the coefficients a, b And With. The young Norwegian mathematician N. Abel (1802–1829) proved that it is impossible to obtain common decision equations of degree above 4 using a finite number of algebraic operations. However, there are many equations of a special form of degree higher than 4 that admit such a solution. On the eve of his death in a duel, the young French mathematician E. Galois (1811–1832) gave a decisive answer to the question of which equations are solvable in radicals, i.e. the roots of which equations can be expressed through their coefficients using a finite number of algebraic operations. Galois theory used substitutions or permutations of roots and introduced the concept of a group, which has found wide application in many areas of mathematics.

Analytic geometry.

Analytical, or coordinate, geometry was created independently by P. Fermat (1601–1665) and R. Descartes in order to expand the capabilities of Euclidean geometry in construction problems. However, Fermat considered his work only as a reformulation of the work of Apollonius. The real discovery - the realization of the full power of algebraic methods - belongs to Descartes. Euclidean geometric algebra for each construction required the invention of its own original method and could not offer quantitative information, necessary for science. Descartes solved this problem: he formulated geometric problems algebraically, solved the algebraic equation, and only then constructed the desired solution - a segment that had the appropriate length. Analytical geometry itself arose when Descartes began to consider indeterminate construction problems whose solutions were not one, but many possible lengths.

Analytic geometry uses algebraic equations to represent and study curves and surfaces. Descartes considered an acceptable curve that could be written using a single algebraic equation with respect to X And at. This approach was an important step forward, because it not only included such curves as conchoid and cissoid among the acceptable ones, but also significantly expanded the range of curves. As a result, in the 17th–18th centuries. many new important curves, such as the cycloid and catenary, entered scientific use.

Apparently, the first mathematician who used equations to prove the properties of conic sections was J. Wallis. By 1865 he had obtained algebraically all the results presented in Book V Began Euclid.

Analytical geometry completely reversed the roles of geometry and algebra. As the great French mathematician Lagrange noted, “As long as algebra and geometry went their separate ways, their progress was slow and their applications limited. But when these sciences united their efforts, they borrowed new vital forces from each other and since then have moved quickly towards perfection.” see also ALGEBRAIC GEOMETRY; GEOMETRY ; GEOMETRY REVIEW.

Mathematical analysis.

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 speed at an instant of time by dividing the path by 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 long years was interrupted with damage to 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.

MODERN MATHEMATICS

The creation of differential and integral calculus marked the beginning of “higher mathematics.” The methods of mathematical analysis, in contrast to the concept of limit that underlies it, seemed clear and understandable. For many years mathematicians, including Newton and Leibniz, tried in vain to give a precise definition of the concept of limit. And yet, despite numerous doubts about the validity of mathematical analysis, it found increasingly widespread use. Differential and integral calculus became the cornerstones of mathematical analysis, which eventually included such subjects as theory differential equations, ordinary and partial derivatives, infinite series, calculus of variations, differential geometry and much more. A strict definition of the limit was obtained only in the 19th century.

Non-Euclidean geometry.

By 1800, mathematics rested on two pillars - the number system and Euclidean geometry. Since many properties of the number system were proven geometrically, Euclidean geometry was the most reliable part of the edifice of mathematics. However, the axiom of parallels contained a statement about straight lines extending to infinity, which could not be confirmed by experience. Even Euclid's own version of this axiom does not at all state that some lines will not intersect. It rather formulates a condition under which they intersect at some end point. For centuries, mathematicians have tried to find a suitable replacement for the parallel axiom. But in each option there was certainly some gap. The honor of creating non-Euclidean geometry fell to N.I. Lobachevsky (1792–1856) and J. Bolyai (1802–1860), each of whom independently published his own original presentation of non-Euclidean geometry. In their geometries through this point it was possible to draw an infinite number of parallel lines. In the geometry of B. Riemann (1826–1866), no parallel can be drawn through a point outside a straight line.

Nobody seriously thought about physical applications of non-Euclidean geometry. Creation by A. Einstein (1879–1955) general theory relativity awakened in 1915 scientific world to an awareness of the reality of non-Euclidean geometry.

Mathematical rigor.

Until about 1870, mathematicians believed that they were acting as the ancient Greeks had designed, applying deductive reasoning to mathematical axioms, thereby providing their conclusions with a reliability no less than that possessed by the axioms. Non-Euclidean geometry and quaternions (an algebra that does not obey the commutative property) forced mathematicians to realize that what they took to be abstract and logically consistent statements were in fact based on an empirical and pragmatic basis.

The creation of non-Euclidean geometry was also accompanied by the awareness of the existence of logical gaps in Euclidean geometry. One of the disadvantages of Euclidean Began was the use of assumptions that were not explicitly stated. Apparently, Euclid did not question the properties that his geometric figures, but these properties were not included in his axioms. In addition, when proving the similarity of two triangles, Euclid used the superposition of one triangle on another, implicitly assuming that the properties of the figures do not change when moving. But besides such logical gaps, in Beginnings There was also some erroneous evidence.

The creation of new algebras, which began with quaternions, gave rise to similar doubts regarding the logical validity of arithmetic and the algebra of the ordinary number system. All numbers previously known to mathematicians had the property of commutativity, i.e. ab = ba. Quaternions, which revolutionized traditional ideas about numbers, were discovered in 1843 by W. Hamilton (1805–1865). They turned out to be useful for solving a number of physical and geometric problems, although the commutativity property did not hold for quaternions. Quaternions forced mathematicians to realize that, apart from the part dedicated to integers and far from perfect, the Euclidean Began, arithmetic and algebra do not have their own axiomatic basis. Mathematicians freely handled negative and complex numbers and performed algebraic operations, guided only by the fact that they worked successfully. Logical rigor gave way to demonstrating the practical benefits of introducing dubious concepts and procedures.

Almost from the very beginning of mathematical analysis, attempts have been made repeatedly to provide rigorous foundations for it. Mathematical analysis introduced two new complex concepts - derivative and definite integral. Newton and Leibniz struggled with these concepts, as well as mathematicians of subsequent generations, who turned differential and integral calculus into mathematical analysis. However, despite all efforts, much uncertainty remained in the concepts of limit, continuity and differentiability. In addition, it turned out that the properties of algebraic functions cannot be transferred to all other functions. Almost all mathematicians of the 18th century. and the beginning of the 19th century. efforts have been made to find a rigorous basis for mathematical analysis, and all have failed. Finally, in 1821, O. Cauchy (1789–1857), using the concept of number, provided a strict basis for all mathematical analysis. However, later mathematicians discovered logical gaps in Cauchy. The desired rigor was finally achieved in 1859 by K. Weierstrass (1815–1897).

Weierstrass initially considered the properties of real and complex numbers self-evident. Later, like G. Cantor (1845–1918) and R. Dedekind (1831–1916), he realized the need to build a theory of irrational numbers. They gave a correct definition of irrational numbers and established their properties, but they still considered the properties of rational numbers to be self-evident. Finally, the logical structure of the theory of real and complex numbers acquired its complete form in the works of Dedekind and J. Peano (1858–1932). The creation of the foundations of the numerical system also made it possible to solve the problems of substantiating algebra.

The task of increasing the rigor of the formulations of Euclidean geometry was relatively simple and boiled down to listing the terms being defined, clarifying the definitions, introducing missing axioms, and filling gaps in the proofs. This task was completed in 1899 by D. Gilbert (1862–1943). Almost at the same time, the foundations of other geometries were laid. Hilbert formulated the concept of formal axiomatics. One of the features of the approach he proposed is the interpretation of undefined terms: they can be understood as any objects that satisfy the axioms. The consequence of this feature was the increasing abstractness of modern mathematics. Euclidean and non-Euclidean geometries describe physical space. But in topology, which is a generalization of geometry, the undefined term "point" can be free of geometric associations. For a topologist, a point can be a function or a sequence of numbers, as well as anything else. Abstract space is a set of such “points” ( see also TOPOLOGY).

Hilbert's axiomatic method was included in almost all branches of mathematics of the 20th century. However, it soon became clear that this method had certain limitations. In the 1880s, Cantor tried to systematically classify infinite sets (for example, the set of all rational numbers, the set of real numbers, etc.) by comparatively quantifying them, attributing to them the so-called. transfinite numbers. At the same time, he discovered contradictions in set theory. Thus, by the beginning of the 20th century. mathematicians had to deal with the problem of their resolution, as well as with other problems of the foundations of their science, such as the implicit use of the so-called. axioms of choice. And yet nothing could compare with the destructive impact of K. Gödel's (1906–1978) incompleteness theorem. This theorem states that any consistent formal system rich enough to contain number theory must necessarily contain an undecidable proposition, i.e. a statement that can neither be proven nor disproved within its framework. It is now generally accepted that there is no absolute proof in mathematics. Opinions differ as to what evidence is. However, most mathematicians tend to believe that the problems of the foundations of mathematics are philosophical. Indeed, not a single theorem has changed due to the newly discovered logically rigorous structures; this shows that mathematics is based not on logic, but on sound intuition.

If the mathematics known before 1600 can be characterized as elementary, then in comparison with what was created later, this elementary mathematics is infinitesimal. Old areas expanded and new ones emerged, both pure and applied branches of mathematical knowledge. About 500 mathematical journals are published. Great amount published results do not allow even a specialist to become familiar with everything that is happening in the field in which he works, not to mention the fact that many results are understandable only to a specialist in a narrow profile. No mathematician today can hope to know more than what is going on in a very small corner of science. see also articles about scientists - mathematicians.

Literature:

Van der Waerden B.L. Awakening Science. Mathematics of Ancient Egypt, Babylon and Greece. M., 1959
Yushkevich A.P. History of mathematics in the Middle Ages. M., 1961
Daan-Dalmedico A., Peiffer J. Paths and labyrinths. Essays on the history of mathematics. M., 1986
Klein F. Lectures on the development of mathematics in the 19th century. M., 1989

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 gives great importance size

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 .

Next, using only differentials, the extremum conditions are formulated and considered big number complex problems related mainly to differential geometry on the plane. At the end of the book, in ch. 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 the differential of the numerator to the differential of the denominator, taken at .

According to L'Hopital'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 relationship of variables according to known connection 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 trigonometric circle, Euler deduces the following from the addition formulas:

Assuming and , he gets

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

Having indicated various expressions for functions that are now called elementary, Euler proceeds to consider curves on the plane drawn free movement hands. 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, every continuous modern sense the curve 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 last third In 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. To others important event The 20th century was the development of non-standard analysis as an alternative approach to substantiating analysis.

Sections of mathematical analysis

Metric space, Topological space

Bibliography

Encyclopedic articles

// Encyclopedic Lexicon: St. Petersburg: type. A. Plushara, 1835-1841. Volume 1-17.

// Encyclopedic Dictionary of Brockhaus and Efron: In 86 volumes (82 volumes and 4 additional ones). - St. Petersburg. , 1890-1907.

Educational literature

Standard textbooks

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

Courant, R. Course of differential and integral calculus (in two volumes). The main methodological discovery of the course: first, the main ideas are simply stated, and then they are given rigorous evidence. Written by Courant while he was a professor at the University of Göttingen in the 1920s under the influence of Klein’s ideas, then transferred to American soil in the 1930s. The Russian translation of 1934 and its reprints gives the text based on the German edition, the translation of the 1960s (the so-called 4th edition) is a compilation from the German and American versions of the textbook and is therefore very verbose.
Fikhtengolts G. M. A course in differential and integral calculus (in three volumes) and a problem book.
Demidovich B. P. Collection of problems and exercises in mathematical analysis.
Lyashko I. I. et al. Reference book for higher mathematics, vol. 1-5.

Some universities have their own analysis guides:

Moscow State University, MechMat:

Arkhipov G. I., Sadovnichy V. A., Chubarikov V. N. Lectures on math. analysis.
Zorich V. A. Mathematical analysis. Part I. M.: Nauka, 1981. 544 p.
Zorich V. A. Mathematical analysis. Part II. M.: Nauka, 1984. 640 p.
Kamynin L. I. Course of mathematical analysis (in two volumes). M.: Moscow University Publishing House, 2001.

V. A. Ilyin, V. A. Sadovnichy, Bl. H. Sendov. Mathematical analysis / Ed. A. N. Tikhonova. - 3rd ed. , processed and additional - M.: Prospekt, 2006. - ISBN 5-482-00445-7

Moscow State University, Faculty of Physics:

Ilyin V. A., Poznyak E. G. Fundamentals of mathematical analysis (in two parts). - M.: Fizmatlit, 2005. - 648 p. - ISBN 5-9221-0536-1
Butuzov V.F. et al. Mat. analysis in questions and tasks

Mathematics in technical university Collection of textbooks in 21 volumes.

St. Petersburg State University, Faculty of Physics:

Smirnov V.I. Course of higher mathematics, in 5 volumes. M.: Nauka, 1981 (6th edition), BHV-Petersburg, 2008 (24th edition).

NSU, Mechanics and Mathematics:

Reshetnyak Yu. G. Course of mathematical analysis. Part I. Book 1. Introduction to mathematical analysis. Differential calculus of functions of one variable. Novosibirsk: Publishing House of the Institute of Mathematics, 1999. 454 with ISBN 5-86134-066-8.
Reshetnyak Yu. G. Course of mathematical analysis. Part I. Book 2. Integral calculus of functions of one variable. Differential calculus of functions of several variables. Novosibirsk: Publishing House of the Institute of Mathematics, 1999. 512 with ISBN 5-86134-067-6.
Reshetnyak Yu. G. Course of mathematical analysis. Part II. Book 1. Fundamentals of smooth analysis in multidimensional spaces. Series theory. Novosibirsk: Publishing House of the Institute of Mathematics, 2000. 440 with ISBN 5-86134-086-2.
Reshetnyak Yu. G. Course of mathematical analysis. Part II. Book 2. Integral calculus of functions of several variables. Integral calculus on manifolds. External differential forms. Novosibirsk: Publishing House of the Institute of Mathematics, 2001. 444 with ISBN 5-86134-089-7.
Shvedov I. A. Compact course of mathematical analysis,: Part 1. Functions of one variable, Part 2. Differential calculus of functions of several variables.

MIPT, Moscow

Kudryavtsev L. D. Course of mathematical analysis (in three volumes).

BSU, physics department:

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. Part 1, Part 2, 1978. (Most of the material relates to TFKP)
Pascal, E.(Napoli). Esercizii, 1895; 2 ed., 1909 // Internet Archive

Textbooks for humanities

A. M. Akhtyamov Mathematics for sociologists and economists. - M.: Fizmatlit, 2004.
N. Sh. Kremer and others. Higher mathematics for economists. Textbook. 3rd ed. - M.: Unity, 2010

Problem books

G. N. Berman. Collection of problems for the course of mathematical analysis: Tutorial for universities. - 20th ed. M.: Science. Main editorial office of physical and mathematical literature, 1985. - 384 p.
P. E. Danko, A. G. Popov, T. Ya. Kozhevnikov. Higher mathematics in exercises and problems. (In 2 parts) - M.: Vyssh.shk, 1986.
G. I. Zaporozhets Guide to solving problems in mathematical analysis. - M.: graduate School, 1966.
I. A. Kaplan. Practical lessons in higher mathematics, in 5 parts.. - Kharkov, Publishing house. Kharkov State Univ., 1967, 1971, 1972.
A. K. Boyarchuk, G. P. Golovach. Differential equations in examples and problems. Moscow. Editorial URSS, 2001.
A. V. Panteleev, A. S. Yakimova, A. V. Bosov. Ordinary differential equations in examples and problems. "MAI", 2000
A. M. Samoilenko, S. A. Krivosheya, N. A. Perestyuk. Differential equations: examples and problems. VS, 1989.
K. N. Lungu, V. P. Norin, D. T. Pismenny, Yu. A. Shevchenko. Collection of problems in higher mathematics. 1 course. - 7th ed. - M.: Iris-press, 2008.
I. A. Maron. Differential and integral calculus in examples and problems (Functions of one variable). - M., Fizmatlit, 1970.
V. D. Chernenko. Higher mathematics in examples and problems: Textbook for universities. In 3 volumes - St. Petersburg: Politekhnika, 2003.

Directories

Classic works

Essays on the history of analysis

Kestner, Abraham Gottgelf. Geschichte der Mathematik . 4 volumes, Göttingen, 1796-1800
Kantor, Moritz. Vorlesungen über geschichte der mathematik Leipzig: B. G. Teubner, - . Bd. 1, Bd. 2, Bd. 3, Bd. 4
History of mathematics edited by A. P. Yushkevich (in three volumes):

Volume 1 From ancient times to the beginning of modern times. (1970)
Volume 2 Mathematics of the 17th century. (1970)
Volume 3 Mathematics XVIII century. (1972)

Markushevich A.I. Essays on the history of the theory of analytic functions. 1951
Vileitner G. History of mathematics from Descartes to the middle of the 19th century. 1960

Notes

Wed., e.g. Cornell Un course
Newton I. Mathematical works. M, 1937.
Leibniz //Acta Eroditorum, 1684. L.M.S., vol. V, p. 220-226. Rus. Transl.: Uspekhi Mat. Sciences, vol. 3, v. 1 (23), p. 166-173.
L'Hopital. Infinitesimal Analysis. M.-L.: GTTI, 1935. (Hereinafter: L'Hopital) // Mat. analysis on EqWorld
L'Hopital, ch. 1, def. 2.
L'Hopital, ch. 4, def. 1.
L'Hopital, ch. 1, requirement 1.
L'Hopital, ch. 1, requirement 2.
L'Hopital, ch. 2, def.

Introduction

L. Euler is the most productive mathematician in history, the author of more than 800 works on mathematical analysis, differential geometry, number theory, approximate calculations, celestial mechanics, mathematical physics, optics, ballistics, shipbuilding, music theory, etc. Many of his works had a significant impact influence on the development of science.

Euler spent almost half his life in Russia, where he energetically helped create Russian science. In 1726 he was invited to work in St. Petersburg. In 1731-1741 and starting from 1766 he was an academician of the St. Petersburg Academy of Sciences (in 1741-1766 he worked in Berlin, remaining an honorary member of the St. Petersburg Academy). He knew the Russian language well and published some of his works (especially textbooks) in Russian. The first Russian academicians in mathematics (S.K. Kotelnikov) and astronomy (S.Ya. Rumovsky) were students of Euler. Some of his descendants still live in Russia.

L. Euler made a very great contribution to the development of mathematical analysis.

The purpose of the essay is to study the history of the development of mathematical analysis in the 18th century.

The concept of mathematical analysis. Historical sketch

Mathematical analysis is a set of branches of mathematics devoted to the study of functions and their generalizations using the methods of differential and integral calculus. With such a general interpretation, functional analysis should also be included in the analysis together with the theory of the Lebesgue integral, comprehensive analysis(TFKP), which studies functions defined on the complex plane, non-standard analysis, which studies infinitesimal and infinitely big numbers, as well as the calculus of variations.

In the educational process, analysis includes

· differential and integral calculus

· theory of series (functional, power and Fourier) and multidimensional integrals

· vector analysis.

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 presentation follows the patterns late XIX century and in particular uses naive set theory.

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, was rather vague for the authors of the idea. The algebraic approach (infinitesimal calculus) begins to appear with 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. Newton I. Mathematical works. M, 1937.

The official date of birth of differential calculus can be considered May 1684, when Leibniz published the first article “A new method of maxima and minima...” Leibniz //Acta Eroditorum, 1684. L.M.S., vol. V, p. 220--226. Rus. Transl.: Uspekhi Mat. Sciences, vol. 3, v. 1 (23), p. 166--173.. This article, in a concise and inaccessible form, set out the principles of a new method called differential calculus.

At the end of the 17th century, a circle emerged around Leibniz, the most prominent representatives of which were the Bernoulli brothers, Jacob and Johann, and L'Hopital. In 1696, using the lectures of I. Bernoulli, L'Hopital wrote the first L'Hopital textbook. Analysis of infinitesimals. M.-L.: GTTI, 1935., expounding new method in application to the theory of plane curves. He called it “Infinitesimal Analysis”, thereby giving one of the names to the new branch of mathematics. The presentation is based on the concept of variable quantities, between which there is some connection, due to which a change in one entails a change in the other. In L'Hôpital, this connection is given using plane curves: if M is a moving point of a plane curve, then its Cartesian coordinates x and y, called the diameter and ordinate of the curve, are variables, and a change in x entails a change in y. The concept of a function is absent: wanting to say that the dependence of the variables is given, L'Hopital says that “the nature of the curve is known.” The concept of differential is introduced as follows:

“The infinitesimal part by which a variable quantity continuously increases or decreases is called its differential... To denote the differential of a variable quantity, which itself is expressed by a single letter, we will use the sign or symbol d. Right there. Chapter 1, definition 2http://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8 %D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7 - cite_note -4#cite_note-4 ... The infinitesimal part by which the differential of a variable value continuously increases or decreases is called ... the second differential.” Right there. Chapter 4, definition 1.

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

It is required that two quantities differing from each other only by an infinitesimal amount can be taken indifferently one instead of the other. L'Hopital. Analysis of infinitesimals. M.-L.: GTTI, 1935. Chapter 1, requirement 1.

dxy = (x + dx)(y + dy) ? xy = xdy + ydx + dxdy = (x + dx)dy + ydx = xdy + ydx

and so on. differentiation rules. The second requirement states:

It is required that one can consider a curved line as a collection of an infinite number of infinitesimal straight lines.

The continuation of each such line is called a tangent to the curve. Right there. Chapter 2. def. Investigating the tangent passing through the point M = (x,y), L'Hopital attaches great importance to the quantity

reaching extreme values at the inflection points of the curve, but the ratio of dy to dx is not given any special significance.

It is noteworthy to find extremum points. If, with a continuous increase in diameter x, the ordinate y first increases and then decreases, then the differential dy is first positive compared to dx, 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, y = x2, then by virtue of the first requirement

2xdx + dx2 = 2xdx;

at zero, the right hand side is zero and the left hand side is not. Apparently it should have been said that dy can be transformed in accordance with the first requirement so that at the maximum point dy = 0. In the examples everything is self-explanatory, and only in the theory of inflection points L'Hopital writes that dy is equal to zero at the maximum point, being divided by dx L'Hopital. Analysis of infinitesimals. M.-L.: GTTI, 1935 § 46.

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 ch. 10, sets out what is now called L'Hopital's rule, although in an unusual form. Let the ordinate y of the curve be expressed as a fraction whose numerator and denominator vanish at x = a. Then the point of the curve with x = a has a ordinate y equal to the ratio of the differential of the numerator to the differential of the denominator taken at x = a.

According to L'Hopital'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 Method of Integral” Bernulli, Johann. Die erste Integrelrechnunug. Leipzig-Berlin, 1914. Here a method is given for taking most elementary integrals and methods for solving many first-order differential equations are indicated.