XI Международная студенческая научная конференция Студенческий научный форум - 2019

Leonard Euler was born in 1707 into the family of a Basel pastor, a friend of the Bernoulli family. Early discovered mathematical abilities. Primary education received at home under the guidance of his father, who once studied m

athematics under Jacob Bernoulli. The pastor was preparing his eldest son for a spiritual career, but he studied with him and mathematics, both as entertainment and for developing logical thinking. Along with studying at the gymnasium, the boy enthusiastically studied mathematics under the guidance of Jacob Bernoulli, and in recent years at the gymnasium he attended university lectures of his younger brother Jacob, Johann Bernoulli.

On October 20, 1720, 13-year-old Leonard Euler became a student at the Faculty of Arts at the University of Basel. But the love of mathematics sent Leonard on a different path. Soon a capable boy attracted the attention of Professor Johann Bernoulli. He handed over to the gifted student maths articles for study, and on Saturdays he invited me to come to his house in order to work together to sort out the incomprehensible. At his teacher’s home, Euler met and became friends with Bernoulli’s sons, Daniel and Nicholas, who were also enthusiastically engaged in mathematics.

On June 8, 1724, 17-year-old Leonard Euler made a speech in Latin about the comparison of the philosophical views of Descartes and Newton and was awarded the degree of master.

In the next two years, the young Euler wrote several scientific papers. One of them, “Thesis in Physics on Sound”, which received a favorable review, was submitted to the competition to replace the post of professor of physics (1725), which was unexpectedly vacant at the University of Basel. But, despite the positive feedback, 19-year-old Euler was considered too young to be included in the list of candidates for the professorship. It should be noted that the number of scientific vacancies in Switzerland was quite small. Therefore, the brothers Daniel and Nikolay Bernoulli went to Russia, where the Academy of Sciences was just being organized; they promised to pat there and about the position for Euler.

At the beginning of the winter of 1726, Euler was reported from St. Petersburg: on the recommendation of the Bernoulli brothers, he was invited to the post of physiology adjunct with a salary of 200 rubles. forever left Switzerland.

Number theory

P. L. Chebyshev wrote: "Euler was the beginning of all research that constituted the general theory of numbers." Most 18th century mathematicians were engaged in the development of analysis, but Euler carried the fascination with ancient arithmetic throughout his life. Thanks to his works, interest in number theory was revived by the end of the century.

Euler continued his research on Fermat, who had previously expressed (under the influence of Diophantus) a series of separate hypotheses about natural numbers. Euler strictly proved these hypotheses, considerably generalized them and combined them into a meaningful number theory. He introduced the extremely important “Euler function” into mathematics and formulated with it the “Euler theorem”. Euler created a theory of comparisons and quadratic residues, specifying Euler's criteria for the latter.

He refuted Fermat's hypothesis that all numbers of the form F_n = 2 ^ {2 ^ n} +1 are simple; it turned out that F5 is divisible by 641.

Proved Fermat's statement on the representation of an odd prime number as the sum of two squares.

Gave one of the solutions to the four cubes problem.

Euler proved the Great Fermat Theorem for n = 3 and n = 4, created a complete theory of continued fractions, investigated various classes of Diophantine equations, the theory of partitioning numbers into terms.

He discovered that in the theory of numbers the application of methods of mathematical analysis is possible, giving rise to the analytical theory of numbers. It is based on the Euler identity and the general method of generating functions.

Euler introduced the concept of primitive root and put forward the hypothesis that for any prime number p there is a primitive root modulo p; he failed to prove it; later Legendre and Gauss proved the theorem. Of great importance in the theory was another Euler hypothesis - the quadratic law of reciprocity, also proved by Gauss.

Mathematical analysis

One of the main merits of Euler before science is the monograph Introduction to the Analysis of the Infinitely Small (1748). In 1755, the augmented "Differential Calculus" was published, and in 1768-1770, three volumes of the "Integral Calculus". In the aggregate, this is a fundamental, well-illustrated course with a well thought-out terminology and symbolism, from which much has passed into modern textbooks. Actually modern methods of differentiation and integration were published in these works.

The basis of natural logarithms has been known since the days of Napier and Jacob Bernoulli, but Euler gave such a deep study of this important constant that since then she bears his name. Another constant he studied was the Euler – Mascheroni constant.

He shares with Lagrange the honor of discovering the calculus of variations by writing out the Euler-Lagrange equations for the general variational problem. In 1744, Euler published the first book on the calculus of variations ("The method of finding curves with the properties of maximum or minimum").

Euler significantly advanced the theory of series and extended it to a complex area, while obtaining the famous Euler formula. The mathematical world was greatly impressed by the series, which were first summed up by Euler, including a series of inverse squares that could not be given to anyone:

The modern definition of exponential, logarithmic and trigonometric functions is also his merit, as well as their symbolism and generalization to a complex case. Formulas, often referred to in textbooks as the Cauchy – Riemann conditions, would be more correct to call the Dalambert conditions Euler.

He first gave a systematic theory of integration and the technical methods used there, found important classes of integrable differential equations. He discovered Euler integrals — valuable classes of special functions arising during integration: the beta function and the Euler gamma function. Simultaneously with Klero, he deduced the integrability conditions for linear differential forms of two or three variables (1739). The first introduced double integrals. Received serious results in the theory of elliptic functions, including the first addition theorems.

From a later point of view, Euler’s actions with infinite rows cannot always be considered correct (the analysis was justified only half a century later), but phenomenal mathematical intuition almost always prompted him the correct result.

However, it was not only intuition, Euler acted here quite consciously, in many important respects his understanding of the meaning of divergent series and operations with them exceeded the standard understanding of the XIX century and served as the basis for the modern theory of divergent series developed in the late XIX - early XX century.

Geometry

In elementary geometry, Euler discovered several facts that were not noticed by Euclid:

The three heights of the triangle intersect at one point (orthocenter).

In the triangle, the orthocenter, the center of the circumscribed circle and the center of gravity lie on one straight line - the “Euler line”.

The bases of the heights of an arbitrary triangle, the midpoints of its three sides and the midpoints of the three segments connecting its vertices with the orthocenter, lie all on one circle (Euler's circle).

The number of vertices (C), faces (F) and edges (P) of any convex polyhedron is connected by a simple formula: C + D = P + 2.

The second volume of “Introduction to the Analysis of the Infinitely Small” (1748) is the world's first textbook on analytical geometry and the basics of differential geometry. The term affine transformations was first introduced in this book along with the theory of such transformations.

In 1760, the fundamental "Investigations on the Curvature of Surfaces" were published. Euler found that at each point of a smooth surface there are two normal sections with minimum and maximum curvature radii, and their planes are mutually perpendicular. He derived a formula for the connection of the curvature of the section of the surface with the main curvatures.

1771: published essay "On bodies, the surface of which can be deployed on a plane." In this paper, the concept of an unfolding surface is introduced, that is, a surface that can be superimposed on a plane without folds and gaps. Euler, however, gives here a completely general theory of the metric, on which the entire internal geometry of a surface depends. Later, the study of the metric becomes his main tool in the theory of surfaces.

Combinatorics

Euler's Magic Square

Euler paid much attention to the representation of natural numbers as sums of a special type and formulated a series of theorems for calculating the number of partitions.

He investigated the algorithms for constructing magic squares using a chess knight bypass.

In solving combinatorial problems, he deeply studied the properties of combinations and permutations, introduced Euler numbers into consideration.

Other areas of mathematics

Graph theory began with Euler solving the problem of the seven bridges of Königsberg.

The method of broken Euler.

Mechanics and Mathematical Physics

Many of Euler's works are devoted to mathematical physics: mechanics, hydrodynamics, acoustics, and others. In 1736, the treatise “Mechanics, or the science of motion, in analytical presentation,” marked the new stage in the development of this ancient science. 29-year-old Euler abandoned the traditional geometric approach to mechanics and summed up a rigorous analytical foundation. Essentially, from this point on, mechanics becomes an applied mathematical discipline.

In 1755, the General Principles of Fluid Motion were published, which laid the foundation for theoretical hydrodynamics. Basic hydrodynamic equations (Euler's equation) for a fluid without viscosity are derived. Disassembled system solutions for different special cases.

In 1765, in his book The Theory of Motion of Solids, Euler mathematically described the kinematics of a solid body of finite size (before it was mainly studied the motion of a point). He introduced the Euler angles and the rotation theorem into mathematics. His name also carries the kinematic formula for the distribution of velocities in a solid body, the equations (Euler – Poisson) of the dynamics of a rigid body, an important case of integrability in the dynamics of a solid body.

Euler summarized the principle of least action, rather confusingly described by Maupertuis, and pointed out its fundamental importance in mechanics. Unfortunately, he did not reveal the variational nature of this principle, but still attracted the attention of physicists, who later clarified its fundamental role in nature.

Astronomy

Euler worked hard in the field of celestial mechanics. He laid the foundation for the perturbation theory, later completed by Laplace, and developed a very accurate theory of the motion of the moon. This theory turned out to be suitable for solving the urgent problem of determining longitude at sea, and the English Admiralty paid a special prize for it to Euler.

The main works of Euler in this area:

• "Theory of the Moon", 1753.

• "Theory of the motion of planets and comets" (Latin: Theoria motus planetarum et cometarum), 1774.

• "The New Theory of the Motion of the Moon", 1772.

Euler investigated the field of not only spherical, but also ellipsoidal bodies, which was a significant step forward.

reference

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Перевод

Артемьева Т. В. Леонард Эйлер как философ - www.ideashistory.org.ru/pdfs/07euler.pdf // Философия в Петербургской Академии наук XVIII века. — СПб., 1999. — 182 с.

Башмакова И. Г., Юшкевич А. П. Леонард Эйлер // Историко-математические исследования. — М.: ГИТТЛ, 1954. — № 7. — С. 453-512.

Белл Э. Т. Творцы математики - www.math.ru/lib/book/djvu/istoria/bell.djvu. — М.: Просвещение, 1979.

Гиндикин С. Г. Рассказы о физиках и математиках - www.mccme.ru/free-books/gindikin/index.html. — 3-е изд., расш. — М.: МЦНМО, 2001. — 465 с. — ISBN 5-900916-83-9

Делоне Б. Н. Леонард Эйлер - kvant.mccme.ru/1974/05/leonard_ejler.htm // Квант. — 1974. — № 5.

Котек В. В. Леонард Эйлер. — М.: Учпедгиз, 1961.

К 250-летию со дня рождения Л. Эйлера. — Сборник. — Изд-во АН СССР, 1958.

Бурья А. Смерть Леонарда Эйлера. - mikv1.narod.ru/text/Buria1958.htm С. 605-607.

Летопись Российской Академии наук. — М.: Наука, 2000. — Т. 1: 1724—1802. — ISBN 5-02-024880-0

Математика XVIII столетия - ilib.mccme.ru/djvu/istoria/istmat3.htm / Под редакцией А. П. Юшкевича. — М.: Наука, 1972. — Т. 3. — (История математики в 3-х томах).

Полякова Т. С. Леонард Эйлер и математическое образование в России. — КомКнига, 2007. — 184 с. — ISBN 978-5-484-00775-2

Развитие идей Леонарда Эйлера и современная наука. Сб. статей. — М.: Наука, 1988. — 525 с. — ISBN 5-02-000002-7

Прудников В. Е. Русские педагоги-математики XVIII—XIX веков. — 1956.

Юшкевич А. П. История математики в России. — М.: Наука, 1968.

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