Рефераты. Выдающаяся роль Леонарда Эйлера в развитии алгебры, геометрии и теории чисел

Выдающаяся роль Леонарда Эйлера в развитии алгебры, геометрии и теории чисел

МИНИСТЕРСТВО ОБРАЗОВАНИЯ РОССИЙСКОЙ ФЕДЕРАЦИИ

ФАКУЛЬТЕТ МАТЕМАТИКИ И КОМПЬЮТЕРНЫХ НАУК

КАФЕДРА АЛГЕБРЫ И МАТЕМАТИЧЕКСКОЙ ЛОГИКИ

Выдающаяся роль Леонарда Эйлера в развитии алгебры, геометрии и теории чисел.

(дипломная работа)

Т - 2004

Содержание

Содержание 2

Введение. 3

Chapter I. Biography of Leonard Euler. 5
Глава II. Вклад Эйлера в развитие алгебры. 13
§2.1. Алгебраические доказательства основной теоремы алгебры. 13
§2.2 Числовые приближенные методы решения уравнений. 16
п.2.2.1. Метод рекуррентных рядов. 16
п.2.2.2. Еще два оригинальных метода. 19
§2.3. Общая теория уравнений. 21
Глава III. Выдающиеся достижения Леонарда Эйлера в области геометрии и тригонометрия. 23
§3.1. Развитие аналитической геометрии, начиная с систематического исследования высших порядков. 23
§3. 2. Поверхности второго и высших порядков. 28
§3.3. Второй том «введения в анализ бесконечных» 33
§3.4. Специальные плоские кривые. 38
§3.5. Геодезические линии 39
§3.6. Общие пространственные кривые и развертывающиеся поверхности 42
§3.7. Общие поверхности 45
§3.8. Заслуги Эйлера в преобразовании и дальнейших успехах тригонометрии. 52
ГЛАВА IV. Влияние Леонарда Эйлера на развитие теории чисел. 58
§4.1. Целочисленное решение неопределенных уравнений. 58
§4.2. Теорема Эйлера. 62
§4.3. Вычеты . 63
§4.4. Разложение на простые множители. 64

Заключение. 68
Список литературы. 69

Введение

Математика есть самая удивительная и загадочная сфера деятельности человеческой мысли. Развитие области фундаментальных знаний исторически неотъемлемо связано с развитием человеческого социума. Это значит, что основные грандиозные вехи развития этой изящной науки связаны с жизнью, без сомнения, гениальных умов человечества. В пантеон бессмертия выписаны имена математиков, чьи титанические труды обогатили людские знания всесущей.

Настоящая работа посвящена освещению биографии великого Леонарда Эйлера и его трудов, привнесших огромный вклад в развитие математики, и, прежде всего, в приложении её к практической деятельности.

Необыкновенная интуиция, точный и искусный ум вели Леонарда Эйлера к изящным и удивительным открытиям, ныне кажущиеся столь простыми и естественными, что не вызывают никакого сомнения в их неприкасаемой правоте. В настоящее время многочисленные отрасли математики, механики, физики, астрономии до сих пор используют научные достижения трудов Эйлера, признанные, как основополагающие.

В первой главе представлено описание основных моментов жизни, повлиявших на творчество и развитие Леонарда Эйлера как крупного ученого и просветителя того времени. Большинство источников о биографии и деятельности Леонарда Эйлера были переведены, в основном, с немецкого на английский и французские языки, в частности, и на русский. В виду целесообразности первая глава представлена на английском языке, дабы быть ближе к первоисточникам.

Вторая глава повествует о существенном вкладе таланта Леонарда Эйлера в развитие алгебры XVIII столетия. В ней представлена работа, связанная с доказательством основной теоремы алгебры и методах приближенных решений алгебраических уравнений п-ой степени.

Третья глава посвящена выдающимся достижениям Леонарда Эйлера в области геометрии и тригонометрии. В нее включены работы по исследованию поверхностей второго и высших порядков, а так же специальных плоских кривых и геодезических линий. Леонард Эйлер написал первый систематизированный учебник по геометрии, общепризнанный классическим. Это второй том «Введения в анализ бесконечно малых». В данном учебнике развит единый метод для классификации плоских алгебраических кривых любого порядка и систематизированы практически все общие методы исследования таких кривых.

Четвертая глава повествует о крупнейших открытиях в теории диофантовых уравнений, занимавшей своей сложностью и изяществом прогрессивные умы математиков многих столетий. В XVIII веке Л. Эйлер, работая в Петербургской академии наук, издал большую часть своих работ по теории чисел и диофантовых уравнений. Он обобщил основной результат ферма для случая делимости на составные числа, создал общую теорию так называемых степенных вычетов, получил очень большое число разнообразных результатов о представимости чисел в виде форм определенного типа, исследовал ряд систем неопределенных уравнений и получил интересные результаты о разбиение чисел на слагаемые. У Эйлера мы впервые встречаемся с идеей применения методов математического анализа к задачам теории чисел. Рассмотрение бесконечных рядов и произведений являлось у Эйлера действенным орудием для получения теоретико-числовых результатов.

Нет, пожалуй, ни одной значительной области математики, в которой не оставил бы след один из величайших математиков 18 столетия Леонард Эйлер, чья жизнь и работа стимулируют творчество многие поколения математиков.

Chapter I. Biography of Leonard Euler.

Pic. 1 (Poster of Eule)

Leonhard Euler's father was Paul Euler. Paul Euler had studied theology at the University of Basel and had attended Jacob Bernoulli's lectures there. In fact Paul Euler and Johann Bernoulli had both lived in Jacob Bernoulli's house while undergraduates at Basel. Paul Euler became a Protestant minister and married Margaret Brucker, the daughter of another Protestant minister. Their son Leonhard Euler was born in Basel, but the family moved to Riehen when he was one year old and it was in Riehen, not far from Basel, that Leonard was brought up. Paul Euler had, as we have mentioned, some mathematical training and he was able to teach his son elementary mathematics along with other subjects.

Leonhard was sent to school in Basel and during this time he lived with his grandmother on his mother's side. This school was a rather poor one, by all accounts, and Euler learnt no mathematics at all from the school. However his interest in mathematics had certainly been sparked by his father's teaching, and he read mathematics texts on his own and took some private lessons. Euler's father wanted his son to follow him into the church and sent him to the University of Basel to prepare for the ministry. He entered the University in 1720, at the age of 14, first to obtain a general education before going on to more advanced studies. Johann Bernoulli soon discovered Euler's great potential for mathematics in private tuition that Euler himself engineered. Euler's own account given in his unpublished autobiographical writings, see [1] is as follows:-

... I soon found an opportunity to be introduced to a famous professor Johann Bernoulli. ... True, he was very busy and so refused flatly to give me private lessons; but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could; if I came across some obstacle or difficulty, I was given permission to visit him freely every Sunday afternoon and he kindly explained to me everything I could not understand ...

In 1723 Euler completed his Master's degree in philosophy having compared and contrasted the philosophical ideas of Descartes and Newton. He began his study of theology in the autumn of 1723, following his father's wishes, but, although he was to be a devout Christian all his life, he could not find the enthusiasm for the study of theology, Greek and Hebrew that he found in mathematics. Euler obtained his father's consent to change to mathematics after Johann Bernoulli had used his persuasion. The fact that Euler's father had been a friend of Johann Bernoulli's in their undergraduate days undoubtedly made the task of persuasion much easier.

Euler completed his studies at the University of Basel in 1726. He had studied many mathematical works during his time in Basel, and Calinger [4] has reconstructed many of the works that Euler read with the advice of Johann Bernoulli. They include works by Varignon, Descartes, Newton, Galileo, von Schooten, Jacob Bernoulli, Hermann, Taylor and Wallis. By 1726 Euler had already a paper in print, a short article on isochronous curves in a resisting medium. In 1727 he published another article on reciprocal trajectories and submitted an entry for the 1727 Grand Prize of the Paris Academy on the best arrangement of masts on a ship.

The Prize of 1727 went to Bouguer, an expert on mathematics relating to ships, but Euler's essay won him second place which was a fine achievement for the young graduate. However, Euler now had to find himself an academic appointment and when Nicolaus(II) Bernoulli died in St Petersburg in July 1726 creating a vacancy there, Euler was offered the post which would involve him in teaching applications of mathematics and mechanics to physiology. He accepted the post in November 1726 but stated that he did not want to travel to Russia until the spring of the following year. He had two reasons to delay. He wanted time to study the topics relating to his new post but also he had a chance of a post at the University of Basel since the professor of physics there had died. Euler wrote an article on acoustics, which went on to become a classic, in his bid for selection to the post but he was nor chosen to go forward to the stage where lots were drawn to make the final decision on who would fill the chair. Almost certainly his youth (he was 19 at the time) was against him. However Calinger [4] suggests:-

This decision ultimately benefited Euler, because it forced him to move from a small republic into a setting more adequate for his brilliant research and technological work.

As soon as he knew he would not be appointed to the chair of physics, Euler left Basel on 5 April 1727. He travеlled down the Rhine by boat, crossed the German states by post wagon, then by boat from Lьbeck arriving in St Petersburg on 17 May 1727. He had joined the St. Petersburg Academy of Science two years after it had been founded by Catherine I the wife of Peter the Great. Through the requests of Daniel Bernoulli and Jakob Hermann, Euler was appointed to the mathematical-physical division of the Academy rather than to the physiology post he had originally been offered. At St Petersburg Euler had many colleagues who would provide an exceptional environment for him [1]:-

Nowhere else could he have been surrounded by such a group of eminent scientists, including the analyst, geometer Jakob Hermann, a relative; Daniel Bernoulli, with whom Euler was connected not only by personal friendship but also by common interests in the field of applied mathematics; the versatile scholar Christian Goldbach, with whom Euler discussed numerous problems of analysis and the theory of numbers; F Maier, working in trigonometry; and the astronomer and geographer J-N Delisle.

Euler served as a medical lieutenant in the Russian navy from 1727 to 1730. In St Petersburg he lived with Daniel Bernoulli who, already unhappy in Russia, had requested that Euler bring him tea, coffee, brandy and other delicacies from Switzerland. Euler became professor of physics at the academy in 1730 and, since this allowed him to became a full member of the Academy, he was able to give up his Russian navy post.

Daniel Bernoulli held the senior chair in mathematics at the Academy but when he left St Petersburg to return to Basel in 1733 it was Euler who was appointed to this senior chair of mathematics. The financial improvement which came from this appointment allowed Euler to marry which he did on 7 January 1734, marrying Katharina Gsell, the daughter of a painter from the St Petersburg Gymnasium. Katharina, like Euler, was from a Swiss family. They had 13 children altogether although only five survived their infancy. Euler claimed that he made some of his greatest mathematical discoveries while holding a baby in his arms with other children playing round his feet.

Страницы: 1, 2, 3, 4, 5, 6, 7, 8