Бакалавриат 2022/2023

Математический анализ 2

Статус: Курс обязательный (Прикладной анализ данных)

Направление: 01.03.02. Прикладная математика и информатика

Кто читает: Департамент больших данных и информационного поиска

Где читается: Факультет компьютерных наук

Когда читается: 2-й курс, 1, 2 модуль

Формат изучения: без онлайн-курса

Охват аудитории: для своего кампуса

Преподаватели: Коновалов Григорий Вадимович, Медведев Владимир Олегович, Томащук Корней Кириллович

Язык: английский

Кредиты: 5

Контактные часы: 64

Course Syllabus

Abstract

This course covers specific topics of advanced calculus, such as numeric and functional series, infinite products, Eulerian integrals, multiple integrals. The convergence and functional properties of power series are considered along with their applications to some problems of discrete mathematics involving the generating functions. Prerequisites: High school algebra and trigonometry, basic concepts of calculus (e. g., sequences, limits and continuity, derivatives, integrals).

Learning Objectives

• Students will understand the concept of сonvergence and divergence of infinite series and infinite products; the rate of convergence.
• Students will understand the concept of pointwise and uniform convergence of the functional series; the functional properties of their sums.
• Students will understand the concept of representing functions by power series; Taylor series of the most common elementary functions.
• Students will understand the concept of generating functions and their applications for solving linear recurrence relations.
• Students will understand the concept of representing functions by trigonometric Fourier series.
• Students will understand the concept of integrals depending on a parameter; beta and gamma functions.
• Students will understand the concept of double and triple integrals; reduction to iterated integrals (Fubini’s theorem).
• Students will understand the concept of the change of variables in multiple integrals; polar, cylindrical and spherical coordinate systems.
• Studentswill understand the concept of improper integrals.

Expected Learning Outcomes

• Students should be able to calculate sums of series using the methods of partial sums, power series, Fourier series.
• Students should be able to compute double and triple integrals by means of Fubini’s theorem, using a change of variables if necessary.
• Students should be able to determine the radius and the domain of convergence of power series.
• Students should be able to apply the properties of the Eulerian integrals for the calculation of specific integrals.
• Students should be able to calculate areas of regions, volumes of solids and surface areas.
• Students should be able to calculate sums of series using the methods of partial sums, power series, Fourier series.
• Students should be able to compute double and triple integrals by means of Fubini’s theorem, using a change of variables if necessary.
• Students should be able to determine the radius and the domain of convergence of power series.

Course Contents

• Infinite series.
• Series of Nonnegative Terms. Convergence Tests.
• Alternating series. Absolute and conditional convergence.
• Products of series. Infinite Products.
• Uniform convergence.
• Power series.
• Generating functions.
• Fourier series.
• Integrals depending on a parameter.
• Eulerian integrals.
• Double integrals.
• Triple integrals. Applications of double and triple integrals.
• Improper integrals. Multiple integrals.

Assessment Elements

• Written test
  At the end of the first module the students pass a written test. The test consists of a selection of problems similar to those from the seminar/homework exercise lists. Students solve the problems in written form during 2 academic hours and pass the solutions to the teacher.
• Written exam
  The exam may be carried out online via distance learning platforms. The exam may be carried out online via distance learning platforms. At the end of the second module the students pass a written exam. The exam consists of a selection of problems similar to those from the seminar/homework exercise lists. Students solve the problems in written form during 2 academic hours and pass the solutions to the teacher.
• Regular quizzes
  This is a completely online activity. A small workshop about handlng the testing system (Yandex.Contest) is provided in the first lecture.
• Seminar points
  Several more complicated problems are marked as «bonus». In order to earn bonus points for a problem student should report the solution in class. In case of online report a small presentation is advisable.
• Homeworks

Interim Assessment

• 2022/2023 2nd module
  ROUND(MIN( 0.25HomeWorks + 0.2Test + 0.3Exam + 0.1SeminarMark + 0.25Quizzes and Tasks, 10))

Bibliography

Recommended Core Bibliography

• Calculus early transcendentals, Stewart, J., 2012
• Friedman, A. (2007). Advanced Calculus (Vol. Dover edition). Mineola, N.Y.: Dover Publications. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1153250

Recommended Additional Bibliography

• Ronald L. Graham, Donald E. Knuth, & Oren Patashnik. (1994). Concrete Mathematics : A Foundation for Computer Science. [N.p.]: Addison-Wesley Professional. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1601594

Authors

• MEDVEDEV VLADIMIR OLEGOVICH
• Галевская Софья Андреевна
• LYTKIN SERGEY MIKHAYLOVICH