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# Calculus 2

2020/2021
Учебный год
ENG
Обучение ведется на английском языке
4
Кредиты
Статус:
Курс обязательный
Когда читается:
2-й курс, 1, 2 модуль

### 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. Most of the developed concepts are illustrated with Matplotlib. 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.
• 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 improper integrals and 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.

#### Expected Learning Outcomes

• Students will be able to calculate sums of series using the methods of partial sums, power series, Fourier series.
• Students should be able to determine the radius and the domain of convergence of power series.
• Students should be able to find the Fourier series of a given function and justify its convergence.
• Students should be able to calculate sums of series using the methods of partial sums, power series, Fourier series.
• Students should be able to study given series and products for convergence.
• Students should be able to examine a given functional sequence or series for uniform convergence on a given interval.
• Students should be able to calculate areas of regions, volumes of solids and surface areas.
• 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 apply the properties of the Eulerian integrals for the calculation of specific integrals.
• Students will develop skills in applying Python to perform numerical calculations of series and integrals.
• Students should be able to represent a given analytic function by convergent power series.
• Students will develop skills in applying Python to make visualizations in order to achieve a deeper understanding of a certain concept.
• Students will develop skills in applying Python to find analytical and numerical solutions to the same problem and compare them to each other.
• Students should be able to solve linear differential equations of the first and the second order.

#### Course Contents

• Infinite series.
Finite sums and products. Harmonic numbers. Convergent and divergent series. Examples of series: telescoping series, geometric series, decimal fractions, p-series, alternating series. Necessary condition for convergence. Integral test. Tail of a series.
• Series of Nonnegative Terms. Convergence Tests.
Series of nonnegative terms. Comparison test. Limit comparison test. Ratio and Root tests, their relationship. Rate of convergence. Gauss test.
• Alternating series. Absolute and conditional convergence.
Cauchy criterion. Alternating series test. Dirichlet/Abel tests. Absolute and conditional convergence. Sine and cosine sums. Conditionally convergent alternating and trigonometric series.
• Products of series. Infinite Products.
Rearrangement of series, Cauchy’s and Riemann’s theorems. Product of series, Cauchy products. Convergence and divergence of infinite products, reduction to series. Wallis product. Stirling’s formula. Sine product formula.
• Uniform convergence.
Sums of functions. Pointwise and uniform convergence of functional sequences and series, their relationship. Cauchy criterion for the uniform convergence. Tests for the uniform convergence of series: alternating series test, Weierstrass M-test, Dirichlet/Abel tests. Interchange of limits, continuity of a limit function. Term-by-term integration and differentiation of uniformly convergent series. Riemann zeta function.
• Power series.
Examples of power series. Radius and interval of convergence of power series. Cauchy–Hadamard formula. Uniform convergence of power series. Term-by-term differentiation and integration of power series. Abel’s theorem. Products of power series. Uniqueness of power series expansion. Taylor series of common functions. Binomial series. Analytic functions. Complex power series. Euler’s formula.
• Generating functions.
Examples of generating functions. Operations with generating functions. First-order and second-order difference equations. Fibonacci numbers. Convolutions, Catalan numbers. Exponential generating functions. Binomial convolutions, Bernoulli numbers. Tangent and cotangent power expansions.
• Fourier series.
Trigonometric series. Fourier coefficients and Fourier series. Parseval’s identity. Piecewise functions. Riemann–Lebesgue lemma. Dirichlet kernel. Dini’s conditions for convergence of Fourier series. Pointwise convergence of the Fourier series of a 2π-periodic piecewise continuously differentiable function. Applications of Fourier series.
• Integrals depending on a parameter.
Proper integrals depending on a parameter, their properties. Complete elliptic integrals. Convergence tests of the improper integrals. Improper integrals depending on a parameter (IIDP). Uniform convergence, Weierstrass M-test, Dirichlet/Abel tests. Properties of IIDP. Dirichlet integral.
• Eulerian integrals.
Beta and gamma functions, their properties and relationship. Gauss representation formula. Euler’s reflection formula. Euler–Poisson integral. Digamma function.
• Double integrals.
Riemann sums. Double integrals over rectangles. Lower and upper Darboux sums. Darboux criterion. Properties of double integrals. Fubini’s theorem, reduction to iterated integrals. Double integrals over general regions. Change of variables in double integrals. Polar coordinate system.
• Triple integrals. Applications of double and triple integrals.
Fubini’s theorem, reduction to iterated integrals. Change of variables in triple integrals. Cylindrical and spherical coordinate systems. Calculating areas of domains, volumes of solids, areas of surfaces.
• Improper integrals. Multiple integrals.
Exhaustions. Improper double integrals. Volumes of the standard simplex and an n-ball. Gaussian integral.

#### 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 online 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.
• Python and Jupyter notebook tasks
• Homeworks

#### Interim Assessment

• Interim assessment (2 module)
ROUND(MIN( 0.25HomeWorks + 0.2Test + 0.2Exam + 0.2SeminarMark + 0.25Quizzes and ProgrammingTasks, 10))

#### Recommended Core Bibliography

• Calculus early transcendentals, Stewart, J., 2012