Course detail

# Mathematical Analysis

The course provides an introduction to the theory of multiple, path, and surface integrals, series of functions and the theory of differential equations. These branches form the theoretical background in the study of many physical and engineering problems. The course deals with the following topics: Multiple integrals. Path integrals. Surface integrals. Power series. Taylor series. Fourier series. Ordinary differential equations and their systems. Higher order linear differential equations.

Pouze domovské fakulty

Learning outcomes of the course unit

Students will be familiarized with integral calculus of functions of more variables, path and surface integrals. They will be able to apply this knowledge in various engineering problems. They will master solving of problems of expansions of functions into power and Fourier series. Students will acquire knowledge of basic types of differential equations (DEs). They will be enlightened on DEs as mathematical models. They will acquire skills for analytical and numerical solving of problems involving DEs, as well as for qualitative analysis of DEs. After completing the course students will be equipped with knowledge that are needed for the study of physics, mechanics, and other technical disciplines.

Prerequisites

Linear algebra, differential calculus of functions of one and several variables, integral calculus of functions of one variable, sequences and series of real numbers, fundamentals of function series, first order ordinary differential equations.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

J. Stewart, Calculus, 7th Edition, Cengage Learning, 2012. (EN)
W. E. Boyce, R. C. DiPrima, Elementary Differential Equations, 9th Edition, Wiley, 2008. (EN)

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

Course-unit credit is awarded on the following conditions: Active participation in seminars. Fulfilment of all conditions of the running control of knowledge. At least half of total maximum points in both check tests is required. If a student does not fulfil this condition, the teacher can set an alternative one.

Examination: The examination tests the knowledge of definitions and theorems (especially the ability of their application to the given problems) and practical skills in solving particular problems. The exam has written and oral part.

The final grade reflects the results of the written and oral part of the exam, and the results achieved in seminars. Grading scheme is as follows: excellent (90-100 points), very good (80-89 points), good (70-79 points), satisfactory (60-69 points), sufficient (50-59 points), failed (0-49 points).

Language of instruction

English

Work placements

Not applicable.

Aims

The goal of the course is to acquaint the students with the basics of multiple, path, and surface integrals, Taylor and Fourier series. The course also aims to explaining basic notions and methods of solving ordinary and partial differential equations. The task is to show that knowledge of the theory of integrals, series, and differential equations can be utilized especially in physics and technical branches.

Specification of controlled education, way of implementation and compensation for absences

Attendance at lectures is recommended, attendance at seminars is obligatory and checked. Lessons are planned according to the week schedules. Absence from seminars may be compensated for by the agreement with the teacher.

Classification of course in study plans

• Programme N-ENG-A Master's, 1. year of study, winter semester, 7 credits, compulsory

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Syllabus

1. Multiple integrals. Fubini's theorem. Change of variables.

2. Curves. Path integrals. Path-independence. Green's theorem.

3. Surfaces. Surface integrals. Divergence theorem. Stokes's theorem.

4. Power series. Taylor series. Power series expansions.

5. Trigonometric Fourier series. Convergence and expansions of functions.

6. Systems of first order ordinary differential equations (ODE). Basic notions. Initial value problem. Structure of the solution set.

7. Methods of solving of homogeneous systems of linear ODEs with constant coefficients.

8. Nonhomogeneous systems of linear ODEs. The variation of constants method.

9. Higher order linear differential equations with constant coefficients. Method of solving.

10. Stability of solutions of ODEs and their systems. The Laplace transform and its use in ODEs. Boundary value problems.

11. Numerical methods for ODEs. The method of power series for ODEs.

12. Partial differential equations (PDE). Basic notions. Classification of second order PDEs.

13. Equations of mathematical physics. Methods of solving of PDEs.

Exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

1. Differentiation and integration - revision.

2. Multiple integrals.

3. Path integrals.

4. Surface integrals.

5. Power series.

6. Fourier series.

7. Analytical methods of solving of systems of linear ODEs.

8. Analytical methods of solving of systems of linear ODEs (continuation).

9. Analytical methods of solving of higher order linear ODEs.

10. Analytical methods of solving of higher order linear ODEs (continuation).

11. Stability of ODEs. The Laplace transform.

12. Numerical methods for ODEs.

13. Methods of solving of PDEs.