Course detail
Mathematical Analysis
FSI-UMA-A Acad. year: 2021/2022 Winter semester
The course provides an introduction to the theory of multiple, curve, 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.
Supervisor
Department
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.
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: Project consisting of assigned problems. Active participation in seminars (unless the student attends the course in the form of consultations).
Examination: The exam has written and oral part. 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 final grade reflects the results of the written and oral part of the exam.
Language of instruction
English
Aims
The goal of the course is to acquaint the students with the basics of multiple, curve, and surface integrals, Taylor and Fourier series. The course also aims to explaining basic notions and methods of solving ordinary 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.
The study programmes with the given course
Programme N-ENG-A: Mechanical Engineering, Master's
branch ---: no specialisation, 7 credits, compulsory
Type of course unit
Lecture
39 hours, optionally
Teacher / Lecturer
Syllabus
1. Double integrals. Fubini's theorem. Change of variables. Applications.
2. Triple integrals. Fubini's theorem. Change of variables. Applications.
3. Vector calculus. Curves. Curve (line) integrals. Applications. Green's theorem. Potential function.
4. Surfaces. Surface integrals. Applications. Divergence theorem. Stokes's theorem.
5. Series of real numbers, function series – revision. Power series.
6. Taylor series. Power series expansions. Trigonometric Fourier series.
7. First order ordinary differential equations (ODE) – revision. Higher order ODEs, basic notions. Structure of the solution set of linear equations.
8. Method of solving of higher order linear ODEs. Method of variation of parameters, method of undetermined coefficients.
9. Systems of first order ODEs, basic notions. Structure of the solution set of linear systems.
10. Methods of solving of homogeneous systems of linear ODEs with constant coefficients.
11. Methods of solving of non-homogeneous systems of linear ODEs. Method of variation of parameters, method of undetermined coefficients.
12. Autonomous systems of ODEs. Stability of solutions of ODEs and their systems.
13. The method of power series for ODEs. The Laplace transform and its use in ODEs.
Exercise
26 hours, compulsory
Teacher / Lecturer
Syllabus
1. Double integrals.
2. Triple integrals.
3. Curve (line) integrals.
4. Surface integrals.
5. Curves and surface integrals – continuation.
6. Series of real numbers, function series – revision. Power series.
7. Taylor series. Power series expansions.
8. Trigonometric Fourier series.
9. Analytical methods of solving of higher order linear ODEs.
10. Analytical methods of solving of higher order linear ODEs – continuation.
11. Analytical methods of solving of homogeneous systems of first order linear ODEs.
12. Analytical methods of solving of homogeneous systems of first order linear ODEs – continuation.
13. Analytical methods of solving of non-homogeneous systems of first order linear ODEs.