Selected chapters in mathematics

Lecturer:

Tutor:

Supervisor:

Department of Biomedical Technology

Synopsis:

Overview of mathematics as a scientific branch and its historical development. Partial differential equations, wave equation. Introduction into complex analysis.

Requirements:

Assessment:

Maximum 3 absences during the semester for serious reasons like sickness, injury etc. (medical certificate required).

Minimum 50% (i.e. 10 pts) evaluation at each of the 2 tests, each test consisting of 4 tasks, a task evaluated max. 5 pts each (the tests are taken in 6th and 13th week of the semester).

Exam:

1. Assessment recorded in „KOS? and signed by respective teacher in student?s “Index?,

2. Minimum 50% evaluation at the exam test. Exam test comprises of 10 tasks, a single task evaluated max 10% each.

Evaluation scale: less than 50% - F, 50-59% - E, 60-69% - D, 70-79% - C, 80-89% - B, 90-100% - A.

Syllabus of lectures:

1. What is mathematics?

2. Numbers and arithmetics.

3. Principles of reasoning

4. Calculus. Mathematics of movement.

5. Geometry and we

6. What has it all in common? Algebra and abstract systems

7. Topology - abstraction of a strukture

8. Differential equations as a model of physical phenomena

9. Topology of n-dimensional real space, classes of continuously differentiable functions

10. Introduction to partial differential equations

11. Wave equation. D´Alambert´s solution, Fourier´s solution

12. Laplace and Poissson equation

13. Heat transfer equation

14. Numerical solution of PDEs.

Syllabus of tutorials:

1. Set of complex numbers, functions of a complex variable

2. Limit a derivative

3. Cauchy-Riemann conditions

4. Holomorphic functions

5. Elementary and multivalue functions - an overview

6. Curved integral, Cauchy theorem

7. Expansions of a holomorphic function in a power or Laurent series

8. Isolated singular points, residues

9. Residual theorem

10. Intergral calculation using residual theorem

11.-14. Practical exercises

Study Objective:

The goal of the subject is to present mathematics as a science, to describe its means and structure, fundamentals of complex analysis, the residual theorem, Fourier transform, partial differential equations, wave equation.

Study materials:

[1] Devlin, K.: The Language of Mathematics: Making the Invisible Visible. Holt Paperbacks, London, 1998;

[2] Ahlfors, L.V.: Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable. 3rd ed. McGraw-Hill, New York, 1979;

[3] Strauss, W. A. Partial Differential Equations: An Introduction. Wiley, New York, 1992

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans:

• Navazující magisterský studijní obor Přístroje a metody pro biomedicínu - prezenční v angličtině (compulsory course)