Czech

doc. Mgr. Jaroslav Hrdina, Ph.D.

Institute of Mathematics

Students get acquainted with a broad range of mathematical concepts occurring in physical applications, both from mathematical analysis and from algebra. They will be made familiar with derivatives and partial derivatives and their use when investigating extremes. A further topic are indefinite, definite, and more-dimensional integrals in the sense of Riemann and of Lebesgue. A next part of the programme are functions of a complex variable. Last but not least, students will revise important notions from linear algebra.

Linear algebra, differential and integral calculus.

The course is taught through lectures explaining the basic principles and theory of the discipline.

The course is finished by an oral examination. The examiner verifies the knowledge of definitions, theorems, and algorithms and the ability of their use in concrete applications.

The aim of the subject is a summarization, extension, and enlargement of knowledge of mathematics from bachelor´s and master´s studies with a view to applications, especially in physical engineering.

Attendance at lectures is recommended. The lessons are planned on the basis of a weekly schedule. It is possible to study individually according to the recommended literature with the use of consultations.

Programme CEITEC-AMN-EN-K: Advanced Materials and Nanosciences, Doctoral, recommended course

Programme CEITEC-AMN-CZ-K: Advanced Materials and Nanosciences, Doctoral, recommended course

Programme CEITEC-AMN-CZ-P: Advanced Materials and Nanosciences, Doctoral, recommended course

Programme CEITEC-AMN-EN-P: Advanced Materials and Nanosciences, Doctoral, recommended course

Programme CEITEC-AMN-EN-Z: Advanced Materials and Nanosciences, Doctoral, recommended course

Programme D-FIN-P: Physical Engineering and Nanotechnology, Doctoral, recommended course

Programme D-FIN-K: Physical Engineering and Nanotechnology, Doctoral, recommended course

20 hours, optionally

(the choice will issue from specializations of of individual students)

Differential and integral calculus of functions of one variable
- Derivative, its geometrical and physical meaning
- Investigation of a function
- Taylor´s series
- Primitive function
- Evaluation of integrals by a substitution and by parts
- Riemann´s definite integral – geometrical and physical meaning
- Lebesgue´s integral
- Delta function and theory of distributuions

Differential and inegral calculus of functions of more variables
- Partial derivatives
- Total differential – applications in physics
- Extremes and saddle points
- Differential operators: gradient, divergence, curl, and Laplacian – applications in physics
- Geometrical and physical meaning of double and triple integral
- Transformation of coordinates – Jacobian
- Line integral – independence of the path of integration
- Surface integral
- Green´s, Gauss´, and Stokes´ theorems – applications in physics

Series
- Numerical series
- Functional series
- Fourier series

Analysis in complex domain
- Holomorphic functions
- Integral in complex domain, Cauchy´s theorem
- Taylor´s and Laurent´s series, theory of residues
- Hilbert transform

Differential equations
- Ordinary linear differential equations
- Systems of ordinary linear differential equations with constant coefficients
- Partial differential equations (Fourier method, method of characteristics)

Algebra
- Systems of linear equations
- Matrices and determinants
- Polynomials and solution of algebraic equations in complex domain
- Groups

Elements of functional analysis
- Metric, vector, unitary, and Hilbert spaces
- Spaces of functions
- Orthogonal systems, orthogonal (Fourier) transform

Elements of calculus of variations