Course detail

Mathematics for Applications

FSI-9MPAAcad. year: 2022/2023

The exposition will face across the traditional classification of mathematical branches so that it will respect students´ needs and options. It will be directed in an interactive form in order to respond to suggestions of students. A global view of problems enables students to see connections among apparently remote branches of mathematics.

Language of instruction

Czech

Mode of study

Not applicable.

Guarantor

doc. Mgr. Jaroslav Hrdina, Ph.D.

Department

Institute of Mathematics (ÚM)

Learning outcomes of the course unit

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.

Prerequisites

Linear algebra, differential and integral calculus.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

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

Assesment methods and criteria linked to learning outcomes

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.

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

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.

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

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.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

A. A. Howard: Elementary Linear Algebra, Wiley 2002 (EN)
G. B. Arfken, V. J. Walker: Mathematical Methods for Physicists (4th ed.). Academic Press, 1995. (EN)
G. B. Thomas, R. L. Finney: Calculus and Analytic Geometry, Addison Wesley 2003 (EN)

Recommended reading

J. Karásek, L. Skula: Lineární algebra. Cvičení, Cerm 2005
J. Karásek, L. Skula: Lineární algebra. Teoretická část, Cerm 2005
J. Karásek, L. Skula: Obecná algebra, Cerm 2008
J. Karásek: Matematika II., Cerm 2002
J. Nedoma: Matematika I., Cerm 2001
M. Druckmüller, A. Ženíšek: Funkce komplexní proměnné, PC-Dir 2000

Classification of course in study plans

  • Programme D-FIN-K Doctoral 1 year of study, summer semester, recommended course
  • Programme D-FIN-P Doctoral 1 year of study, summer semester, recommended course

Type of course unit

 

Lecture

20 hod., optionally

Teacher / Lecturer

doc. Mgr. Jaroslav Hrdina, Ph.D.

Syllabus

(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