Mathematics for Applications (FSI-9MPA)

Academic year 2020/2021
Supervisor: doc. Mgr. Jaroslav Hrdina, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech or English
Aims of the course unit:
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.
Learning outcomes and competences:
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.
Course contents:
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.
Teaching methods and criteria:
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.
Controlled participation in lessons:
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.
Type of course unit:
    Lecture  10 × 2 hrs. optionally                  
Course curriculum:
    Lecture (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
Literature - fundamental:
1. G. B. Arfken, V. J. Walker: Mathematical Methods for Physicists (4th ed.). Academic Press, 1995.
2. G. B. Thomas, R. L. Finney: Calculus and Analytic Geometry, Addison Wesley 2003
3. A. A. Howard: Elementary Linear Algebra, Wiley 2002
Literature - recommended:
1. J. Nedoma: Matematika I., Cerm 2001
2. J. Karásek: Matematika II., Cerm 2002
3. J. Karásek, L. Skula: Lineární algebra. Teoretická část, Cerm 2005
4. J. Karásek, L. Skula: Lineární algebra. Cvičení, Cerm 2005
5. J. Karásek, L. Skula: Obecná algebra, Cerm 2008
6. M. Druckmüller, A. Ženíšek: Funkce komplexní proměnné, PC-Dir 2000
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
D4F-P full-time study D-FMI Physical and Materials Engineering F Physical Engineering DrEx 0 Recommended course 3 1 S