Course: Comprehensive Master Examination in Mathematics

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Course title Comprehensive Master Examination in Mathematics
Course code MU/01012
Organizational form of instruction no contact
Level of course Master
Year of study not specified
Semester Summer
Number of ECTS credits 6
Language of instruction Czech
Status of course Compulsory
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
  • KOČAN Zdeněk, doc. RNDr. Ph.D.
Course content
REQUIREMENTS FOR THE COMPREHENSIVE EXAMINATION IN MATHEMATICS - Master Level (Mathematics - Mathematical Analysis) 1. Sets and mappings, binary relations (set operations, image, preimage, surjections, injections, bijections, equivalence, ordering). 2. Matrices and determinants (matrix operations, properties of determinants, rank of a matrix and its applications, eigenvalues of a matrix, Jordan normal form of a square matrix, examples). 3. Vector spaces, linear maps (linear dependence, bases, subspaces, expressing a linear map with respect to a basis, basis change matrix, examples of vector spaces and linear mappings). 4. Inner product and norm (bilinear and quadratic forms, normed vector spaces and inner product spaces, examples of such spaces, orthonormal systems of functions, trigonometric orthonormal systems). 5. Diagonalization of linear operators on a finite dimensional vector space (eigenvalues, first and second Jordan decomposition of linear map, orthogonal and symmetric operators on a real inner product space and their diagonalization, principal axes theorem, spectral theorem, canonical representation of a quadratic form). 6. Linear algebraic equations (homogeneous and non-homogeneous systems, solution methods). 7. Polynomials (fundamental theorem of algebra, methods of finding roots). 8. Basic algebraic structures (groups, rings, fields, vector spaces, examples of these). 9. Basic notions of topology (open sets, interior, exterior, boundary, closure, continuity and limits of a mapping, compactness, connectedness, metric topology, Euclidean space topology, examples of topological spaces, of continuous and discontinuous mappings). 10. The domain of real nukbers (algebraic and topological properties). 11. Sequences and series (sequences and series of real numbers, absolute and non-absolute convergence, sequences and series of functions, pointwise and uniform convergence, power series, Taylor series, Fourier series, applications to solving differential equations). 12. Functions of one and several real variables (continuity and limits, basic theorems on continuity, examples of continuous and discontinuous functions). 13. Derivatives of functions of one and several real variables, partial and directional derivatives (basic properties of derivatives, basic theorems on derivatives). 14. Derivatives of higher order, the Taylor polynomial (Taylor theorem for functions of one or several variables, applications). 15. Derivatives of mappings of Euclidean spaces (basic properties of derivatives, chain rule, derivatives of inverse functions, implicit function theorem). 16. Extrema of functions of one or several variables, constrained extrema. 17. Integration of functions of one or several variables (basic theorems on integrals, applications of integrals in geometry and physics, improper integral). 18. Computation of integrals (relation between integrals and primitives, Fubini's theorem, change of variable theorem). 19. Ordinary differential equations (existence and uniqueness theorems for solutions, method of successive approximations, elementary solution methods). 20. Systems of linear differential equations of first order (properties of solutions, variation of constants, elementary methods of solutions for systems with constant coefficients, applications to a single equation of higher order). 21. Basic types of partial differential equations (heat equation, wave equation, initial and boundary conditions, separation of variables, Fourier method, examples). 22. Integration of forms, contour and surface integrals, Stokes theorem. 23. Curves in three-dimensional Euclidean space (curves, Frenet's frame, curvature and torsion, Frenet-Serret formulas). 24. Differential forms (algebra of differential forms on a manifold, the theorem on local exactness of a closed differential form).

Learning activities and teaching methods
Recommended literature
  • A. P. Mattuck. Introduction to Analysis. Prentice Hall, New Jersey, 1999.
  • B. Budinský. Analytická a diferenciální geometrie. SNTL, Praha, 1983.
  • D. K. Fadejev, I. S. Sominskij. Algebra. Fizmatgiz, Moskva, 1980.
  • D. Krupka. Úvod do analýzy na varietách. SPN, Praha, 1986.
  • G. Birkhoff, T. O. Bartee. Aplikovaná algebra. Alfa, Bratislava, 1981.
  • I. G. Petrovskij. Lekcii ob uravnenijach s častnymi proizvodnymi. Mir, Moskva, 1961.
  • J. Kurzweil. Obyčejné diferenciální rovnice. SNTL, Praha, 1978.
  • L. Klapka. Geometrie. MÚ SU, Opava, 1999.
  • M. Greguš, M. Švec, V. Šeda. Obyčajné diferenciálne rovnice. Alfa-SNTL, Bratislava-Praha, 1985.
  • M. Marvan. Algebra II. MÚ SU,, Opava, 1999.
  • M. Marvan. Algebra I. MÚ SU, Opava, 1999.
  • M. Spivak. Matematičeskij analiz na mnogoobrazijach. Mir, Moskva, 1968.
  • V. Jarník. Diferenciální počet I. ČSAV, Praha, 1963.
  • V. Jarník. Diferenciální počet II. ČSAV, Praha, 1963.
  • V. Jarník. Integrální počet I. ČSAV, Praha, 1963.
  • V. Jarník. Integrální počet II. ČSAV, Praha, 1963.
  • W. Rudin. Analýza v reálném a komplexním oboru. Academia, Praha, 1987.

Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Mathematical Institute in Opava Mathematical Analysis (1-IVT) Mathematics courses 2 Summer