MU01012 Comprehensive Master Examination in Mathematics

Mathematical Institute in Opava
Summer 2018
Extent and Intensity
0/0. 6 credit(s). Type of Completion: zk (examination).
Guaranteed by
doc. RNDr. Zdeněk Kočan, Ph.D.
Mathematical Institute in Opava
Prerequisites (in Czech)
(MU00004 || MU01004 Mathematical Analysis IV ) && MU01005 Algebra I && MU01006 Algebra II && MU01007 Geometry && MU01008 Laboratory in Mathematics and && MU01009 Laboratory in Mathematics and && MU01001 Mathematical Analysis I && MU01002 Mathematical Analysis II && (MU00003 || MU01003 Mathematical Analysis III )
Course Enrolment Limitations
The course is offered to students of any study field.
Course objectives
The comprehensive master examination in mathematics contains basic parts of the calculus, linear algebra and geometry.
Syllabus
  • 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).
Literature
    recommended literature
  • M. Marvan. Algebra I. MÚ SU, Opava, 1999. URL info
  • M. Marvan. Algebra II. MÚ SU,, Opava, 1999. URL info
  • L. Klapka. Geometrie. MÚ SU, Opava, 1999. info
  • A. P. Mattuck. Introduction to Analysis. Prentice Hall, New Jersey, 1999. info
  • W. Rudin. Analýza v reálném a komplexním oboru. Academia, Praha, 1987. info
  • D. Krupka. Úvod do analýzy na varietách. SPN, Praha, 1986. info
  • M. Greguš, M. Švec, V. Šeda. Obyčajné diferenciálne rovnice. Alfa-SNTL, Bratislava-Praha, 1985. info
  • B. Budinský. Analytická a diferenciální geometrie. SNTL, Praha, 1983. info
  • G. Birkhoff, T. O. Bartee. Aplikovaná algebra. Alfa, Bratislava, 1981. info
  • D. K. Fadejev, I. S. Sominskij. Algebra. Fizmatgiz, Moskva, 1980. info
  • J. Kurzweil. Obyčejné diferenciální rovnice. SNTL, Praha, 1978. info
  • M. Spivak. Matematičeskij analiz na mnogoobrazijach. Mir, Moskva, 1968. info
  • V. Jarník. Diferenciální počet I. ČSAV, Praha, 1963. info
  • V. Jarník. Diferenciální počet II. ČSAV, Praha, 1963. info
  • V. Jarník. Integrální počet I. ČSAV, Praha, 1963. info
  • V. Jarník. Integrální počet II. ČSAV, Praha, 1963. info
  • I. G. Petrovskij. Lekcii ob uravnenijach s častnymi proizvodnymi. Mir, Moskva, 1961. info
Language of instruction
Czech
Further comments (probably available only in Czech)
The course can also be completed outside the examination period.
General note: původní hodnocení: souborná zkouška.
Teacher's information
This examination consists of two parts - writing and oral. There are two members in the examining board.
The course is also listed under the following terms Summer 1998, Summer 1999, Winter 1999, Summer 2000, Winter 2000, Summer 2001, Winter 2001, Summer 2002, Winter 2002, Summer 2003, Winter 2003, Summer 2004, Winter 2004, Summer 2005, Winter 2005, Summer 2006, Winter 2006, Summer 2007, Summer 2008, Summer 2009, Summer 2010, Summer 2011, Summer 2012, Summer 2013, Summer 2014, Summer 2015, Summer 2016, Summer 2017, Summer 2019.
  • Enrolment Statistics (Summer 2018, recent)
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