MU16141 Comprehensive Bachelor 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. Marta Štefánková, Ph.D.
Mathematical Institute in Opava
Prerequisites (in Czech)
MU01001 Mathematical Analysis I && MU01002 Mathematical Analysis II && ( MU20003 Mathematical Analysis III || MU01003 Mathematical Analysis III ) && ( MU20004 Mathematical Analysis IV || MU01004 Mathematical Analysis IV ) && ( MU01005 Algebra I || MU01015 Algebra I ) && ( MU01006 Algebra II || MU01016 Algebra II ) && ( MU20007 Geometry || MU01007 Geometry || MU01017 Geometry ) && ( MU01021 Analysis in the Complex Domain || MU01022 Analysis in the Complex Domain ) && MU01008 Laboratory in Mathematics and && MU01009 Laboratory in Mathematics and && ( MU20009 Probability and Statistics I || MU01133 Probability and Statistics ) && ( MU20010 Numerical Methods || MU01136 Numerical Methods )
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
The comprehensive bachelor examination in mathematics contains basic parts of the calculus, linear algebra, and probability theory.
Syllabus
  • REQUIREMENTS FOR THE COMPREHENSIVE EXAMINATION IN MATHEMATICS - Bachelor level
    (for study field Mathematics - General Mathematics)
    1. Sets and mappings, binary relations (set operations, image, preimage, surjections, injections and 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 matrix, basis change matrices, examples of vector spaces and linear maps).
    4. Scalar 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 a linear operator on a finite dimensional vector space (eigenvalues, first and second - Jordan - decomposition of a linear operator, orthogonal and symmetric operators on real inner product spaces 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 all these).
    9. Basic notions of topology (open sets, interior, exterior, boundary and closure of a set, continuity and limit of maps, compactness, connectedness, metric topologies, Euclidean space topology, examples of topological spaces, of continuous and discontinuous maps).
    10. Domain of real numbers (algebraic and topological properties).
    11. Sequences and series (sequences and series of real numbers, absolutely and conditionally convergent series, sequences and series of functions, pointwise and uniform convergence, power series, Taylor series, Fourier series, applications to solutions of differential equations).
    12. Functions of one and several real variables (continuity and limit, basic theorems on continuity, examples of continuous and discontinuous functions).
    13. Derivatives of functions of one or several real variables, partial and directional derivatives (basic properties of derivatives, basic theorems on derivatives).
    14. Derivatives of higher order, Taylor polynomial (Taylor theorem for functions of one or several real variables, applications).
    15. Derivatives of mappings of Euclidean spaces (basic properties of derivatives, chain rule, derivative of inverse function, implicit function theorem).
    16. Extrema of functions of one or several real variables, constrained extrema.
    17. Integrals of functions of one or several real variables (basic theorems, applications of the integral in geometry and physics, improper integrals).
    18. Computations of integrals (relationship between the integral and the primitive, Fubini's theorem, change of variable theorem).
    19. Ordinary differential equations (existence and uniqueness theorems, method of successive approximations, elementary solution methods).
    20. Systems of linear differential equations of first order (properties of solutions, variation of constants, elementary solution methods for systems with constant coefficients, application to the linear system of higher order.
    21. Curves in three-dimensional Euclidean space (curve, Frenet frame, curvature and torsion, Frenet-Serret formulae).
    22. Differential forms (algebra of differential forms on a manifold, theorem about local exactness of closed differential forms).
Literature
    recommended literature
  • M. Marvan. Algebra I. MÚ SU, Opava, 1999. URL info
  • M. Marvan. Algebra II. MÚ SU,, Opava, 1999. URL 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 2008, Summer 2009, Summer 2010, Summer 2011, Summer 2012, Summer 2013, Summer 2014, Summer 2015, Summer 2016, Summer 2017, Summer 2019, Summer 2020, Summer 2021, Summer 2022.
  • Enrolment Statistics (Summer 2018, recent)
  • Permalink: https://is.slu.cz/course/sumu/summer2018/MU16141