MU22141 Comprehensive Bachelor Examination in Mathematics

Mathematical Institute in Opava
Summer 2016
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 ) && 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 - Bachelor level
    (for study fields: Mathematics - Applied Mathematics, Mathematical Methods in Economics, Applied Mathematics for Crises Management)
    1. Matrices and determinants (matrix operations, properties of determinants, rank of matrix, eigenvalues of matrix, Jordan normal form of a square matrix, examples).
    2. Vector spaces, linear maps (linear dependence, bases, subspaces, representing linear map with respect to a basis, examples of vector spaces and linear maps).
    3. Scalar product (bilinear and quadratic forms, inner product spaces, angle of subspaces, orthogonality, examples of inner product spaces, orthogonal matrices).
    4. Linear algebraic equations (homogeneous and non-homogeneous systems, solution methods, inetrative methods and computer aided solutions).
    5. Polynomials (methods of root finding, numerical solution of algebraic equations on computer).
    6. Sequences and series (of numbers and functions, convergence criteria).
    7. Functions of one or several real variables (coninuity and limit, basic theorems on continuity, uniform continuity, Lipschitz condition).
    8. Derivatives and differentials (definition and basic properties, directional and partial derivatives, derivatives and differentials of higher order).
    9. Extrema of functions of one or several real variables, constrained extrema.
    10. Taylor polynomial and Taylor series (in one or several real variables, Taylor remainder, Taylor series of functions of one complex variable).
    11. Elementary functions (trigonometric functions, exponential function, the logarithm in the real and complex domain).
    12. Riemann integral of a function of one or several real variables (definition and basic properties, contour integrals).
    13. Computation of integrals (relationship betweem integral and primitive, integration by parts, change of variable in the integral, integrating rational functions, integrals that can be reduced to integration of rational functions, Fubini's theorem, numerical integration).
    14. Implicit function theorem (solvin functional equations involving one or several unknon functions).
    15. Ordinary differential equations of first order (separation of variables, method of successive approximations, approximate solution methods, linear equations).
    16. Ordinary differential equations of higher order, systems of ordinary differential equations (properties of solution sets, solving equations with constant coefficients).
    17. Approximation and interpolation (least squares method, spline approximation).
    18. Basic properties of functions of complex variable (continuity and limit, derivative with respect to a complex variable, Cauchy-Riemann equations).
    19. Contour integrals and primitives of functions of a complex variable.
    20. Holomorphic functions (definition, basic properties, singularities).
    21. Basics of probability theory (probability, dependent and independent phenomena, conditional probability).
    22. Random variables (basic characteristics, relationships between random variables, law of large numbers).
    23. Basics of mathematical statistics (basic notions, estimation theory).
    24. Testing statistical hypothesis (examples of applications).
Literature
    recommended literature
  • M. Marvan. Algebra I. MÚ SU, Opava, 1999. URL info
  • M. Marvan. Algebra II. MÚ SU,, Opava, 1999. URL info
  • A. P. Mattuck. Introduction to Analysis. Prentice Hall, New Jersey, 1999. info
  • M. Jůza. Vybrané partie z matematické analýzy. MÚ SU, Opava, 1997. info
  • W. Rudin. Analýza v reálném a komplexním oboru. Academia, Praha, 1987. info
  • Z. Riečanová a kol. Numerické metody a matematická štatistika. Alfa, Bratislava, 1987. ISBN 063-559-87. info
  • G. Birkhoff, T. O. Bartee. Aplikovaná algebra. Alfa, Bratislava, 1981. info
  • K. Rektorys a spolupracovníci. Přehled užité matematiky. SNTL, Praha, 1968. info
  • V. Jarník. Diferenciální počet I. ČSAV, Praha, 1963. info
  • V. Jarník. Integrální počet I. ČSAV, Praha, 1963. 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 has 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 2017, Summer 2018, Summer 2019, Summer 2020, Summer 2021.
  • Enrolment Statistics (Summer 2016, recent)
  • Permalink: https://is.slu.cz/course/sumu/summer2016/MU22141