Gurantor department | Department of Mathematics | Credits | 4 |

Subject guarantor | doc. RNDr. Pavel Kreml, CSc. | Subject version guarantor | doc. RNDr. Pavel Kreml, CSc. |

Study level | undergraduate or graduate | Requirement | Compulsory |

Year | 1 | Semester | summer |

Study language | Czech | ||

Year of introduction | 2019/2020 | Year of cancellation | |

Intended for the faculties | FAST | Intended for study types | Bachelor |

Instruction secured by | |||
---|---|---|---|

Login | Name | Tuitor | Teacher giving lectures |

DUB02 | RNDr. Viktor Dubovský, Ph.D. | ||

KRE40 | doc. RNDr. Pavel Kreml, CSc. |

Extent of instruction for forms of study | ||
---|---|---|

Form of study | Way of compl. | Extent |

Part-time | Credit and Examination | 16+0 |

Mathematics is essential part of education on technical universities.
It should be considered rather the method in the study of technical
courses than a goal. Thus the goal of mathematics is train logical
reasoning than mere list of mathematical notions, algorithms and
methods.
Students should learn how to
analyze problems,
distinguish between important and unimportant,
suggest a method of solution,
verify each step of a method,
generalize achieved results,
analyze correctness of achieved results with respect to given conditions,
apply these methods while solving technical problems,
understand that mathematical methods and theoretical advancements
outreach the field mathematics.

Lectures

Individual consultations

Tutorials

Other activities

Integral calculus of function of one real variable: the indefinite and definite
integrals, properties of the indefinite and definite integrals, application in
the geometry and physics. Differential calculus of functions of several
independent variables. Ordinary differential equations of the first and the
second order.

Kreml, Pavel: Mathematics II, VŠB – TUO, Ostrava 2005, ISBN 80-248-0798-X
http://mdg.vsb.cz/portal/en/Mathematics2.pdf

Harshbarger, R.J.-Reynolds, J.J.: Calculus with Applications.
D.C.Heath and Company, Lexington1990, ISBN 0-669-21145-1
James, G.: Modern Engineering Mathematics. Addison-Wesley, 1992.
ISBN 0-201-1805456

Passing the course, requirements
Course-credit
-participation on tutorials is obligatory,
-elaborate programs,
Point classification: 5-20 points.
Exam
Practical part of an exam is classified by 0 - 60 points. Practical part is successful if student obtains at least
25 points.
Theoretical part of the exam is classified by 0 - 20 points. Theoretical part is successful if student obtains
at least 5 points.
Point quantification in the interval 100 - 86 85 - 66 65 - 51 50 - 0
National grading scheme excellent very good satisfactory failed
1 2 3 4
List of theoretical questions:
1. Antiderivatives, primitive functions.
2. Integration by substitution.
3. Integration by parts.
4. Integration of rational functions, polynomials in denominator have different real roots.
5. Integration of rational functions, polynomials in denominator have k-fold roots.
6. Integration of rational functions, polynomials in denominator have complex conjugate roots.
7. Integration of functions of the type R(sin x)cos x.
8. Integration of functions of the type R(cos x)sin x.
9. Integration of functions of the type sin^m x cos^n x.
10. Integration of functions of the type R(sin x, cos x). Universal trigonometric substitution.
11. Newton-Leibnitz theorem for calculation of definite integrals.
12. Integration by substitution for definite integrals.
13. Integration by parts for definite integrals.
14. Application of definite integrals - area. Explicit and parametric representation.
15. Application of definite integrals - arc length. Explicit and parametric representation.
16. Application of definite integrals - volume of a solid of revolution. Explicit and parametric representation.
17. Application of definite integrals - lateral surface of a solid of revolution. Explicit and parametric representation.
18. Definition of functions of n real variables.
19. Partial derivatives.
20. Geometrical meaning of partial derivatives of functions of two variables.
21. Equation of a tangent plane to a graph of functions of two variables.
22. Equation of a normal to a graph of functions of two variables.
23. Second order partial derivative.
24. Total differential of functions of more variables.
25. Necessary condition for existence of extremum of functions of more variables, Fermat theorem.
26. Sufficient condition for existence of extremum of functions of more variables.
27. Implicit functions, derivation of implicit functions.
28. Ordinary differential equations.
29. General and particular solution of differential equations.
30. Separable differential equation, general form and method of solution.
31. Homogeneous differential equation, general form and method of solution.
32. Linear differential equation, general form and method of solution.
33. Linear differential equation, method of variation of arbitrary constant.
34. Linearly independent functions, Wronskian.
35. 2nd order linear differential equations with constant coefficients, general form, method of solution.
36. 2nd order linear differential equations with constant coefficients, characteristic equation.
37. LDE, independent solutions for different real roots of characteristic equation.
38. LDE, independent solutions for 2-fold real roots of characteristic equation.
39. LDE, independent solutions for complex conjugate roots of characteristic equation.
40. 2nd order linear differential equations with constant coefficients, method of variation of arbitrary constants.
41. 2nd order LDE, write a particular solution for a special right-hand side f(x)=Pm(x).
42. 2nd order LDE, write a particular solution for a special right-hand side f(x)=e^(ax) Pm(x).
43. 2nd order LDE, write a particular solution for a special right-hand side f(x)=x^2 e^x cos3x.
44. 2nd order LDE, write a particular solution for a special right-hand side f(x)=x e^x sin3x.
45. 2nd order LDE, write a particular solution for a special right-hand side f(x)=x sin3x.
46. 2nd order LDE, write a particular solution for a special right-hand side f(x)=x e^(5x).
47. 2nd order LDE, write a particular solution for a special right-hand side f(x)=e^2x sin2x.
48. 2nd order LDE, principle of superposition.

http://www.studopory.vsb.cz
(Česky)
http://mdg.vsb.cz/portal/en/Mathematics2.pdf

At least 70% attendance at the exercises. Absence, up to a maximum of 30%, must be excused and the apology must be accepted by the teacher (the teacher decides to recognize the reason for the excuse).

Subject has no prerequisities.

Subject has no co-requisities.

Syllabus of lecture
Integral calculus of functions of one variable. Antiderivatives and indefinite integral. Integration of elementary
functions.
Integration by substitutions, integration by parts.
Integration of rational functions.
Definite integral and methods of integration.
Geometric and physical application of definite integrals.
Differential calculus of functions of two or more real variables. Functions of two or more variables, graph, partial
derivatives of the 1-st and higher order.
Total differential of functions of two variables, tangent plane and normal to a surface, derivation of implicit
functions.
Extrema of functions.
Ordinary differential equations. General, particular and singular solutions. Separable and homogeneous equations.
Linear differential equations of the first order, method of variation of arbitrary constant. Exact differential
equations.
2nd order linear differential equations with constant coefficients, linearly independent solutions, Wronskian,
fundamental system of solutions.
2nd order LDE with constant coefficients - method of variation of arbitrary constants.
2nd order LDE with constant coefficients - method of undetermined coefficients.

Task name | Type of task | Max. number of points
(act. for subtasks) | Min. number of points |
---|---|---|---|

Credit and Examination | Credit and Examination | 100 (100) | 51 |

Credit | Credit | 20 | 5 |

Examination | Examination | 80 (80) | 30 |

Písemná zkouška | Written examination | 60 | 25 |

Ústní zkouška | Oral examination | 20 | 5 |

Show history

Academic year | Programme | Field of study | Spec. | Zaměření | Form | Study language | Tut. centre | Year | W | S | Type of duty | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

2021/2022 | (B0732A260001) Civil Engineering | K | Czech | Ostrava | 1 | Compulsory | study plan | |||||

2020/2021 | (B0732A260001) Civil Engineering | K | Czech | Ostrava | 1 | Compulsory | study plan | |||||

2019/2020 | (B0732A260001) Civil Engineering | K | Czech | Ostrava | 1 | Compulsory | study plan |

Block name | Academic year | Form of study | Study language | Year | W | S | Type of block | Block owner |
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