Gurantor department | Department of Mathematics | Credits | 5 |

Subject guarantor | RNDr. Petr Volný, Ph.D. | Subject version guarantor | RNDr. Petr Volný, Ph.D. |

Study level | undergraduate or graduate | ||

Study language | English | ||

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

Intended for the faculties | FMT, USP, FEI, FBI, FS, FAST, HGF | Intended for study types | Bachelor |

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

Login | Name | Tuitor | Teacher giving lectures |

VOL06 | RNDr. Petr Volný, Ph.D. |

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

Form of study | Way of compl. | Extent |

Full-time | Credit and Examination | 2+2 |

Goals and competence
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
analyse problems,
distinguish between important and unimportant,
suggest a method of solution,
verify each step of a method,
generalize achieved results,
analyse 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.
It is necessary to complete Mathematics 1 course or its equivalent.

Lectures

Individual consultations

Tutorials

Other activities

Mathematics II is connected with Mathematics I.
We have to stress that student can enrol in this course only if he passed the course Mathematics I or an equivalent course.
Mathematics II is divided in three parts:
1. Integral calculus of functions of one real variable,
2. Differential calculus of functions of two real variables,
3. Ordinary differential equations.

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

Doležalová Jarmila: Mathematics I, VŠB - TUO, Ostrava 2005, 80-248-0796-3.
Harshbarger, R.J.-Reynolds, J.J.: Calculus with Applications, D.C.Heath and Company, Lexington1990, 0-669-21145-1.
James, G.: Modern Engineering Mathematics, Addison-Wesley, 1992, 0-201-1805456.

Passing the course, requirements
Course-credit
-participation on tutorials is obligatory, 20% of absence can be apologized,
-elaborate programs,
-pass the written tests,
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 - 91 90 - 81 80 - 71 70 - 61 60 - 51 50 - 0
ECTS grade A B C D E F
Point quantification in the interval 100 - 86 85 - 66 65 - 51 50 - 0
National grading scheme excellent very good satisfactory failed
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 two 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 two 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 of the 1st order, general form and method of solution.
33. Linear differential equation of the 1st order, 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.

There are no other requirements on students.

Subject has no prerequisities.

Subject has no co-requisities.

Syllabus of lecture
1. Integral calculus of functions of one variable. Antiderivatives and indefinite integral. Integration of elementary functions.
2. Integration by substitutions, integration by parts.
3. Integration of rational functions.
4. Definite integral and methods of integration.
5. Geometric and physical application of definite integrals.
6. 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.
7. Total differential of functions of two variables, tangent plane and normal to a surface, derivation of implicit functions.
8. Extrema of functions.
9. Ordinary differential equations. General, particular and singular solutions. Separable and homogeneous equations.
10. Linear differential equations of the first order, method of variation of arbitrary constant. Exact differential equations.
11. 2nd order linear differential equations with constant coefficients, linearly independent solutions, Wronskian, fundamental system of solutions.
12. 2nd order LDE with constant coefficients - method of variation of arbitrary constants.
13. 2nd order LDE with constant coefficients - method of undetermined coefficients.
14. Reserve.
Syllabus of tutorial
1. Course of a function of one real variable.
2. Integration by a direct method. Integration by substitution.
3. Integration by substitution. Integration by parts.
4. Integration of rational functions.
5. 1st test (basic methods of integration). Definite integrals.
6. Applications of definite integrals.
7. Functions of more variables, domain, partial derivatives.
8. Equation of a tangent plane and a normal to a graph of functions of two variables. Derivation of implicit functions.
9. Extrema of functions. 2nd test (functions of two variables).
10. Differential equations, separable and homogeneous differential equations.
11. Linear differential equations of 1st order. Exact differential equations.
12. 2nd order linear differential equations with constant coefficients.
13. Method of undetermined coefficients. 3rd test (differential equations).
14. Reserve.

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. | Form | Study language | Tut. centre | Year | W | S | Type of duty |
---|

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

ECTS - MechEng - Bachelor Studies | 2020/2021 | Full-time | English | Choice-compulsory | 301 - Study and International Office | stu. block | |||

ECTS FCE Bc-Mgr | 2019/2020 | Full-time | English | Choice-compulsory | 200 - Faculty of Civil Engineering - Dean's Office | stu. block | |||

FMT-new subjects | 2019/2020 | Full-time | English | Optional | 600 - Faculty of Materials Science and Technology - Dean's Office | stu. block | |||

Additional Subjects | 2019/2020 | Full-time | English | Choice-compulsory | 301 - Study and International Office | stu. block |