230-0202/08 – Mathematics II (BcM II)

 Gurantor department Department of Mathematics Credits 4 Subject guarantor RNDr. Petr Volný, Ph.D. Subject version guarantor RNDr. Petr Volný, Ph.D. Study level undergraduate or graduate Requirement Compulsory Year 1 Semester summer Study language English Year of introduction 2019/2020 Year of cancellation Intended for the faculties FAST Intended for study types Bachelor
Instruction secured by
KRE40 doc. RNDr. Pavel Kreml, CSc.
VOL06 RNDr. Petr Volný, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 2+2

Subject aims expressed by acquired skills and competences

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.

Teaching methods

Lectures
Individual consultations
Tutorials
Other activities

Summary

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.

Compulsory literature:

Hass, J.R.; Heil, C.E.; Bogacki, P.; Weir, M.D.: Thomas' Calculus, 15th Ed., Pearson, 2023. Volná, J.; Volný, P.: Worksheets for Mathematics II, VŠB-TUO, 2021; http://mdg.vsb.cz/portal/en/Mathematics2_worksheets.pdf

Recommended literature:

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. Kreml, Pavel: Mathematics II, VŠB – TUO, Ostrava 2005, http://mdg.vsb.cz/portal/en/Mathematics2.pdf

Way of continuous check of knowledge in the course of semester

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 - 86 85 - 66 65 - 51 51 - 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.

E-learning

http://www.studopory.vsb.cz (in Czech language) http://mdg.vsb.cz/portal/en/Mathematics2.pdf

Other requirements

At least 20% 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).

Prerequisities

Subject has no prerequisities.

Co-requisities

Subject has no co-requisities.

Subject syllabus:

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.

Conditions for subject completion

Full-time form (validity from: 2019/2020 Winter semester)
Min. number of pointsMax. počet pokusů
Credit and Examination Credit and Examination 100 (100) 51
Credit Credit 20  5
Examination Examination 80 (80) 30 3
Písemná zkouška Written examination 60  25
Ústní zkouška Oral examination 20  5
Mandatory attendence participation: 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).

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Conditions for subject completion and attendance at the exercises within ISP: Mandatory participation in the course is not required. Other conditions for subject completion will respect the individual needs of the student.

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Occurrence in study plans

Academic yearProgrammeBranch/spec.Spec.ZaměřeníFormStudy language Tut. centreYearWSType of duty
2024/2025 (B0732A260002) Civil Engineering P English Ostrava 1 Compulsory study plan
2023/2024 (B0732A260002) Civil Engineering P English Ostrava 1 Compulsory study plan
2022/2023 (B0732A260002) Civil Engineering P English Ostrava 1 Compulsory study plan
2021/2022 (B0732A260002) Civil Engineering P English Ostrava 1 Compulsory study plan
2020/2021 (B0732A260002) Civil Engineering P English Ostrava 1 Compulsory study plan
2019/2020 (B0732A260002) Civil Engineering P English Ostrava 1 Compulsory study plan

Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner