714-0867/01 – Mathematics 2 (Math 2)

Gurantor departmentDepartment of Mathematics and Descriptive GeometryCredits5
Subject guarantorRNDr. Petr Volný, Ph.D.Subject version guarantorRNDr. Petr Volný, Ph.D.
Study levelundergraduate or graduate
Study languageEnglish
Year of introduction2009/2010Year of cancellation
Intended for the facultiesHGF, FMT, FEI, USP, FAST, FS, FBIIntended for study typesBachelor
Instruction secured by
LoginNameTuitorTeacher giving lectures
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

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.

Teaching methods

Lectures
Individual consultations
Tutorials
Other activities

Summary

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.

Compulsory literature:

Kreml, Pavel: Mathematics II, Ostrava 2005, 80-248-0798-X.

Recommended literature:

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.

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 - 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.

E-learning

Další požadavky na studenta

There are no other requirements on students.

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: 2014/2015 Winter semester)
Task nameType of taskMax. number of points
(act. for subtasks)
Min. number of points
Exercises evaluation and Examination Credit and Examination 100 (100) 51
        Exercises evaluation Credit 20  5
        Examination Examination 80 (80) 30
                Písemná zkouška Written examination 60  25
                Ústní zkouška Oral examination 20  5
Mandatory attendence parzicipation:

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Occurrence in study plans

Academic yearProgrammeField of studySpec.FormStudy language Tut. centreYearWSType of duty

Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner
ECTS - MechEng - Bachelor Studies 2019/2020 Full-time English Choice-compulsory 301 - Study and International Office stu. block
FMST 2019/2020 Full-time English Compulsory 601 - Study Office stu. block
ECTS - MechEng - Bachelor Studies 2018/2019 Full-time English Choice-compulsory 301 - Study and International Office stu. block
FBI_ECTS 2018/2019 Full-time Czech Optional 010 - Faculty of Safety Engineering - Dean's Office stu. block
FMMI 2018/2019 Full-time English Compulsory 601 - Study Office stu. block
ECTS FCE Bc-Mgr 2018/2019 Full-time English Choice-compulsory 200 - Faculty of Civil Engineering - Dean's Office stu. block
ECTS FMG-VSB 2018/2019 Full-time English Optional 501 - Study Office stu. block
ECTS - MechEng - Bachelor Studies 2017/2018 Full-time English Choice-compulsory 301 - Study and International Office stu. block
FBI_ECTS 2017/2018 Full-time English Optional 010 - Faculty of Safety Engineering - Dean's Office stu. block
USP 2017/2018 Full-time English Optional USP - University Study Programmes stu. block
FBI_ECTS 2017/2018 Full-time Czech Optional 010 - Faculty of Safety Engineering - Dean's Office stu. block
ECTS FCE Bc-Mgr 2017/2018 Full-time English Choice-compulsory 200 - Faculty of Civil Engineering - Dean's Office stu. block
FMMI 2017/2018 Full-time English Compulsory 601 - Study Office stu. block
ECTS FMG-VSB 2017/2018 Full-time English Optional 501 - Study Office stu. block
ECTS - MechEng - Bachelor Studies 2016/2017 Full-time English Choice-compulsory 301 - Study and International Office stu. block
FMMI 2016/2017 Full-time English Compulsory 601 - Study Office stu. block
ECTS FCE Bc-Mgr 2016/2017 Full-time English Choice-compulsory 200 - Faculty of Civil Engineering - Dean's Office stu. block
FBI_ECTS 2016/2017 Full-time Czech Optional 010 - Faculty of Safety Engineering - Dean's Office stu. block
USP 2016/2017 Full-time English Optional USP - University Study Programmes stu. block
ECTS FMG-VSB 2016/2017 Full-time English Optional 501 - Study Office stu. block
USP 2015/2016 Full-time English Optional USP - University Study Programmes stu. block
V - ECTS - VSB 2015/2016 Full-time English Optional 401 - Study Office stu. block
FMMI 2015/2016 Full-time English Compulsory 601 - Study Office stu. block
ECTS - MechEng - Bachelor Studies 2015/2016 Full-time English Choice-compulsory 301 - Study and International Office stu. block
FBI_ECTS 2015/2016 Full-time Czech Optional 010 - Faculty of Safety Engineering - Dean's Office stu. block
ECTS FMG - VSB 2015/2016 Full-time Czech Optional 501 - Study Office stu. block
V - ECTS - VSB 2014/2015 Full-time Czech Optional 401 - Study Office stu. block
FBI_ECTS 2014/2015 Full-time Czech Optional 010 - Faculty of Safety Engineering - Dean's Office stu. block
FMMI_N 2014/2015 Full-time Czech Compulsory 601 - Study Office stu. block
USP 2014/2015 Full-time Czech Optional USP - University Study Programmes stu. block
O - ECTS FMG - Bc. 2014/2015 Full-time Czech Optional 501 - Study Office stu. block
ECTS - MechEng 2014/2015 Full-time Czech Choice-compulsory 301 - Study and International Office stu. block
V - ECTS - VSB 2013/2014 Full-time Czech Optional 401 - Study Office stu. block
O - ECTS FMG - Bc. 2013/2014 Full-time Czech Optional 501 - Study Office stu. block
USP 2013/2014 Full-time Czech Optional USP - University Study Programmes stu. block
FMMI 2013/2014 Full-time Czech Compulsory 601 - Study Office stu. block
ECTS - MechEng 2013/2014 Full-time Czech Choice-compulsory 301 - Study and International Office stu. block
ECTS - MechEng 2012/2013 Full-time Czech Choice-compulsory 301 - Study and International Office stu. block
O - ECTS FMG - Bc. 2012/2013 Full-time Czech Optional 501 - Study Office stu. block
FMMI 2012/2013 Full-time Czech Compulsory 601 - Study Office stu. block
USP 2012/2013 Full-time Czech Optional USP - University Study Programmes stu. block