# 714-0267/01 – Mathematics II (BcM2)

 Gurantor department Department of Mathematics and Descriptive Geometry Credits 5 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 1999/2000 Year of cancellation Intended for the faculties FAST Intended for study types Bachelor
Instruction secured by
TUZ006 RNDr. Michaela Bobková, Ph.D.
DUB02 RNDr. Viktor Dubovský, Ph.D.
JAR71 Mgr. Marcela Jarošová
KRC20 RNDr. Břetislav Krček, CSc.
KRC76 Mgr. Jiří Krček
KRC23 Mgr. Jitka Krčková, Ph.D.
KRE40 doc. RNDr. Pavel Kreml, CSc.
OTI73 Mgr. Petr Otipka
POL12 RNDr. Jiří Poláček, 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

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 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:

Kreml, Pavel: Mathematics II, VŠB – TUO, Ostrava 2005, ISBN 80-248-0798-X

### 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. ISBN 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 - 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 http://mdg.vsb.cz (in Czech language)

Requiirements are in the part passing the course.

### Prerequisities

Subject codeAbbreviationTitleRequirement
714-0266 BcM1 Mathematics I Compulsory

### 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

Conditions for completion are defined only for particular subject version and form of study

### Occurrence in study plans

Academic yearProgrammeField of studySpec.FormStudy language Tut. centreYearWSType of duty
2017/2018 (B3607) Civil Engineering P Czech Ostrava 1 Compulsory study plan
2016/2017 (B3607) Civil Engineering P Czech Ostrava 1 Compulsory study plan
2015/2016 (B3607) Civil Engineering P Czech Ostrava 1 Compulsory study plan
2014/2015 (B3607) Civil Engineering P Czech Ostrava 1 Compulsory study plan
2013/2014 (B3607) Civil Engineering P Czech Ostrava 1 Compulsory study plan
2012/2013 (B3607) Civil Engineering P Czech Ostrava 1 Compulsory study plan
2011/2012 (B3607) Civil Engineering P Czech Ostrava 1 Compulsory study plan
2010/2011 (B3607) Civil Engineering P Czech Ostrava 1 Compulsory study plan
2009/2010 (B3607) Civil Engineering P Czech Ostrava 1 Compulsory study plan
2008/2009 (B3607) Civil Engineering P Czech Ostrava 1 Compulsory study plan
2007/2008 (B3607) Civil Engineering (3607R999) Společné studium FAST P Czech Ostrava 1 Compulsory study plan
2006/2007 (B3502) Architecture and Construction (3501R011) Architecture and Construction P Czech Ostrava 1 Compulsory study plan
2006/2007 (B3607) Civil Engineering (3607R999) Společné studium FAST P Czech Ostrava 1 Compulsory study plan
2005/2006 (B3607) Civil Engineering (3607R999) Společné studium FAST P Czech Ostrava 1 Compulsory study plan
2005/2006 (B3651) Stavební inženýrství (3651R999) Společné studium FAST P Czech Ostrava 1 Compulsory study plan
2005/2006 (B3502) Architecture and Construction (3501R011) Architecture and Construction P Czech Ostrava 1 Compulsory study plan
2004/2005 (B3651) Stavební inženýrství (3651R999) Společné studium FAST P Czech Ostrava 1 Compulsory study plan
2004/2005 (B3502) Architecture and Construction (3501R011) Architecture and Construction P Czech Ostrava 1 Compulsory study plan
2003/2004 (B3651) Stavební inženýrství (3651R999) Společné studium FAST P Czech Ostrava 1 Compulsory study plan
2003/2004 (B3502) Architecture and Construction (3501R011) Architecture and Construction P Czech Ostrava 1 Compulsory study plan

### Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner