714-0867/01 – Mathematics 2 (Math 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.

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.

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.