## Mathematics II

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Course Unit Code Number of ECTS Credits Allocated Type of Course Unit * Level of Course Unit * Year of Study * 230-0202/01 5 ECTS credits Compulsory First Cycle First Year Summer Semester Face-to-face Czech, English Course succeeds to compulsory courses of previous semester KRE40 doc. RNDr. Pavel Kreml, CSc. VOL06 RNDr. Petr Volný, Ph.D. 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. 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. 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. Kreml, Pavel: Mathematics II, VŠB – TUO, Ostrava 2005, ISBN 80-248-0798-X http://mdg.vsb.cz/portal/en/Mathematics2.pdf Vrbenská, H.: Základy matematiky pro bakaláře II. Skripta VŠB - TU, Ostrava 1998. Pavelka, L. – Pinka, P.: Integrální počet funkce jedné proměnné. Skripta VŠB- TU, Ostrava 1999. Vlček, J. – Vrbický, J.: Diferenciální rovnice. Skripta VŠB-TU, Ostrava 1997. Píšová, D. a kol.: Diferenciální počet funkcí více proměnných. Skripta VŠB, Ostrava 1989. Škrášek, J. a kol.: Základy aplikované matematiky I. a II. SNTL, Praha 1986. mdg.vsb.cz/portal/ www.studopory.vsb.cz/ 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 http://mdg.vsb.cz/portal/ http://www.studopory.vsb.cz/studijnimaterialy/MatematikaII/m2.pdf http://mdg.vsb.cz/portal/m2/PV_PracovniListyM2.pdf http://www.studopory.vsb.cz/studijnimaterialy/Sbirka_uloh/pdf/suzm.pdf Lectures, Individual consultations, Tutorials, Other activities 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