714-0369/04 – Mathematics IV (MIV)

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.

Systems of n ordinary linear differential equations of the first order for n functions: definition, representation at matrix form, methods of solution of systems of 2 equations for 2 functions, Euler method for homogeneous systems of n equations for n functions. Integral calculus of functions of several independent variables: two-dimensional integrals, three-dimensional integrals, vector analysis, line integral of the first and the second kind, surface integral of the first and second kind. Infinite series: number series, series of functions, power series.

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 James, G.: Advanced Modern Engineering Mathematics. Addison-Wesley, 1993. ISBN 0-201-56519-6

Seminar A student will be awarded 10-20 points for the attendance at the consultations. Moreover, a student can obtain 5 points for elaborating the home project. A maximum number of points awarded is 20. Examination The exam consists of two parts: I) Practical part - tests the ability of solving practical problems. (60 points is a maximum, but at least 20 points are necessary) II) Theoretical part - examines the understanding of the underlying theoretical concepts. (60 points is a maximum, but at least 20 points are necessary) Classifications Points obtained ECTS Grade 100-91 A 90-81 B 80-71 C 70-61 D 60-51 E 50-0 F Points obtained National grading scheme 100-86 1 (excellent) 85-66 2 (very good) 65-51 3 (good) 50-0 4 (failed) Topics for the theoretical part of the exam Systems of n ordinary linear differential equations of the first order for n functions: definition, matrix representation Elimination method for the systems of LDE Euler method for the homogeneous systems of LDE Two-dimensional integral on a rectangle Two-dimensional integral on a bounded subset of R2 Transformation - polar coordinates Geometrical and physical applications of the two-dimensional integral Three-dimensional integrals on a cube, on a bounded subset of R3 Transformation - cylindrical and spherical coordinates, Geometrical and physical applications of the three-dimensional integral Scalar field, gradient Vector field, divergence, rotation (curl) Line integral of the first and of the second kind Green´s theorem Path independence for the line integral, potential Geometrical and physical applications of the line integral Infinite number series Necessary condition for convergence Geometric series Harmonic series, generalized harmonic series, Leibniz series Infinite series of functions, power series