# 714-0369/03 – Mathematics IV (MIV)

 Gurantor department Department of Mathematics and Descriptive Geometry Credits 5 Subject guarantor doc. RNDr. Jarmila Doležalová, CSc. Subject version guarantor Mgr. Arnošt Žídek, Ph.D. Study level undergraduate or graduate Study language English Year of introduction 2015/2016 Year of cancellation 2019/2020 Intended for the faculties FS Intended for study types Follow-up Master
Instruction secured by
DOL30 doc. RNDr. Jarmila Doležalová, CSc.
LUN44 Mgr. Milena Luňáčková, Ph.D.
VAV14 RNDr. Eva Vavříková, Ph.D.
ZID76 Mgr. Arnošt Žídek, Ph.D.  Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 2+3
Part-time Credit and Examination 16+4

### Subject aims expressed by acquired skills and competences

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

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.

### Compulsory 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

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

### Way of continuous check of knowledge in the course of semester

Seminar Attending the seminar is required, only 20 % of missed lessons will be excused. It is also necessary to elaborate the home project in a form specified by the lecturer and a student is awarded 5 points for the project. Several tests will be carried out during the semester, a student can acquire up to 15 points. 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

### E-learning

http://www.studopory.vsb.cz http://mdg.vsb.cz

### Other requirements

No more requirements are put on the student.

### Prerequisities

Subject has no prerequisities.

### Co-requisities

Subject has no co-requisities.

### Subject syllabus:

Syllabus of lecture 1 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 2 Euler method for homogeneous systems of n equations for n functions 3 Integral calculus of functions of several independent variables: two-dimensional integrals on coordinate rectangle, on bounded subset of R2 4 Transformation - polar coordinates, geometrical and physical applications 5 Three-dimensional integrals on coordinate cube, on bounded subset of R3 6 Transformation - cylindrical and spherical coordinates, geometrical and physical applications 7 Vector analysis, gradient 8 Divergence, rotation 9 Line integral of the first and of the second kind 10 Green´s theorem, potential 11 Geometrical and physical applications 12 Infinite number series 13 Infinite series of functions, power series Syllabus of seminar 1 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 2 Euler method for homogeneous systems of n equations for n functions 3 Euler method for homogeneous systems of n equations for n functions, test 4 Integral calculus of functions of several independent variables: two-dimensional integrals on coordinate rectangle, on bounded subset of R2 5 Transformation - polar coordinates 6 Geometrical and physical applications 7 Three-dimensional integrals on coordinate cube, on bounded subset of R3 8 Transformation - cylindrical and spherical coordinates 9 Geometrical and physical applications, test 10 Vector analysis, gradient 11 Divergence, rotation 12 Line integral of the first kind 13 Line integral of the second kind, test 14 Geometrical and physical applications

### 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.ZaměřeníFormStudy language Tut. centreYearWSType of duty ### Occurrence in special blocks

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