# 714-0868/01 – Mathematics 3 (Math 3)

 Gurantor department Department of Mathematics and Descriptive Geometry Credits 5 Subject guarantor Mgr. Arnošt Žídek, Ph.D. Subject version guarantor Mgr. Arnošt Žídek, Ph.D. Study level undergraduate or graduate Study language English Year of introduction 2010/2011 Year of cancellation 2019/2020 Intended for the faculties FS, FAST, FEI, FMT, FBI, USP, HGF Intended for study types Master, Bachelor
Instruction secured by
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+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 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 and Mathematics 2 courses or their equivalents.

### Teaching methods

Lectures
Individual consultations
Other activities

### Summary

Mathematics 3 is connected with Mathematics 1,2. We have to stress that student can enrol in this course only if he passed the course Mathematics 1 and 2 or an equivalent course. - Integral calculus of functions of more than one variable - Double and volume integral. Fubini's Theorem: integrating over regular regions. - Transformation of variables, polar, cylindrical and spherical coordinates. - Practical applications of double and volume integral. - Curves and their orientation, line integral of a scalar function and its geometrical applications. - Line integral of a vector function and its physical applications. - Path independence, Green's theorem.

### Compulsory literature:

http://mdg.vsb.cz/wiki/public/Double_Integral.pdf

### Recommended literature:

Neustupa J., Kračmar S.: Mathematics II. ČVUT, Praha 1998. http://www.studopory.vsb.cz/studijnimaterialy/MatematikaIII/Matematika3_obsah.pdf (in czech language)

### 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 - 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. Definition of the double integral over rectangular region. 2. Dirichlet's Theorem for calculation of the double integral over rectangular region. 3. Elementary region relative to the x-axis. 4. Elementary region relative to the y-axis. 5. Fubini's Theorem for calculation of the double integral over regular region. 6. Transformation of the double integral to the polar coordinates. 7. Regular mapping and its Jacobian. Derive Jacobian for the transformation to the polar coordinates. 8. Transformation to generalized polar coordinates. 9. Calculation of the volume of perpendicular cylinder over the region (using double integral). 10. Calculation of the size of a regular plane area (using double integral). 11. Calculation of the size of an area over the region (using double integral). 12. Calculation of coordinates of center of mass (using double integral). 13. Calculation of moments of inertia about the x- (y-) axis. (using double integral). 14. Definition of the volume integral on a rectangular region. 15. Dirichlet's Theorem for calculation of the volume integral on a rectangular region. 16. Elementary region relative to the xy-plane. 17. Elementary region relative to the xz-plane. 18. Elementary region relative to the yz-plane. 19. Fubini's Theorem for calculation of the volume integral on a regular region. 20. Calculation of the volume integral by transformation to the cylindrical coordinates. 21. Calculation of the volume integral by transformation to the spherical coordinates. 22. Derive Jacobian for the transformation to the cylindrical coordinates. 23. Derive Jacobian for the transformation to the spherical coordinates. 24. Calculation of the volume of regular 3-D region (using volume integral). 25. Calculation of static moments of 3-D region with about the coordinate planes (using volume integral). 26. Calculation of moments of inertia of 3-D region about the coordinate axis (using volume integral). 27. Calculation of the center of mass of 3-D region (using volume integral). 28. Equation of the curve in R^3. Positive and negative orientation of a closed curve. 29. Line integral of a scalar function and its calculation. 30. Geometrical meaning of a line integral of a scalar function. 31. Line integral of a vector function and its calculation. 32. Physical meaning of a line integral of a vector function. 33. Green's Theorem. 34. Calculation of the size of an area using line integral.

### E-learning

http://www.studopory.vsb.cz/studijnimaterialy/MatematikaIII/Matematika3_obsah.pdf (in czech language) 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:

Course description (weekly lessons): 1. Double integral over rectangular region. 2. Double integral over regular region. Fubini's Theorem. 3. Transformation of variables. Mapping and its Jacobian. Polar coordinates. 4. Practical applications of double integral. 5. Volume integral over rectangular region. 6. Volume integral over regular region. 7. Transformation to the cylindrical coordinates. 8. Transformation to the spherical coordinates. 9. Practical applications of volume integral. 10. Curves in R^3. Their equations and orientation of closed curves. 11. Line integral of a scalar function. 12. Line integral of a vector function. 13. Path independence. Green's Theorem. 14. Practical applications of line integrals of both kinds.

### Conditions for subject completion

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

### Occurrence in study plans

Academic yearProgrammeBranch/spec.Spec.ZaměřeníFormStudy language Tut. centreYearWSType of duty

### Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner
ECTS - MechEng - Bachelor Studies 2019/2020 Full-time English Choice-compulsory 301 - Study and International Office stu. block
ECTS - MechEng - Bachelor Studies 2018/2019 Full-time English Choice-compulsory 301 - Study and International Office stu. block
FBI_ECTS 2018/2019 Full-time Czech Optional 010 - Faculty of Safety Engineering - Dean's Office stu. block
FMMI 2018/2019 Full-time English Compulsory 601 - Study Office stu. block
ECTS FCE Bc-Mgr 2018/2019 Full-time English Choice-compulsory 200 - Faculty of Civil Engineering - Dean's Office stu. block
ECTS - MechEng - Bachelor Studies 2017/2018 Full-time English Choice-compulsory 301 - Study and International Office stu. block
FBI_ECTS 2017/2018 Full-time English Optional 010 - Faculty of Safety Engineering - Dean's Office stu. block
USP 2017/2018 Full-time English Optional USP - University Study Programmes stu. block
FBI_ECTS 2017/2018 Full-time Czech Optional 010 - Faculty of Safety Engineering - Dean's Office stu. block
ECTS FCE Bc-Mgr 2017/2018 Full-time English Choice-compulsory 200 - Faculty of Civil Engineering - Dean's Office stu. block
FMMI 2017/2018 Full-time English Compulsory 601 - Study Office stu. block
ECTS FMG-VSB 2017/2018 Full-time English Optional 501 - Study Office stu. block
ECTS - MechEng - Bachelor Studies 2016/2017 Full-time English Choice-compulsory 301 - Study and International Office stu. block
FMMI 2016/2017 Full-time English Compulsory 601 - Study Office stu. block
ECTS FCE Bc-Mgr 2016/2017 Full-time English Choice-compulsory 200 - Faculty of Civil Engineering - Dean's Office stu. block
FBI_ECTS 2016/2017 Full-time Czech Optional 010 - Faculty of Safety Engineering - Dean's Office stu. block
USP 2016/2017 Full-time English Optional USP - University Study Programmes stu. block
ECTS FMG-VSB 2016/2017 Full-time English Optional 501 - Study Office stu. block
USP 2015/2016 Full-time English Optional USP - University Study Programmes stu. block
V - ECTS - VSB 2015/2016 Full-time English Optional 401 - Study Office stu. block
FMMI 2015/2016 Full-time English Compulsory 601 - Study Office stu. block
ECTS - MechEng - Bachelor Studies 2015/2016 Full-time English Choice-compulsory 301 - Study and International Office stu. block
FBI_ECTS 2015/2016 Full-time Czech Optional 010 - Faculty of Safety Engineering - Dean's Office stu. block
ECTS FMG - VSB 2015/2016 Full-time Czech Optional 501 - Study Office stu. block
V - ECTS - VSB 2014/2015 Full-time Czech Optional 401 - Study Office stu. block
FBI_ECTS 2014/2015 Full-time Czech Optional 010 - Faculty of Safety Engineering - Dean's Office stu. block
FMMI_N 2014/2015 Full-time Czech Compulsory 601 - Study Office stu. block
USP 2014/2015 Full-time Czech Optional USP - University Study Programmes stu. block
O - ECTS FMG - Bc. 2014/2015 Full-time Czech Optional 501 - Study Office stu. block
ECTS - MechEng 2014/2015 Full-time Czech Choice-compulsory 301 - Study and International Office stu. block
V - ECTS - VSB 2013/2014 Full-time Czech Optional 401 - Study Office stu. block
O - ECTS FMG - Bc. 2013/2014 Full-time Czech Optional 501 - Study Office stu. block
USP 2013/2014 Full-time Czech Optional USP - University Study Programmes stu. block
FMMI 2013/2014 Full-time Czech Compulsory 601 - Study Office stu. block
ECTS - MechEng 2013/2014 Full-time Czech Choice-compulsory 301 - Study and International Office stu. block
ECTS - MechEng 2012/2013 Full-time Czech Choice-compulsory 301 - Study and International Office stu. block
O - ECTS FMG - Bc. 2012/2013 Full-time Czech Optional 501 - Study Office stu. block
FMMI 2012/2013 Full-time Czech Compulsory 601 - Study Office stu. block
USP 2012/2013 Full-time Czech Optional USP - University Study Programmes stu. block
V - ECTS - VSB 2012/2013 Full-time Czech Optional 401 - Study Office stu. block

### Assessment of instruction

Předmět neobsahuje žádné hodnocení.