# 310-2303/01 – 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 2019/2020 Year of cancellation Intended for the faculties FMT, FBI, FS, FAST, HGF, FEI, USP Intended for study types Bachelor, Master
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
Seminars
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. - Theory of the field. Scalar and vector fields. - 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:

Lectures in English on school Moodle system - lms.vsb.cz

### 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

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

Full-time form (validity from: 2019/2020 Winter semester)
Min. number of points
Credit and Examination Credit and Examination 100 (100) 51
Credit Credit 20  5
Examination Examination 80 (80) 30
Writing test Written test 60  25
Theoretical part of the exam Oral examination 20  5
Mandatory attendence parzicipation: 80%

Show history

### Occurrence in study plans

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

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner FMT+9360 2020/2021 Full-time English  Optional 600 - Faculty of Materials Science and Technology - Dean's Office stu. block
ECTS FCE Bc-Mgr 2020/2021 Full-time English  Choice-compulsory 200 - Faculty of Civil Engineering - Dean's Office stu. block
ECTS - MechEng - Bachelor Studies 2020/2021 Full-time English  Choice-compulsory 301 - Study and International Office stu. block
ECTS FCE Bc-Mgr 2019/2020 Full-time English  Choice-compulsory 200 - Faculty of Civil Engineering - Dean's Office stu. block
Additional Subjects 2019/2020 Full-time English  Choice-compulsory 301 - Study and International Office stu. block
FMST 2019/2020 Full-time English  Compulsory 601 - Study Office stu. block