714-0868/01 – Mathematics 3 (Math 3)

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.

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.

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

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.