Mathematics II
Code  Completion  Credits  Range  Language 

17PBOMA2  Z,ZK  5  2P+2C  Czech 
 Vztahy:
 The course 17PBOMA2 can be graded only after the course 17PBOMA1 has been successfully completed.
 Garant předmětu:
 Lecturer:
 Tutor:
 Supervisor:
 Department of Natural Sciences
 Synopsis:

The subject is an introduction to integral calculus and integral transforms.
Integral calculus: definition of an indefinite integral, properties and methods of integration (integration by substitution, integration by parts, partial fractions), definite integral, properties, NewtonLeibnitz fundamental theorem, simple applications of both indefinite and definite integrals, improper integral, solving differential equations (ODEs) (1st order ODEs with separable variables, linear 1st order homogenous as well as nonhomogenous ODEs, 2nd order linear ODEs homogenous and nonhomogenous with constant coefficients),intro to multiple integrals, particularly applications of double integral.
Integral transforms: Laplace transform  definition and properties, inverse Laplace transform, application of Laplace transform for solving nth order linear ODEs with constant coefficients,
Ztransform  definition and properties, inverse Ztransform, application of Ztransform for solving nth order linear difference equations.
 Requirements:

Prerequisites
Seminars assesment:
a) Compulsory attendance of all seminars.
b) Activities at seminars will be checked by minitests. There will be 10 minitests during the semester, evaluated by 5 points each. Total sum MT = 0  50 points.
c) Midterm tests
1st midterm test in 7th week and 2nd midterm test in 13th week of the semester.
Midterm test consists of 4 tasks, each task evaluated by 5 points (maximum 20 points at each midterm test).
A student must gain at least 9 points at one midterm test, but minimum gained from both midterm tests must be at least 20 points. So, total sum VT=2040 points.
Student's grading from seminars transferred for the exam:
MT/10+VT/4=minimum 5 maximum 15 points.
Exam:
Assesment of Seminars signed by respective teacher in student's Index and registered in KOS.
Exam is only in written form, lasts 90 minutes.
It is forbidden to use a calculator or a mobile phone during the exam.
The exam consists of
a) 7 tasks, evaluated by 10 points each, in total maximum 70 points
b) 5 tests, evaluated by 3 points each, in total maximum 15 points
c) transferred points maximum 15 points
Total maximum 100 points
Exam grading
A: 90100, B: 8089, C: 7079, D: 6069, E: 5059, F: less than 50
 Syllabus of lectures:

1. Introduction to indefinite integral, basic properties, elementary functions integration, integration by parts, integration by substitution.
2. Rational functions integration, partial fraction technique.
3. Integration of trigonometric functions, combined techniques of integration.
4. Introduction to definite integral, simple geometrical applications (area, volume of rotational bodies, curve length).
5. Improper integral, introduction to differential equations, general solution.
6. Differential equations, initial value problem for ODEs, 1st order ODE with separable variables, linear 1st order ODEs homogenous and nonhomogenous, method of variation of constant, homogenous ODEs (substitution z=y/x).
7. nth order linear ODEs with constant coefficients and their solution.
8. Double integral, introduction and elementary methods of its calculating.
9. Jacobian and substitution in double integral, polar coordinates, geometrical applications of double integral.
10. Laplace transform definition, properties and examples.
11. Inverse Laplace transform, application of Laplace transform for solving IVP for homogenous and nonhomogenous nth order linear ODEs with constant coefficients.
12. Ztransform  definition, properties and examples.
13. Inverse Ztransform, Test No. 2
14. Ztransform for solving linear difference equations.
 Syllabus of tutorials:

1. Elementary functions integration, integration by parts, integration by substitution.
2. Rational functions integration, partial fraction technique.
3. Integration of trigonometric functions, combined techniques of integration.
4. Definite integral, simple geometrical applications (area, volume of rotational bodies, curve length).
5. Improper integral, simple examples of improper integrals due to the function or due to the infinite interval of integration, introduction to differential equations, general solution.
6. 1st order ODE with separable variables examples, linear 1st order ODEs homogenous and nonhomogenous, method of variation of constant, examples.
7. Homogenous ODEs (substitution z=y/x), nth order linear ODEs with constant coefficients and their solution.
8. Double integral, introduction and elementary methods of its calculating.
9. Jacobian and substitution in double integral, polar coordinates, geometrical applications of double integral.
10. Laplace transform properties and examples.
11. Inverse Laplace transform and application of Laplace transform for solving IVP for homogenous and nonhomogenous nth order linear ODEs with constant coefficients.
12. Ztransform properties and examples.
13. Inverse Ztransform. Simple examples.
14. Ztransform for solving linear difference equations.
 Study Objective:

The goal of the subject is to gain necessary theoretical background in the field of integral calculus and integral transforms. By the end of the course students will be able to apply the knowledge in solving various practical problems of fundamental integral calculus, and integral transforms.
 Study materials:

[1] Neustupa J.: Mathematics 1, skriptum ČVUT, 2004
[2] Bubeník F.: Problems to Mathematics for Engineers, skriptum ČVUT, 2007
 Note:
 Further information:
 No timetable has been prepared for this course
 The course is a part of the following study plans:

 Optics and Optometry  fulltime (compulsory course)