Description:

The subject is an introduction to integral calculus and integral transforms.
Integral calculus: anti-derivative, indefinite integral, properties and methods of integration (integration by parts and by substitution, partial fractions), definite integral, properties, Newton-Leibnitz 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 non-homogenous ODEs, 2nd order linear homogenous and non-homogenous ODEs with constant coefficients),intro to multiple integrals, particularly double integral and applications.
Integral transforms: Laplace transform and inverse Laplace transform and their application for solving nth order linear ODEs with constant coefficients.
Z-transform and inverse Z-transform, their application for solving nth order linear difference equations.

Contents:

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 non-homogenous, 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 non-homogenous nth order linear ODEs with constant coefficients.
12. Z-transform - definition, properties and examples.
13. Inverse Z-transform, Test No. 2
14. Z-transform for solving linear difference equations.

Seminar contents:

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 non-homogenous, 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 non-homogenous nth order linear ODEs with constant coefficients.
12. Z-transform properties and examples.
13. Inverse Z-transform. Simple examples.
14. Z-transform for solving linear difference equations.

Recommended literature:

Study materials
[1] Neustupa J.: Mathematics 1, skriptum ČVUT, 2004
[2] Bubeník F.: Problems to Mathematics for Engineers, skriptum ČVUT, 2007
[3] Stewart, J. : Calculus, Brooks/Cole, 2012
[4]

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar