Course detail

Mathematics 2

CESA-SMA2Acad. year: 2022/2023

Differential analysis of functions of several variables, domain, limit, continuity, partial and directional derivatives, gradient, differential, tangent plane, implicit functions. Ordinary differential equations, existence and uniqueness of solutions, equations of the first order with separated variables and linear equations of the first order, equations of the nth order with constant coefficients. Analysis in the complex domain, holomorphic functions, derivation, curve parameterization, curve integral, Cauchy's theorem, Cauchy's formula, Laurent series, singular points, residues, residue theorem. Laplace transform, forward and inverse, solution of differential equation with initial conditions. Signals and impulses, special and generalized functions, Laplace images of signals with finite impulses. Fourier series of periodic functions, orthogonal system of functions, trigonometric system of functions, Fourier series in complex form. Fourier transform, forward and inverse, Fourier images of special functions. Z-transformation, direct and inverse, solution of differential equation with initial conditions. 

Language of instruction

Czech

Number of ECTS credits

6

Mode of study

Not applicable.

Guarantor

doc. RNDr. Zdeněk Šmarda, CSc.

Department

Department of Mathematics (UMAT)

Learning outcomes of the course unit

At the end of the course students should be able to know the basic concepts and corresponding context, as well as:

- be able to find and draw the domain of the function of two variables;
- compute partial derivatives of arbitrary order for any (even implicitly) function of several variables;
- find the tangent plane to the surface specified by the function of two variables;
- solve separated and linear first order differential equations;
- solve the n-th order differential equation with constant coefficients including the special right-hand side;
- decompose a complex function into a real and imaginary component and determine the functional values ​​of complex functions;
- find the second component of a complex holomorphic function and determine this function in a complex variable including its derivative;
- calculate the integral of a complex function across a curve by parameterizing the curve, Cauchy theorem or Cauchy formula;
- be able to find singular points of complex functions and calculate their residues;
- calculate the integral of a complex function by means of a residual theorem;
- solve by the Laplace transform the n-th order differential equation with constant coefficients;
- find the Fourier series of the periodic function;
- solve by means of Z-transformation n-th order differential equation with constant coefficients;

Prerequisites

The knowledge on the secondary school level and the course of Mathematics 1 is required. In order to master the subject matter it is necessary to be able to determine the definition fields of common functions of one variable, understand the concept of limits of one variable function, numerical sequence and its limits. Further it is necessary to know the rules for derivation of real functions of one variable, knowledge of basic methods and methods of integration (decomposition into partial fractions, integration by parts, substitution method) for indefinite and definite integral and to be able to apply them to problems in the scope of Mathematics 1. Knowledge of infinite number series and basic criteria of their convergence as well as power series and search for fields of their convergence are also required.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods include lectures, fundamentals seminar (numerical exercises) and computer-assisted exercise. 

Assesment methods and criteria linked to learning outcomes

Maximum 30 points per semester for three written tests. The criterium of course-unit credit is awarded on condition of having at least 10 points in sum from these three written tests.

The condition for passing the exam is to obtain at least 50 points out of a total of 100 possible (30 can be obtained for work in the semester, 70 can be obtained at the final written exam).

Course curriculum

1. Differential calculus of functions of several variables. Domain, limit, continuity, partial and directional derivatives, gradient, differential, tangent plane, implicit functions.

2. Ordinary differential equations of the first order. Basic concepts, existence and uniqueness of solutions, geometric interpretation of equations, equations with separated variables and linear equations.

3. Ordinary differential equations of the nth order. Basic concepts, linear differential equations of the nth order with constant coefficients including a special right-hand side.

4. Introduction to complex analysis. Complex numbers and basic operations in the complex field, important sets of the complex plane.

5. Complex function, its limit, continuity and derivative. Special cases of complex functions, algebraic decomposition of a function, elementary complex functions, holomorphic functions, Cauchy-Rieman conditions, L'Hospital's rule.

6. Integral calculus in a complex field - Part I. Curve in the complex plane, parametrization of known curves, integral of a complex function along a curve, calculation of the integral along a curve by parametrizing the curve.

7. Integral calculus in a complex field - II. part. Calculating the integral using Cauchy's theorem and Cauchy's formulas.

8. Integral calculus in a complex field - III. part. Laurent series, singular points and their classification, concept of residue and calculation of integral using residue theorem.

9. Forward and inverse Laplace transform. Properties of the transformation, use of the Laplace transform in solving differential equations. 

10. Signals and impulses, special and generalized functions. Finite and Dirac impulses, Heaviside function, needle function, generalized derivative, finding Laplace images of simple signals with finite impulses.

11. Fourier series of periodic functions. Periodic functions, infinite orthogonal system of functions, Fourier series for functions with special and general period, Fourier series in complex form.

12. Forward and inverse Fourier transform. Properties of transformation, search for Fourier images of some special functions (signals), use of Fourier transformation in solving differential equations.

13. Forward and inverse Z-transformation. Transformation properties, differential equations and the use of the Z-transform in solving differential equations. 

Work placements

Not applicable.

Aims

The aim of the course is to acquaint students with basic differential calculus of functions of several variables and with general methods of solving ordinary differential equations. Another point is to teach students how to use mathematical transformations (Laplace, Fourier and Z-transformation) and thus give them a guide to alternative solutions of differential and difference equations that are widely used directly in technical fields. To learning an elements of complex analyzes (especially basic methods of integration in a complex field) offers a good tool for solving specific problems in electrical engineering. 

Specification of controlled education, way of implementation and compensation for absences

Fundamentals seminar (i.e., numerical exercises) and computer-assisted exercise are required. Any absence must be duly apologized and the studied subject must be completed. During the semester three written tests with a total of 30 points are written.  The criterium of course-unit credit is awarded on condition of having together at least 10 points from three written tests.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Kolářová, E.: Matematika 2, Sbírka úloh, FEKT VUT v Brně, 2009, s. 1-83. (CS)
Svoboda, Z., Vítovec, J.: Matematika 2, FEKT VUT v Brně, 2014, s. 1-189. (CS)

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme SPC-STC Bachelor's 1 year of study, summer semester, compulsory

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

Mgr. Jiří Vítovec, Ph.D.

Syllabus

Completely identical to the curriculum (see above).   

Fundamentals seminar

8 hod., compulsory

Teacher / Lecturer

Mgr. Jiří Vítovec, Ph.D.

Syllabus

1. Differential calculus of functions of two variables. Domain, partial derivative, implicit function, tangent plane, gradient.

2. Ordinary differential equations of the first order - part I. Equations with separated variables.

3. Ordinary differential equations of the first order - part II. Linear equation.

4. Ordinary differential equations of the nth order. Equations with constant coefficients including a special right-hand side.

5. Introduction to complex analysis. Complex numbers and basic operations with complex numbers, complex functions and their algebraic decomposition, including determining functional values of complex functions.

6. Derivation in a complex field. Cauchy-Rieman conditions and determination of the second component of a holomorphic function.

7. Integral calculus in a complex field - part I. Curve in the complex plane, parametrization of known curves, calculation of the integral along the curve by parametrization of the curve.

8. Integral calculus in a complex field - part II. Calculating the integral using Cauchy's theorem and Cauchy's formulas.

9. Integral calculus in a complex field - part III. Singular points and their classification, residue of a function and calculation of an integral using the residue theorem.

10. Forward and inverse Laplace transform. Properties of the transformation, use of the Laplace transform in solving differential equations.

11. Fourier series of periodic functions. Fourier series for functions with special and general period.

12. Forward and inverse Z-transformation. Transformation properties, differential equations and the use of the Z-transform in solving differential equations. 

Computer-assisted exercise

18 hod., compulsory

Teacher / Lecturer

Mgr. Jiří Vítovec, Ph.D.

Syllabus

Copies the outline of fundamentals seminar (numerical exercises).