Přístupnostní navigace
E-application
Search Search Close
Course detail
CESA-SMA2Acad. year: 2023/2024
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
Number of ECTS credits
Mode of study
Guarantor
Department
Entry knowledge
Rules for evaluation and completion of the course
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.
Study aids
Prerequisites and corequisites
Basic literature
Recommended literature
Elearning
Classification of course in study plans
Lecture
Teacher / Lecturer
Syllabus
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.
Fundamentals seminar
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
Copies the outline of fundamentals seminar (numerical exercises).
E-learning texts