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Course detail
FEKT-BPC-MA2AAcad. year: 2022/2023
Differential analysis of functions of several variables, domain, limit, continuity, partial and directional derivatives, gradient, differential, tangent plane, implicit function. 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.
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Department
Learning outcomes of the course unit
Prerequisites
Co-requisites
Planned learning activities and teaching methods
Assesment methods and criteria linked to learning outcomes
Course curriculum
1. Differential calculus of functions of several variables. Domain, limit, continuity, partial and directional derivatives, gradient, differential, tangent plane, implicit function.
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 - part II. Calculating the integral using Cauchy's theorem and Cauchy's formulas.
8. Integral calculus in a complex field - part III. 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
Aims
Specification of controlled education, way of implementation and compensation for absences
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Prerequisites and corequisites
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Recommended reading
Classification of course in study plans
branch BIT , 2. year of study, summer semester, elective
Lecture
Teacher / Lecturer
Syllabus
Identical with the line of teaching.
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).
Project