Course detail
Mathematics 2
FEKT-BKC-MA2Acad. year: 2024/2025
Functions of many variables, gradient. Ordinary differential equations, basic terms, exact methods, examples of use. Differential calculus in the complex domain, derivative, Caucy-Riemann conditions, holomorphic functions. Integral calculus in the complex domain, Cauchy theorem, Cauchy formula, Laurent series, singular points, residue theorem. Laplace transform, convolution, applications. Fourier transform, relation to the Laplace transform, practical usage. Z transform, discrete systems, difference equations.
Language of instruction
Czech
Number of ECTS credits
6
Mode of study
Not applicable.
Guarantor
Department
Entry knowledge
The subject knowledge on the secondary school level and BMA1 course. For having a facility for subject matter is needed to can determined domains of usual functions of one real variable, to understanding of a concept ot the limit of functions of one real variable and a concept numerical sequences and their limits and to solve conrete standard tasks. Further there is needed the knowledge of rules for derivations of real functions of one variable, the knowledge of basic methods of integrations - the integration per partes, by the method of substitution at the indefinit and definit integral have a facilitu for applications on tasks with respect to extent of the teaching text BMA1. Knowledges of infinite numerical series and some basic criteria of their convergence is also required.
Rules for evaluation and completion of the course
Maximum 20 points for individual assignments during the semester (at least 5 points for the course-unit credit); maximum 80 points for a written exam.
Tutorials are not compulsory.
Tutorials are not compulsory.
Aims
To extent the student knowlidges on methods of functions of several variables and onto application of partial derivatives. Further, in the other part, to aquiant students with some elementary methods for solving the ordinary differential equations and to make possible a deeper inside into the theory of functions of a complex vairiable, the methods of which are a necessary theoretical equipment of a student of all electrotechnical disciplines. Finally, to provide students by abillity to solve usual tasks by methods of Laplace, Fourier and Z transforms.
The absolvent of the subject is able:
- to compute the partial derivatives of the functions of more variables and use the formulas for the gradient and the tangential plane;
- to distinguish between the separable and linear differential equations and also to solve them;
- to solve linear differential equations of higher order with a special right hand side;
- to figure out from the Cauchy Riemann conditions, if the complex function is holomorfic or not, and to derive the holomorfic funcions;
- to compute, using the definition, the integral from the complex function through a curve, to apply the Cauchy theorem for the integral of the holomorfic funcion;
- to establish the poles and to calculate a residue at asimple and at a pole of higher order, to apply the residue theorem for the integral of the meromorfic funcion;
- to solve differential equations by the Laplace transform;
- to find the real Fourier series of an odd, even and a general function, expand a function to sine or cosine series;
- to solve difference equations by the Z- transform.
The absolvent of the subject is able:
- to compute the partial derivatives of the functions of more variables and use the formulas for the gradient and the tangential plane;
- to distinguish between the separable and linear differential equations and also to solve them;
- to solve linear differential equations of higher order with a special right hand side;
- to figure out from the Cauchy Riemann conditions, if the complex function is holomorfic or not, and to derive the holomorfic funcions;
- to compute, using the definition, the integral from the complex function through a curve, to apply the Cauchy theorem for the integral of the holomorfic funcion;
- to establish the poles and to calculate a residue at asimple and at a pole of higher order, to apply the residue theorem for the integral of the meromorfic funcion;
- to solve differential equations by the Laplace transform;
- to find the real Fourier series of an odd, even and a general function, expand a function to sine or cosine series;
- to solve difference equations by the Z- transform.
Study aids
Not applicable.
Prerequisites and corequisites
Not applicable.
Basic literature
KOLÁŘOVÁ, E., Matematika 2, Sbírka úloh, FEKT VUT v Brně 2009 (CS)
SVOBODA, Z., VÍTOVEC, J., Matematika 2, FEKT VUT v Brně 2015 (CS)
SVOBODA, Z., VÍTOVEC, J., Matematika 2, FEKT VUT v Brně 2015 (CS)
Recommended reading
Not applicable.
Classification of course in study plans
Type of course unit
Lecture
39 hod., optionally
Teacher / Lecturer
Syllabus
1. Multivariable functions (limit, continuity). Partial derivatives, gradient.
2. Ordinary differential equations of order 1 (separable equation, linear equation, variation of a constant).
3. Linear differential equation of order n with constant coefficients.
4. Function of complex variable - transform of a complex plane. Basic transcendental functions.
5. Differential calculus in the complex domain, Caychy-Riemann conditions, holomorphic funkction.
6. Integral calculus in the complex domain, the Cauchy theorem, the Cauchy formula.
7. Laurent series, singular points and their classification, residues and residue theorem.
8. Direct Laplace transform, convolution, grammar of the transform.
9. Inverse Laplace transform, pulses, electric circuits.
10. Fourier series (trigonometric and exponential forms, basic properties).
11. Direct and inverse Fourier transforms, relation to the Laplace transform, the pulse nad spectrum widths.
12. Direct and inverse Z transforms.
13. Difference eqautions solved using Z transform.
2. Ordinary differential equations of order 1 (separable equation, linear equation, variation of a constant).
3. Linear differential equation of order n with constant coefficients.
4. Function of complex variable - transform of a complex plane. Basic transcendental functions.
5. Differential calculus in the complex domain, Caychy-Riemann conditions, holomorphic funkction.
6. Integral calculus in the complex domain, the Cauchy theorem, the Cauchy formula.
7. Laurent series, singular points and their classification, residues and residue theorem.
8. Direct Laplace transform, convolution, grammar of the transform.
9. Inverse Laplace transform, pulses, electric circuits.
10. Fourier series (trigonometric and exponential forms, basic properties).
11. Direct and inverse Fourier transforms, relation to the Laplace transform, the pulse nad spectrum widths.
12. Direct and inverse Z transforms.
13. Difference eqautions solved using Z transform.
Computer-assisted exercise
14 hod., compulsory
Teacher / Lecturer
Syllabus
Individual topics in accordance with the lecture.
Project
6 hod., compulsory
Teacher / Lecturer
Syllabus
Individual topics in accordance with the lecture.