Czech

Not applicable.

1. Students will be endowed with the elementary knowledge on infinite series and they will be able to apply them computing integrals and differential equations which are not solvable by elementary methods. The knowledge can be also applied in possiible future studies of numerical analysis.
2. Applying the knowledge of the previous topic, students make acquaintance with the foundations of the theory of functions of complex variable and follow some principle differences in contrary to the functions of real variable. Students will obtain skills on elementary functions, followd the concept of a holomorphis function and manage the computation of curve integrals and primitive functions.
3. Students make aquaintance with both of the continous and discrete least square methods. They will be able to to apply those methods in the numerical approximation and by the evaluation of measurmaents. Students will follow its significance for the construction of Fourier trigonometric polynomials and Fourier series.
4. Students manage Fourier series, obtains computational skills on them and will be apply them in the modelling of periodical processes.
5. Students manage the concept of Fourier transform (including the discrete and fast ones) theoretically and computationally. They follows the concept of Dirac distribution. They will follow the meaning of Fourier transform in the theory of signals and in the spectroscopy.
6. Students will manage the concept of a tensor and a tensor field including the basic operations on them. They will make aquaintance of tensor calculus in material sciences and physics.

Linear algebra, differential and integral calculus of function of one and more variables.

Not applicable.

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

The examination consists of test (50 percent) and oral parts (50 percent of the total marking). The final classification of the subject is given by the examination.

1. Numerical and functional series, the concepts of point-wisw and uniform convergency, Weierstrass criterion.
2. Drivative and integration series theorems. Power and Taylor series, the concept of the analytical function. Algebraic operations on power series.
3. Integration and solution of differntial equations by means of power series.
4. Complex numbers, algebraic and metric properties, stereographic projection. Elementary complex functions of complex variable, Euler formulas.
5. The concepts of the derivative and the holomorphic function, Cauchy-Riemann conditions, harmonic functions.
6. The concept of a curve and the curve integral, Cauchy formulas. The independency of a curve integral on integration path, the concept of a primitive function.
7. The least square method, orthogonal systems of functions, The concepts of a harmonic function and a trigonometric polynomial, Fourier trigonometric polynomial, Fourier series.
8. Computation of Fourier series and its applications. Fourier transform - the vocabulary, the convolution theorem, the concept of distribution.
9. Dirac distribution and its properties, applications of Fourier transform in spectroscopy, deconvolution methods, apodizing curves. Discrete and fast Fourier transform.
10. Dual vector spaces, tensor product of vector spaces. Tensors (covariant, contravariant and mixed), algebraic operations on them, tensor algebra.
11. The cocepts of a manifold and submanifold, tangent and cotangent bundle. Tensor fields on manifolds, examples (vector product, volume form, tensor strnght and deformation, metric tensor).
12. Symmetric and antisymmetric tensors, differential form, the operation of differentiating, Poincaré lemma and its applications in the field theory.
13. Metric tensor in general relativity, informatively the concept of a covariant derivative, torsion and curvature,the tensor character of physical laws.

Not applicable.

The first aim is obtaining the elementary knowledege on infinite series an on the foundations of the theory of functions of complex variable. Another aim of the course is getting the elementary knowledge concerning Fourier series and Fourier transforms, theoreticaly and computationaly and make aquitance with their applications. The final aim is gettingt the knowledge on tensors accenting some special kinds of them together with algebraic operations, In addition to that, tensor fields with operators and physical applications are explained.

The participation in lectures is not obligatory.

Not applicable.

Not applicable.

Klíč A., Dubcová M.: Základy tensorového počtu s aplikacemi. VŠCHT v Praze, Praha 1998. (CS)
Klíč A., Volek K., Dubcová M.: Fourierova transformace, VŠCHT v Praze, Praha 2002. (CS)
Koukal S., Křížek M., Potůček R.: Fourierovy trigonometrické řady a metoda konečných prvků v komplexním oboru. Academia, Praha 2002. (CS)
Havelka, J. Veverka, J.: Matematika - Dif. rovnice - Nekonečné řady (CS)
Eliáš, J., Horváth, J., Kajan, J., Šulka, R.: Zbierka úloh z vyššej matematiky IV, 1972 (CS)

Doupovec M.: Diferenciální geometrie a tensorový počet. FSI VUT Brno, Brno 1999. (CS)
Griffiths P. R.: Chemical Infrared Fourier Transform Spectroscopy. John Wiley, New York 1975. (CS)
Novák, V.: Nekonečné řady, skripta Přf MU (CS)
Novák, V.: Analýza v komplexním oboru, skripta Přf. MU (CS)