|
|
|
||
|
The third semester of the four-semester course on Applied Mathematics. Vector calculus. Fourier series and Fourier transformation. Eigenvalues and eigenvectors of matrices.
Last update: Houfek Karel, doc. RNDr., Ph.D. (14.05.2023)
|
|
||
|
The course credit is awarded at practicals after passing three brief (60 min.) tests, one from each major topic in the syllabus: 1) Line and surface integrals, 2) Fourier series and transform, 3) Eigenvalues and eigenvectors. Passing each test means gaining at least 50% of points from it.
After getting course credit at practicals, students can attend final exams. These exams consist of written and oral parts and take place during the examination period. The written part (60 min.) comprises solving 2 practical examples from topics 1) —3). The oral part (60 min.) is a discussion of theoretical concepts (definitions and theorems from lectures) related to the examples in the written part.
Last update: Holubec Viktor, doc. RNDr., Ph.D. (25.06.2024)
|
|
||
|
M. Corral, Vector Calculus, LibreTexts Mathematics. G. Strang, Introduction to Linear Algebra, Fifth Edition (2016). R. T. Seeley, An Introduction to Fourier Series and Integrals, Dover Publications 2014. Lecture notes and materials for practical exercises. Last update: Holubec Viktor, doc. RNDr., Ph.D. (25.06.2024)
|
|
||
|
The requirements for the exam correspond to the course syllabus to the extent that was given in the lectures and exercises.
For more details, see Moodle https://dl1.cuni.cz/course/view.php?id=16748. Last update: Holubec Viktor, doc. RNDr., Ph.D. (02.10.2024)
|
|
||
|
Line integral of scalar and vector field, vector potential, field with zero curl. Surface integral of scalar and vector field, Gauss-Green's theorem, Stokes‘ theorem. Integral form of divergence and curl. Fourier series, Bessel’s inequality, Parseval’s identity. Differentiation and integration of Fourier series. Fourier transformation of functions, Fourier inversion theorem, applications. Eigenvalues and eigenvectors of matrices, characteristic polynomial. Jordan normal form, basis of the eigenspace.
Last update: Houfek Karel, doc. RNDr., Ph.D. (02.05.2023)
|
|
||
|
After successful completion of the course, the student is able to:
1) Line and Surface Integrals, Vector Analysis
Explain the concepts of line integrals of the first and second kind and apply them in computations.
Describe the potential of a vector field and characterize vector fields with zero curl.
Define surface integrals of the first and second kind and use them in problem solving.
Formulate and apply the Gauss–Green theorem and Stokes’ theorem.
2) Fourier Series and Fourier Transform
Expand functions into Fourier series and work with their properties.
Explain and apply Bessel’s inequality and Parseval’s identity.
Differentiate and integrate Fourier series under appropriate conditions.
Define the Fourier transform of functions and use it in basic applications (solving differential equations, signal analysis, computation of convolutions).
3) Linear Algebra – Spectral Theory of Matrices
Determine eigenvalues and eigenvectors of matrices.
Use eigenvalues and eigenvectors to diagonalize matrices.
Apply matrix diagonalization to compute matrix functions, in particular the matrix exponential, and to solve differential and difference equations.
Explain the concept of the Jordan canonical form and transform a matrix into this form. Last update: Holubec Viktor, doc. RNDr., Ph.D. (13.01.2026)
|