516-0874/01 – Selected Chapters of Mathematical Physics II (VKMFII)
| Gurantor department | Institute of Physics | Credits | 5 |
| Subject guarantor | prof. RNDr. Richard Dvorský, Ph.D. | Subject version guarantor | prof. RNDr. Richard Dvorský, Ph.D. |
| Study level | undergraduate or graduate | Requirement | Compulsory |
| Year | 2 | Semester | summer |
| | Study language | Czech |
| Year of introduction | 2003/2004 | Year of cancellation | 2014/2015 |
| Intended for the faculties | HGF | Intended for study types | Bachelor |
Subject aims expressed by acquired skills and competences
Deepen their understanding of mathematical methods in describing complex physical probléme
Define and characterize the basic mathematical concepts in the field of vector analysis and tenzor analysis
Solve the fundamental problems with the applications of differential equations of mathematical physics
Teaching methods
Individual consultations
Tutorials
Summary
The aim of reciting problemes, in sequence on the subject CHOSOSED CHAPTERS
FROM MATHEMATIC FYSICS I, is to introduce listeners with basics of vectorial
and tensor analysis, with problems of using diferenciál equations in fysics
and problems of thé function and operators in Gilberts space. Structure of thé
subject is specialized on ephesizing fysical motivation, leading to use
particular mathematic applience.
Compulsory literature:
[1] Kvasnica J.: Matematický aparát fyziky. Academia 1997 (in Czech)
[2] Rektorys K. a spol.: Základy užité matematiky. SNTL 1968 (in Czech)
Recommended literature:
[3] Arsenin V.A.: Matematická fyzika. Alfa 1977 (in Czech)
Way of continuous check of knowledge in the course of semester
E-learning
Other requirements
Systematic off-class preparation.
Prerequisities
Subject has no prerequisities.
Co-requisities
Subject has no co-requisities.
Subject syllabus:
1 Vector and tensor ANALYSIS
1.1 Cartesian coordinates in space E3, their transformations and invariants,
Einstein summation convention (opak.)
1.2 General curvilinear coordinates, covariant and contravariant coordinates
vectors (opak.)
1.3 Tensors in E3, their algebra, reduction of order tensor contractions
tensor invariants
1.4 Physical field as a scalar, vector or tensor function
argument vector in E3
1.5 Derivation of three-dimensional tensor fields, increase and decrease tensor
order derivatives
6.1 Differential operator "nabla" as gradient "grad" field
1.7 Differential divergence operator as a "wonder" field
8.1 Differential rotation operator as "rot" vector field
1.9 Laplace differential operator of second order "Delta"
1.10 Gauss divergence theorem the vector field
1.11 Stokes theorem on the rotation of the vector field
1.12 Directional and substancionální derivation, a generalized balance equation
(Continuity)
2. Differential Equations in Physics
2.1 Description of physical phenomena, methods of infinitesimal calculus, compilation
differential equations based on analysis of physical phenomena
2.2 Newton's equation of motion - movement in the field scalar potential
3.2 Newton's equation of motion - an oscillator, damped and forced oscillations
4.2 Diffusion equation
2.5 Heat equation
6.2 Euler's equation
2.7 Stokes equations
8.2 Laplace equation
2.9 Wave equation
2.10 Schödingerova wave equations - harmonic oscillator
3. FUNCTIONS as vectors in a Hilbert space
3.1 The development function in an infinite number of known elementary functions, Cauch and d'Alembertovo convergence criterion
3.2 Taylor expansion of functions
3.3 Fourier development functions, Fourier integral and Fourier transform
3.4 Geometric interpretation of the development - based on vector elementary functions
Hilbert space, symbolism Dirackova
3.5 Operators in Hilbert space commutation relations, the matrix elements
3.6 Eigenvalues and operator functions
Conditions for subject completion
Occurrence in study plans
Occurrence in special blocks
Assessment of instruction
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