|
Lecturer(s)
|
-
Brázda Michal, Mgr.
-
Říha Jan, Mgr. Ph.D.
-
Richterek Lukáš, Mgr. Ph.D.
|
|
Course content
|
I. Introduction to Tensor Calculus 1. Anisotropic media. Tensor physical quantities and their properties. 2. The concept of a tensor. Algebraic operations on tensors. Transformations of tensor components. 3. Tensors in physics (mechanics, electromagnetic fields). II. Vector Analysis 1. Differential operators in the Cartesian coordinate system 2. Integral theorems 3. Curvilinear orthogonal coordinate systems 4. Spherical coordinate system, cylindrical coordinate system III. Basic Numerical Methods in Physics 1. Nonlinear equations in physics 2. Angle bisection, tangent, and secant methods, and general iterative methods for solving equations 3. Approximation of functions and data sets, interpolation, least squares method 4. Numerical differentiation and integration 5. Fundamentals of numerical solutions to ordinary differential equations, Euler's method, Runge-Kutta methods IV. Mathematical Software and Its Applications in Physics 1. Operations with vectors and tensors in Mathematica. 2. Vector analysis in Mathematica. 3. Operations with matrices in Mathematica (Matlab/GNU Octave, Python) 4. Numerical solution of ordinary differential equations in Mathematica (Excel/Calc, MATLAB/GNU Octave, Python) 5. Processing data files in Mathematica (Excel/Calc, MATLAB/GNU Octave, Python, GNUPlot) 6. Applications in physics.
|
|
Learning activities and teaching methods
|
|
Lecture, Activating (Simulations, Games, Dramatization)
|
|
Learning outcomes
|
The Mathematics for Physicists course builds on the KEF/PMN1 seminar. The aim of the course is to expand students' knowledge of tensor calculus and vector analysis, which are essential for completing courses in theoretical mechanics and electromagnetic field theory. Students will also become familiar with basic numerical methods used in physics. The course also includes continued work with Mathematica (and, depending on preference, with Matlab, GNU Octave, Python, and GNUPlot). For students in teacher education programs, the course also serves to develop, in a formative manner, their ability to view the mathematical framework of physics as a tool for modelling, explaining, and teaching physical phenomena.
The course is primarily focused on developing specialized knowledge and skills in the mathematical apparatus of physics. At the same time, it formatively develops selected KRAAU competencies, particularly in the areas of teaching disciplines and their mediation, learning planning and reflection, feedback, and professional development. The student: a) Explains the concept of a tensor, distinguishes between scalar, vector, and tensor physical quantities, performs basic algebraic operations with tensors, and interprets the transformation properties of tensor components in physics problems. Link to KRAAU: 1.1.1; 1.1.2; 1.2.2. b) Uses differential operators in Cartesian, cylindrical, and spherical coordinates; applies integral theorems of vector calculus; and explains their physical significance in the formulation of field laws. Link to KRAAU: 1.1.1; 1.1.2; 1.2.2. c) Solves basic numerical problems in physics, particularly nonlinear equations, data approximation and interpolation, numerical differentiation, integration, and the solution of ordinary differential equations; estimates the accuracy, stability, and limitations of the method used. Link to KRAAU: 1.1.2; 1.2.2; 4.1.1. d) Uses mathematical software for symbolic and numerical calculations, data processing, and graphical presentation of results; critically verifies software outputs by estimation, analytical solution in a limit case, or an alternative computational method. Link to KRAAU: 1.2.3; 1.2.4; 1.2.5; 6.2.1; 6.2.3. e) Interprets mathematical results in a physical context, distinguishes between mathematical models, numerical algorithms, and physical reality, and works
|
|
Prerequisites
|
Recommended prerequisites: completion of introductory courses in mathematical analysis and linear algebra, or the seminar "Mathematics for Physicists 1"; at least a basic familiarity with computational software or programming is recommended.
|
|
Assessment methods and criteria
|
Student performance
Assessment is based on ongoing seminar work, independent problem-solving, and a final evaluation of students' ability to apply mathematical and computational methods to solve physics problems. The requirements include: a) regular active participation in the seminar (at least 60%); b) ongoing work on computational and conceptual problems in tensor calculus, vector analysis, and numerical methods; c) completion of 3 selected assignments using mathematical software, particularly Mathematica (or, at the student's preference, Matlab/GNU Octave, GNUPlot, or Python), focused on numerical or symbolic solutions to physics problems, data processing, approximation, interpolation, numerical integration, or solving ordinary differential equations; d) the ability to explain the mathematical procedure used, its assumptions, physical significance, and any limitations; e) a final written or combined test verifying understanding of basic concepts, computational procedures, and the ability to apply mathematical tools to specific physical situations. For problems solved using software, the clarity of the procedure, the reproducibility of the calculation, and comments on the commands or algorithm used are also evaluated. Instruction includes formative work on selected KRAAU competencies, particularly in the areas of professional understanding of the subject being taught, the didactic transformation of the mathematical apparatus of physics, planning and reflection on one's own learning, working with feedback, and the critical use of digital tools.
|
|
Recommended literature
|
-
Bořkovec M. a kol. (2023). Kompetenční rámec absolventa a absolventky učitelství. Praha.
-
BRABEC J., HRŮZA B. (1989). Matematická analýza II.. Praha.
-
Brabec J., Martan F., Rozenský Z. (1989). Matematická analýza I. Praha.
-
GARCIA A. (1999). Numerical Methods for Physics. Benjamin Cummings.
-
JIRÁSEK F., ČIPERA S., VACEK M.:. (1989). Sbírka řešených příkladů z matematiky I., II. a III.. Praha.
-
Kulhánek P. (2026). Vybrané kapitoly z teoretické fyziky I (Mechanika, Kvantová teorie, Matematika). Praha.
-
Kvasnica, J. Matematický aparát fyziky. Praha. 2004.
-
LEA S. M. (2004). Mathematics for Physicists. Belmont.
-
Lepil O., Richterek L. Dynamické modelování. Ostrava. 2007.
-
ŠEDIVÝ P. (2010). Modelování pohybů numerickými metodami. Hradec Králové.
|