COURSE GOALS:
- acquire knowledge and understanding of the Fourier analysis and partial differential equations
- acquire operational knowledge from methods used to compute Fourier series and Fourier transforms of functions, solve partial differential equations (separation of variables and Green's functions)
- understand the usage of these mathematical methods in physics
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
Upon completing the degree, students will be able to:
1. KNOWLEDGE AND UNDERSTANDING
1.1 formulate, discuss and explain the basic laws of physics including mechanics, electromagnetism and thermodynamics
1.2 demonstrate a thorough knowledge of advanced methods of theoretical physics including classical mechanics, classical electrodynamics, statistical physics and quantum physics
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.1 identify the essentials of a process/situation and set up a working model of the same or recognize and use the existing models
2.2 evaluate clearly the orders of magnitude in situations which are physically different, but show analogies, thus allowing the use of known solutions in new problems;
2.3 apply standard methods of mathematical physics, in particular mathematical analysis and linear algebra and corresponding numerical methods
4. COMMUNICATION SKILLS
4.3 develop the written and oral English language communication skills that are essential for pursuing a career in physics
5. LEARNING SKILLS
5.1 search for and use physical and other technical literature, as well as any other sources of information relevant to research work and technical project development (good knowledge of technical English is required)
5.2 remain informed of new developments and methods and provide professional advice on their possible range and applications
5.3 carry out research by undertaking a PhD
LEARNING OUTCOMES SPECIFIC FOR THE COURSE:
Upon passing the course on Mathematical methods of physics 2, the student will be able to:
1. Compute the Fourier series of a periodic function
2. Compute the Fourier transform of a function
3. Use delta function in practice
4. Solve linear partial differential equations using separation of variables, the method of characteristics and Fourier transforms
5. Solve linear ordinary and partial differential equations using Green's functions
6. Carry out the separation of variables for partial differential equations in spherical and cylindrical coordinate systems
7. Use special functions (Legendre polynomials, spherical harmonics, Bessel and Neumann functions) in practice
COURSE DESCRIPTION:
The Spring semester (each lesson 45 minutes)
1. Introduction to inner product spaces (definitions of basic objects, properties of the inner product and the norm) [2 lessons]
2. The space of square integrable functions [2 lessons]
3. Orthonormal sets of vectors, projection of vectors onto a subspace, partial sum of the Fourier series, Gibbs phenomenon [3 lessons]
4. Completeness of the inner product space [2 lessons]
5. Classical Fourier series and its convergence [3 lessons]
6. The Fourier transform and its inverse [2 lessons]
7. Plancherel theorem, the principle of uncertainty [2 lessons]
8. Convolution [1 lesson]
9. Delta function [3 lessons]
10. Green's functions for ordinary differential equations [3 lessons]
11. The classification of linear partial differential equations, physical examples [3 lessons]
12. D'Alembert formula for (1+1)-dimensional wave equation [2 lessons]
13. Poisson equation (mean value theorem, Green's function) [3 lessons]
14. The separation of variables for Helmholtz equation in spherical and cylindrical coordinates [5 lessons]
15. Legendre polynomials and spherical harmonics [3 lessons]
16. Bessel and Neumann functions [3 lessons]
17. Calculus of variations [3 lessons]
Exercises follow lectures by content:
The Spring semester (15 weeks)
1st week: Fourier series (basic examples, orthogonality on the space of square integrable functions)
2nd week: Fourier transform
3rd week: Delta function (representations, basic properties, delta function in higher dimensions, Jacobian, densities of matter written with delta function)
4th week: Green's function for ordinary differential equations
5th week: Partial differential equations of 1st order (the method of characteristics)
6th week: Partial differential equations of 2nd order, 1D systems (basic problems)
7th week: Partial differential equations of 2nd order, 1D systems (advanced problems), 2D problems in Cartesian and polar coordinates
8th week: Continuous systems: Green's functions
9th week: Legendre polynomials (basic problems)
10th week: Legendre polynomials (advanced problems)
11th week: Spherical harmonics
12th week: Bessel functions (basic problems)
13th week: Bessel functions (advanced problems)
14th week: Calculus of variations (without constraints)
15th week: Calculus of variations (with constraints)
GRADING AND ASSESSING THE WORK OF STUDENTS:
There are two "shorter" written exams during the semester which are not obligatory and which enable students to a) pass the written part of the exam and b) get some bonus points (5 points per completely solved problem; 20 bonus points max.) for the "standard" written exam. In a case when both of these "shorter" written exams are passed, a student can attend the oral part of the exam. Otherwise, a student has to pass the "standard" written exam in order to attend the oral part. The final grade is the average of the two grades (in written and oral part).
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- Butkov: Mathematical Physics (Addison-Wesley, 1968.)
I. Smolić: skripta za kolegije Matematičke metode fizike 1 i 2 (dostupna u pdf formatu na stranici predavača)
S. Benić, I. Smolić: skripta rješenih zadataka iz Matematičkih metoda fizike 1 i 2 (dostupna u pdf formatu na stranici asistenta)
G.B. Arfken, H. J. Weber: Mathematical Methods for Physicists (Academic Press, 1995.)
Jefferey: Applied Partial Differential Equations (Academic Press, 1995.)
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