1. SylabUZ
2. Wydział Nauk Ścisłych i Przyrodniczych
3. semestr zimowy 2023/2024
4. WFiA - oferta ERASMUS - Program Erasmus
5. Mathematical methods in physics I

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Mathematical methods in physics I - opis przedmiotu

Cel przedmiotu

Acquainting the student with advanced mathematical methods necessary for understanding the contents of main study subjects.

Wymagania wstępne

Zapisz zmiany

Mathematical analysis I and II together with algebraic and geometric methods in physics.

Zakres tematyczny

- Elements of analytical geometry: planar and space curves, tangents and normals to planar curves, various parameterizations of of straight line, conics in Cartesian and polar coordinates, equations of plane in space, surfaces, quadrics and their classifications.

- Differential operators in curvilinear coordinates: planar and spatial Cartesian and curvilinear coordinates, curvilinear orthogonal coordinates, scalar and vector fields, differential operations on scalar and vector fields: gradient, divergence, rotation, Laplace operator in Cartesian coordinates; potential fields, divergence free fields and irrotational fields; gradient, divergence, rotation, Laplace operator in curvilinear orthogonal coordinates. Definition of tensor fields and algebraic operations on them.

- Elements of variational calculus: definition of functional and examples of them, weak and strong extrema, notion of variation of functional, necessary condition for existence of extremum of a functional, Eulera-Lagrange equations and their properties. Applications of variational calculus.

- Functions of complex variable: complex function of complex variable, limit of function, continuity of function, derivative of complex function, Cauchy-Riemanna conditions for the existence of the complex derivative, Cauchy integral formula, Taylor and Laurent series, singular points of a function, residue, calculation of integrals with the help of residue theory.

- Ordinary differential equations: first order differential equations: method of isoclines, finding solutions of various types of differential equations: separable, homogeneous, Bernoulli’s and Riccati’s equations, second order linear homogeneous and non-homogeneous differential equations with constant and variable coefficients, method of constant variations and method of undetermined coefficients.

- Partial differential equations of mathematical physics: vibrating string equation and d'Alembert method, membrane equation and Fouriera method of variables separation, Laplace equation.

Metody kształcenia

Conventional lecture. Computational problems illustrating the lecture material together with its physical applications.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

Lecture: Exam. The course credit is obtained by passing a final written exam composed of tasks of varying degrees of difficulty.

Class: Written test. A student is required to obtain at least the lowest passing grade from the test organized during class.

To be admitted to the exam a student must receive a credit for the class.

Final grade: weighted average of grades from exam (60%) and class (40%).

Literatura podstawowa

[1] R. Leitner, Zarys matematyki wyższej, część I, II i III, WNT, Warszawa 1998.

[2] D. McQuarrie, Matematyka dla przyrodników i inżynierów, T. 1, 2 i 3, PWN, Warszawa 2006.

[3] T. Jurlewicz, Z. Skoczylas, Algebra i geometria analityczna, Oficyna Wydawnicza GiS, Wrocław 2011.

[4] E. Karaśkiewicz, Zarys teorii wektorów i tensorów, PWN, Warszawa 1974.

[5] I. M. Gelfand, S. W. Fomin, Rachunek wariacyjny, PWN, Warszawa 1970.

[6] J. Długosz, Funkcje zespolone, Oficyna Wydawnicza GiS, Wrocław 2005.

[7] M. Gewert, Z. Skoczylas, Równania różniczkowe zwyczajne, Oficyna Wydawnicza GiS, Wrocław 2006.

[8] G. I. Zaporożec, Metody rozwiązywania zadań z analizy matematycznej, WNT, Warszawa 1976.

Literatura uzupełniająca

[1] F. W. Byron, R. W. Fuller, Metody matematyczne w fizyce klasycznej i kwantowej, t. 1-2, PWN, Warszawa 1974, 

eng. F. W. Byron, R. W. Fuller, Mathematics of Classical and Quantum Physics, vol I and II Dover Publications, Inc., New York, 1992

[2] J. Bird, Higher engineering mathematics, Elsevier, Amsterdam 2006.

[3] A. Dubrovin, S. P. Novikov, A.T. Fomenko Modern Geometry. Methods and Applications, Part 1, Springer-Verlag, 1984.

Uwagi

Zmodyfikowane przez dr Marcin Kośmider (ostatnia modyfikacja: 06-02-2023 22:47)