1. SylabUZ
2. Wydział Fizyki i Astronomii
3. semestr zimowy 2020/2021
4. Fizyka - pierwszego stopnia z tyt. licencjata
5. Algebraic and geometrical methods in physics II

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Algebraic and geometrical methods in physics II - opis przedmiotu

Cel przedmiotu

Learning students of more advanced  concepts, facts and methods of linear algebra with chosen elements of abstract algebra and analytical geometry. Obtaining the ability to solve certain typical exercises illustrating introduced notions. The aim of the course is also to develop the students' precise thinking skills and to prepare the methods and techniques of linear algebra in various branches of physics.

Wymagania wstępne

Zapisz zmiany

algebraic and geometric methods in physics

Zakres tematyczny

1. Algebraic structures. Sets, relations, operations (two-arguments), properties of operations, examples. Definition of a group, a ring, a ring with unity and a field, Examples of applications of various algebraic structures with particular emphasis on groups, examples of various groups.
2. Linear spaces. General definition of a linear space, linear subspaces, linear independence, base, dimension, subspace, intersection and the sum of a simple subspace. Examples of various linear spaces
3. Linear mappings and their basic properties. Examples of mappings. Kernel and image of linear mapping. Composition of linear mappings, inverse mapping.
4. Matrix representation of a linear mapping. Definition of matrix representation of linear mapping, mutual uniqueness between linear mappings and matrices. Theorems on the form of the matrix of the composition of linear mappings and the matrix of the inverse mapping to a given automorphism.
5. Matrix of transition and its properties. Theorem on the change of the mapping matrix when changing the domain bases and the
counter-domain.Invariant subspaces. Eigenvectors and eigenvalues

6. Transformation of the linear transformation matrix when changing the vector space base. Diagonalization of the matrix. Jordan’s
theorem. Matrix functions.
7. Euclidean spaces. General definition, scalar product, angle between vectors, orthogonal and orthonormal base, orthogonal
decomposition, Gram-Schmidt orthogonalization.
8. Quadratic forms. Linear transformations of quadratic forms, canonical forms, specificity of forms. Classification of curves and
second degree algebraic surfaces  in R^2 and in R^3.

 

Metody kształcenia

Conventional lecture examples of application of  algebra and analytic geometry in physics.

Calculation classes, within which students solve tasks illustrating the content of the lecture enriched with physical applications.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

Lecture: Positive passing of exam (written).
Classes: Positive passing of all tests (written).
Before taking the exam a student must gain positive grade during the class.

Final grade: the arithmetic average of the exam grades and pass the exercises.

 

Literatura podstawowa

[1] T. Jurlewicz, Z. Skoczylas, Algebra liniowa 2, Oficyna Wydawnicza GiS, Wrocław 2011

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

[3] J. Klukowski, I. Nabiałek, Algebra dla studentów, Wydawnictwo Naukowo-Techniczne, Warszawa 1999.

[4] A. Mostowski, M. Stark, Algebra liniowa, Państwowe Wydawnictwo Naukowe, Warszawa 1977.

[5]Strona:http://wazniak.mimuw.edu.pl/index.php?title=Algebra_liniowa_z_geometrią_analityczną

[6] W.D. Clark, S.L. McCune, Linear Algebra, McGraw-Hill Companies, Inc, 2013

[7] RS. Lipschutz, M. Lipson, Schaum’s Outline of Theory and Problems of Linear Algebra, McGraw-Hill Companies, Inc, 2001

[8] A.V. Pogorelov, Analytical Geometry, Mir Publisher, Moscow, 1980

[9] Materiały udostępnione przez prowadzących zajęcia.

Literatura uzupełniająca

[1] R. Larson, Elementary Linear Algebra, CENGAGE Learning, 2017

[2] 6] E. W. Swokowski, Calculus with Analytic Geometry, Alternate Edition -PWS Publisher 1983.

Uwagi

Zmodyfikowane przez dr hab. Piotr Lubiński, prof. UZ (ostatnia modyfikacja: 03-06-2020 15:54)