1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2022/2023
4. WMIiE - oferta ERASMUS - Program Erasmus
5. Mathematical Analysis 3

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Mathematical Analysis 3 - opis przedmiotu

Cel przedmiotu

 

Acquainting students with differential calculus of functions of several variables and introduction to Fourier analysis. 

Wymagania wstępne

Zapisz zmiany

Mathematical Analysis 1 and  2; Linear Algebra 1 and 2; Logic and Set Theory

Zakres tematyczny

Lecture

I. Applications of integrals of functions of two and three variables

Double integrals in polar coordinates (3 hours);  Application of the integral: area, center of mass, moments of inertia, volume (3 hours) 

II. Fourier series

Trigonometric series (1 hour); Fourier series – representation and properties (2h); Convergence of Fourier series (2h); Fourier series summability. Fejér's theorem (1 hour) 

III. Differentiable functions of several variables 

Directional and partial derivatives. Jacobi matrix and gradient (3 hours); Differentiability and differential (7 hours); Geometric interpretation of differentiability. Tangent lines, normal lines, and tangent planes (3 hours); Regular mappings and diffeomorphisms (2 hours); Implicit function theorem (4h); Extrema (5 hours); Conditional extrema (4 hours); Characterization of  convex  functions (1 hour); Regular mappings and diffeomorphisms between spaces of different dimensions (4 hours)

 

Classes

I. Lebesgue integral

The change of variables theorem for the Lebesgue integral (3h);  Fubini's theorem in tasks (2h)

II. Applications of integrals of functions of two and three variables.

Application of the integral: area, center of mass, moments of inertia, volume (4 hours) 

III. Fourier series.

Determining the expansion of a function into Fourier series (3h). Convergence of Fourier series (3h).

Colloquium (2 hours)

IV. Differential calculus of multivariable functions

Directional and partial derivatives and gradient examples (5 hours); Finding tangents and normals (2 hours); Regular mappings and diffeomorphisms (3 hours);  Implicit function theorem (3 hours); Local extrema of functions (4 hour); Conditional and global extrema (5 hours); Regularity and diffeomorphism of mappings between spaces of different dimensions (4 hours)

Colloquium (2 hours)

 

 

Metody kształcenia

Traditional lecture; exercises in which students solve problems and discuss, as well as prepare biographies of mathematicians whose names appear at the lecture; group work completed with a written study; work with a book and with the help of the Internet. If necessary (determined by the order of the Rector of the University of Zielona Góra), classes can be in online form.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

1. Two tests with exercises of various difficulty levels, allowing to check whether the student has achieved the minimum learning outcomes.
2. Exam in the form of a test with point thresholds.

The grade for the subject is the arithmetic mean of the classes grade and the exam grade. The necessary condition for taking the exam is a positive grade from the classes. The necessary condition for passing the course is a positive grade from the exam.

Literatura podstawowa

1. Charles C. Pugh, Real Mathematical Analysis, Springer 2015.
2. Vladimir A. Zorich, Mathematical Analysis I, Springer 2015.
3. Vladimir A. Zorich, Mathematical Analysis II, Springer 2016.

Literatura uzupełniająca

1. Józef Banaś, Stanisław Wędrychowicz, Zbiór zadań z analizy matematycznej, Wydawnictwo Naukowo-Techniczne, Warszawa, 1993.
2. Walter Rudin, Podstawy analizy matematycznej,  Wydawnictwo Naukowe PWN, Warszawa, 2002.
3. Witold Kołodziej, Analiza matematyczna, Państwowe Wydawnictwo Naukowe, Warszawa, 1986.
4. Andrzej Birkholc, Analiza Matematyczna. Funkcje wielu zmiennych, Wydawnictwo Naukowe PWN, Warszawa, 2002.

Uwagi

Zmodyfikowane przez dr hab. Justyna Jarczyk, prof. UZ (ostatnia modyfikacja: 26-04-2022 23:18)