1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2018/2019
4. Mathematics - pierwszego stopnia z tyt. licencjata
5. Analiza matematyczna 3

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Analiza matematyczna 3 - opis przedmiotu

Cel przedmiotu

To acquaint students with methods of examining extrema of functions in several variables, with integral calculus of multivariable functions, also with the notion of surface integral and basics of Fourier analysis.

Wymagania wstępne

Zapisz zmiany

Mathematical Analysis 1 and 2. Logic and Set Theory. Linear algebra 1 and 2.

Zakres tematyczny

Lecture
I. Differential calculus of functions of several variables II
1. Extrema (3 hours)
2. Extrema subject to a constrain (4 hours)
II. Integral calculus of multivariable functions
1. Double integrals. Iterated integrals. Double integrals in polar coordinates (4 hours)
2. Applications of double integrals: calculating areas of regions in the plane and surface areas in the space, center of mass and moments of inertia (2 hours)
3. Triple integrals and their applications. Convergence and integration (2 hours)
III. Line and surface integrals
1. Regular mappings and diffeomorphisms between spaces of different dimensions (5 hours)
2. Line integral and surface integral (3 hours)
3. Oriented line integral. Green’s theorem (7 hours)
IV. Elements of Fourier analysis (the material should be elaborated in a written form by the student, basing on a literature indicated by the lecturer)
1. Trigonometric series.
2. Fourier series of a function. Criteria of the convergence of Fourier series.
3. Fejér’s theorem.

Class

I. Differential calculus of functions of several variables II
1. Determination of local extrema of a function (4 hours)
2. Finding extrema subject to a constrain and global extrema (5 hours)
II. Integral calculus of multivariable functions
1. Calculating double integrals. Finding areas of regions (3 hours)
2. Calculating triple integrals. Finding volumes of solids (2 hours)
Colloquium (2 hours)
III. Line and surface integrals
1. Studying the regularity and diffeomorphicity of mappings between spaces of different dimensions. Parametrization of curves and surfaces (3 hours)
2. Calculating line integrals. Length of a curve (2 hours)
3. Calculating surface integrals (2 hours)
4. Calculating oriented line integrals (3 hours)
5. Applications of Green’s formula. Calculating areas of regions (2 hours)
Colloquium (2 hours)

Metody kształcenia

Traditional lecture; class where students, leaded by the teacher, solve exercises and discuss; team-work; work over a book; making use of internet.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

1. Verifying the extent of preparation of students and their activity during the classes.
2. Two colloquia with problems of various degree of difficulties, allowing to verify if students attained learning outcomes at the very least.
3. Exam (writ) with indicated point ranges.
The final grade is the arithmetic mean of those of the class and exam. A necessary condition to enter the exam is a positive grade of the classes. A necessary condition to pass the course is a positive grade of the exam.

Literatura podstawowa

1. Witold Jarczyk, Notatki do wykładu z analizy matematycznej, http://www.wmie.uz.zgora.pl/~`wjarczyk/materialy.html
2. Witold Jarczyk, Zadania z analizy matematycznej, http://www.wmie.uz.zgora.pl/~`wjarczyk/materialy.html
3. J. Douglas Faires, Barbara T. Faires, Calculus, Random House, New York

Literatura uzupełniająca

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

Uwagi

Zmodyfikowane przez dr Alina Szelecka (ostatnia modyfikacja: 07-07-2018 10:19)