1. SylabUZ
2. Faculty of Exact and Natural Sciences
3. winter term 2024/2025
4. WMIiE - oferta ERASMUS - Erasmus programme
5. Mathematical Analysis 4

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Mathematical Analysis 4 - course description

Aim of the course

To acquaint the student with the theory of smooth surfaces, the concept of orientation, and then the theory of non-oriented and oriented surface integral;  Stokes theorems and their role in physics, as well as a brief overview of the concepts of divergence and rotation of a vector field.

Prerequisites

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Mathematical Analysis 1, 2 and  3; Linear Algebra 1 and 2; Logic and Set Theory

Scope

Lecture

I. Surfaces 

Smooth surface (2 hours);  tangents space (3 hours); measure on a smooth surface ( 2hours); Orientation and orientability of a smooth surface (3 hours)

II. Surface integrals

Unoriented surface integral (2 hours); Surface integrals of vector fields (3 hours),  Greens theorem (3 hours); Independence of path (1 hour); Surface integral (3 hours); Gauss-Ostrogradsky theorem (3 hours); Stockes' Theorem (3 hours); Vector fields (2 hours)

Classes

I. Surfaces

Examples of smooth surfaces (3 hours); Tangent space (2 hours); Orientation and orientability of a smooth surface, Möbius strip (3 hours)

II. Surface integrals

Parametric description of the curve and surface (3 hours) Unoriented surface integral, Curve length (3 hours); Surface integrals of vector fields (3 hours),  Greens theorem (3 hours); Independence of path (1 hour); Surface integral (3 hours); Gauss-Ostrogradsky theorem (2 hours); Stockes' Theorem (3 hours); Vector fields (2 hours); Line integrals (3 hours); Independence of the integral from the path of integration (3 hours); Application of Green's formula (2 hours); Vector field of divergence and rotation (2 hours)

Colloquium (2 godz.)

Teaching methods

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.

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

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.

Recommended reading

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.

Further reading

1. Józef Banaś, Stanisław Wędrychowicz, Zbiór zadań z analizy matematycznej, Wydawnictwo Naukowo-Techniczne, Warszawa, 1993.
2. Andrzej Birkholc, 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

Notes

Modified by dr Dorota Głazowska (last modification: 18-04-2024 13:08)