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

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Analiza matematyczna 2 - opis przedmiotu

Cel przedmiotu

To acquaint students with differential methods of examining extrema and the convexity of a function, with the notions of the primitive function and Riemann integral. The emphasis is placed on mastering calculating techniques, in particular those of integrating, and also on applications of differential and integral calculus. The next aim is to transfer basics of differential calculus on functions in several variables.

Wymagania wstępne

Zapisz zmiany

Mathematical Analysis 1. Logic and Set Theory. Linear Algebra 1.

Zakres tematyczny

Lecture
I. Elementary differential calculus II
1. Local extrema (1 hour)
2. Characterization of the convexity of a function (1 hour)
3. Relationships of the uniform convergence to differentiating (2 hours)
4. Differentiability of elementary functions (1 hour)
5. Primitive function (2 hours)
6. Algorithm of integrating rational functions (the material should be prepared in student’s own right basing on a literature indicated by the lecturer)
7. Derivative of a function of a convex variable (a brief information) (1 hour)
II. Applications of differential calculus (the material should be elaborated in a written form by teams of students basing on a literature indicated by the lecturer)
1. Straight-line motion.
2. Applications to geometry.
3. Differential and approximate calculation.

4. Newton method.
5. Applications in economics.
III. Elementary integral calculus
1. Riemann integral and area. Basic properties of integral. Mean value theorem for integrals (8 hours)
2. Relationships of differentiation to integration. Newton-Leibniz fundamental theorem of calculus and its consequences (3 hours)
3. Relationships of uniform convergence to integration. Integrating series of functions (2 hours)
4. Improper integral (4 hours)
IV. Techniques of integration
1. Trigonometric substitutions (2 hours)
2. Euler’s substitutions (2 hours)
3. Numerical integration: trapezoidal rule, Simpson’s rule (the material should be prepared in student’s own right basing on a literature indicated by the lecturer)
V. Applications of integral calculus
1. Exemplary applications of integration in geometry: areas of regions in the plane, volumes of solids, area of surfaces (2 hours)
2. Center of mass and moments. Theorems of Pappus (the material should be prepared in student’s own right basing on a literature indicated by the lecturer)
3. Work and pressure (the material should be prepared in student’s own right basing on a literature indicated by the lecturer)
VI. Polar coordinates and parametric equations
1. Polar coordinate system. Curves in polar coordinates. Area of a region bounded by a curve. Length of a curve (3 hours)
2. Parametric equations of a curve on the plane. Tangent line to a curve. Length of a curve (2 hours)

VII. Cartesian spaces
1. Scalars and vectors (1 hour)
2. Cylindrical coordinates and spherical coordinates (1 hour)
VIII. Functions of several variables
1. Level sets of functions of two or three variables (1 hour)
2. Limit and continuity (5 hours)
IX. Differential calculus of functions of several variables I
1. Directional and partial derivatives. Jacobi matrix and gradient (2 hours)
2. Differential and differentiability (7 hours)
3. Geometric interpretation of differentiability. Tangent plane and normal line (2 hours)
4. Regular mappings and diffeomorphisms (2 hours)
5. Implicit function theorem (3 hours)
Class
I. Elementary differential calculus II
1. Determination of local and global extrema. Proving inequalities by finding extrema. Function analysis (6 hours)
2. Examining the uniform convergence of sequences of functions and series of functions (2 hours)
3. Taylor’s expansion of a function (4 hours)

III. Elementary integral calculus, IV. Techniques of integration and V. Applications of integral
calculus
1. Calculating integrals by using definition (2 hours)
2. Integrating by parts and by substitution. Algorithm of integrating rational functions. Making use of Newton-Leibniz fundamental theorem of calculus (10 hours)
Colloquium (2 hours)

3. Convergence and integration. Integrating series of functions (2 hours)
4. Calculating areas of regions in the plane and volumes of solids (3 hours)
5. Determination of the center of mass and calculating the quantity of work (1 hour)
VI. Polar coordinates and parametric equations
1. Changing Cartesian coordinates into polar ones and conversely (2 hours)
2. Calculating areas of regions and length of curves described by polar equations (2 hours)
3. Determination of lines tangent to a curve described parametrically. Calculating areas of regions and length of curves described parametrically (3 hours)
VII. Cartesian spaces
1. Describing surfaces in spherical and cylindrical coordinates (1 hour)
Colloquium (2 hours)
VIII. Functions of several variables
1. Limits and continuity. Iterated limits. Continuity in separated variables (3 hours)
IX. Differential calculus of functions of several variables I
1. Finding directional derivatives, derivative and differential (5 hours)
2. Determination of tangent and normal lines and planes (2 hours)
3. Examining regularity and diffeomorphicity of mappings (3 hours)
4. Studying the problem of implicit functions (3 hours)
Colloquium (2 hours)

Metody kształcenia

Traditional lecture; class where students, leaded by the teacher, solve exercises and discuss; team-work completed with a written composition; 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. Three colloquia with problems of various degree of difficulties, allowing to verify if students attained learning outcomes at the very least.
3. Written compositions elaborated a material indicated by the lecturer and prepared by teams of students.
4. 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 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.

Uwagi

Zmodyfikowane przez dr Alina Szelecka (ostatnia modyfikacja: 08-07-2018 07:04)