Mathematical Analysis 3 - course description

General information

Course name	Mathematical Analysis 3
Course ID	11.1-WK-MATP-AM3-Ć-S14_pNadGenXMYTX
Faculty	Faculty of Mathematics, Computer Science and Econometrics
Field of study	Mathematics
Education profile	academic
Level of studies	First-cycle studies leading to Bachelor's degree
Beginning semester	winter term 2019/2020

Course information

Semester	3
ECTS credits to win	5
Course type	obligatory
Teaching language	polish
Author of syllabus	prof. dr hab. Witold Jarczyk prof. dr hab. Janusz Matkowski

Classes forms

The class form	Hours per semester (full-time)	Hours per week (full-time)	Hours per semester (part-time)	Hours per week (part-time)	Form of assignment
Class	30	2	-	-	Credit with grade
Lecture	30	2	-	-	Exam

Aim of the course

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.

Prerequisites

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

Scope

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)

Teaching methods

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

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

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.

Notes

Modified by dr Robert Dylewski, prof. UZ (last modification: 19-09-2019 13:44)

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