The aim of studying and bibliography

The main aim of study: The aim of the course is to familiarize students with the following topics of mathematical analysis: number series, functional sequence and series, power and Fourier series, limit and continuity of function of many variables, partial derivatives, differentiability of maps, extrema of functions of many variables, implicit functions.

The general information about the module: Semester III, lectures 30 hours, exercises 30 hours, ends with an exam.

Teaching materials: Platforma e-learningowa PRz

Bibliography required to complete the module

Bibliography used during lectures

1	A. Birkholc	Analiza matematyczna. Funkcje wielu zmiennych	Wydawnictwo Naukowe PWN.	2002
2	W. Rudin	Podstawy analizy matematycznej	PWN, Warszawa.	1982
3	A. Sołtysiak	Analiza matematyczna. Część II	Wydawnictwo Naukowe UAM Poznań.	2004

Bibliography used during classes/laboratories/others

1	J. Banaś, S. Wędrychowicz	Zbiór zadań z analizy matematycznej	WNT, Warszawa.	2003
2	M. Gewert, Z. Skoczylas	Analiza matematyczna II. Przykłady i zadania	GiS.	dow

Bibliography to self-study

1	W. Krysicki, L. Włodarski	Analiza matematyczna w zadaniach. Cz. II	PWN, Warszawa.	dow
2	M. Gewert, Z. Skoczylas	Analiza matematyczna II. Definicje, twierdzenia, wzory	GiS.	dow

Basic requirements in category knowledge/skills/social competences

Formal requirements: The student satisfies the formal requirements set out in the study regulations

Basic requirements in category knowledge: Knowledge of the basics on differential and integral calculus of functions of one variable and linear algebra.

Basic requirements in category skills: Ability to calculate derivative, integral, limit, to investigate monotonicity of a function of one variable.

Basic requirements in category social competences: Student is prepared to undertake objective and justified actions in order to solve the posed exercise.

Module outcomes

MEK	The student who completed the module	Types of classes / teaching methods leading to achieving a given outcome of teaching	Methods of verifying every mentioned outcome of teaching	Relationships with KEK	Relationships with PRK
01	can apply the theory of numerical series, especially examine their convergence	lecture, solving classes	written test	K_W02++ K_W03++ K_W04+ K_W05+ K_W07++ K_U10+++ K_U13+ K_U14+ K_K01+	P6S_KK P6S_UW P6S_WG P6S_WK
02	can apply the theory of numerical sequences and numerical series, especially examine their pointwise and uniform convergence	lecture, solving classes	written test	K_W02++ K_W04++ K_W05+ K_U10++ K_U13+ K_K01+	P6S_KK P6S_UW P6S_WG P6S_WK
03	knows the theory of limits of functions of several variables and the methods of the investigation of continuity of those functions	lecture, solving classes	written test	K_W02++ K_W04++ K_W05+ K_U10++ K_K01+	P6S_KK P6S_UW P6S_WG P6S_WK
04	can suitably use knowledge of differential calculus of functions of several variables to calculate extremes and Jacobians	lecture, solving classes	written test	K_W01+++ K_W02+ K_W03+++ K_W07+++ K_U12+++ K_U13++ K_K01+	P6S_KK P6S_UW P6S_WG P6S_WK
05	knows the theory of Fourier series	lecture, solving classes	written test	K_W04+ K_W07+ K_U10+ K_U13+ K_U14++ K_K01+	P6S_KK P6S_UW P6S_WG P6S_WK

Attention: Depending on the epidemic situation, verification of the achieved learning outcomes specified in the study program, in particular credits and examinations at the end of specific classes, can be implemented remotely (real-time meetings).

The syllabus of the module

Sem.	TK	The content	realized in	MEK
3	TK01	Infinite series. Convergence and divergence of series of real numbers and complex numbers. Cauchy condition. series with positive terms and criteria of their convergence. Absolute and conditional convergence.	W01-W06, C01-C06	MEK01
3	TK02	Function sequences and series. Pointwise convergence of a function sequence. The limit function. Uniform convergence. Continuity, differentiability and integrability of a limit function of a function sequence. Function series. Criteria of the uniform convergence of function series. Power series. The radius of convergence of a power series. Taylor and Maclaurin series.	W07-W12, C07-C12	MEK01 MEK02
3	TK03	Limit and continuity of functions of several variables. Limit of a sequences of n coordinates. Limit of functions of several variables. Continuity and uniform continuity of functions of several variables.	W13-W18, C13-C18	MEK03
3	TK04	Differential calculus of functions of several variables. Directional derivative, partial derivatives, the gradient of a function. Differential of a function. Differentiation of a composed function. Differentiation of a mapping of n dimensional space into m dimensional space. The Jacobian of a mapping. Extrema of functions of several variables. Theorem on inverse function and on implicit function.	W19-W26, C19-C26	MEK03 MEK04
3	TK05	Fourier series. Trigonometric series. Fourier coefficients. Test for the convergence of Fourier series. Fourier series of basic functions and their applications.	W27-W30, C27-C30	MEK05

The student's effort

The type of classes	The work before classes	The participation in classes	The work after classes
Lecture (sem. 3)	The preparation for a test: 10.00 hours/sem.	contact hours: 30.00 hours/sem.	complementing/reading through notes: 5.00 hours/sem. Studying the recommended bibliography: 5.00 hours/sem.
Class (sem. 3)	The preparation for a Class: 10.00 hours/sem. The preparation for a test: 30.00 hours/sem.	contact hours: 30.00 hours/sem.	Finishing/Studying tasks: 5.00 hours/sem.
Advice (sem. 3)		The participation in Advice: 5.00 hours/sem.
Credit (sem. 3)	The preparation for a Credit: 10.00 hours/sem.	The written credit: 4.00 hours/sem.

The way of giving the component module grades and the final grade

The type of classes	The way of giving the final grade
Lecture	The grade from lectures is evaluated according to student's presence at lectures.
Class	Student must pass all module outcomes. The grade from the exercises is the arithmetic mean of the all grades of module outcomes, rounded to obligatory scale Activity during exercises can raise a grade.
The final grade	The final grade is the grade from classes

Sample problems

Required during the exam/when receiving the credit
(-)

Realized during classes/laboratories/projects
(-)

Others
(-)

Can a student use any teaching aids during the exam/when receiving the credit : no

The contents of the module are associated with the research profile: yes

1	J. Appell; A. Chlebowicz; S. Reinwand; B. Rzepka	Can one recognize a function from its graph?	2023
2	J. Banaś; A. Chlebowicz; M. Taoudi	On solutions of infinite systems of integral equations coordinatewise converging at infinity	2022
3	A. Chlebowicz	Existence of solutions to infinite systems of nonlinear integral equations on the real half-axis	2021
4	A. Chlebowicz	Solvability of an infinite system of nonlinear integral equations of Volterra-Hammerstein type	2020
5	J. Banaś; A. Chlebowicz; W. Woś	On measures of noncompactness in the space of functions defined on the half-axis with values in a Banach space	2020
6	J. Banaś; A. Chlebowicz	On solutions of an infinite system of nonlinear integral equations on the real half-axis	2019