UNIVERSITATEA BABEŞ-BOLYAI DIN CLUJ-NAPOCA                                                                      Anexa 2.1.3-1                                     FACULTATEA DE MATEMATICĂ ŞI INFORMATICĂ
Domeniul: MATEMATICĂ
Ciclul de studii: MASTERAT
Programul de studii: MATEMATICĂ
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MMA0002	Mathematical Analysis	

Semester	Hours: C+S+L	Category	Status	
3	3+2+0	fundamental	compulsory	

II. Full status faculty members
Name and surname	Scientific	Didactic	Chair	Type	
	Degree	Degree		C	S	L	
 Mureşan Marian	Dr.	Prof.	Analysis and optimization 	*	*		

Associated faculty members
Name and surname	Scientific	Institution	Type of	Type	
	Degree		position	C	S	L	

III. Course objectives
The aim of this course is to introduce the students to get know basic notions and results to classical and modern mathematical analysis. The students are introduced to: the real numbers system,  topology of the real axis, the complex number system, metric spaces, normed spaces, differential and integral  calculus on the real axis and in higher dimensional spaces, sequences and series of numbers and of functions, improper integrals, and integrals depending on parameters. Several applications to finding famous constants are supplied. It is also important the problem solving skills of the students.  	

IV. Course contents
 The course starts with the study of the system of real numbers, intervals, topology on the real axis, the complex number system, numerical sequences, convergence, and the completeness of the real axis.   Then there are studied the real functions of a real variable: limits, continuity, differentiability, mean value theorems, higher order derivatives, Taylor formula, and local extreme problems. The next step consists in the study of the Darboux-Stieltjes integrals for real functions of a real variable. It follows the study of vector valued functions of vector variables: limits, continuity, differentiability, partial derivatives, mean value theorems, higher order partial derivatives and differentials, Taylor formula, extreme problems, inverse function theorem, implicit function theorem, constrained extremes. Then there are studied the double and triple integrals, improper integrals, and integrals depending on parameters. The study of sequences and series of functions ends the topics of these lectures.                 	

V. Bibliography 
1. Arama, L., Morozan, T., Culegere de problema de analiza matematica pentru bacalaureat si admitere in invatamantul superior, Universal Pan, Bucuresti, 1996.
2. Breckner, W. W., Analiza matematica. Topologia spatiului R^n, Universitatea din Cluj-Napoca, 1985.
3. Bucur, G., Campu, E., Gaina, S., Culegere de probleme de calcul differential si integral, II, III,
     Ed. Tehnica, Bucuresti, 1966+1967.
4. Cobzas, S., Analiza matematica (calcul diferential), Presa Universitara Clujeana, Cluj-Napoca, 1997. 
5. Demidovici, B.,  Problem book on differential and integral calculus 
6. Mureşan, M., Mathematical Analysis and Applications, Risoprint, Cluj-Napoca,  2005.
7. Mureşan, M., Mathematical Analysis for Computer Scientists. (to appear)
8. Rudin, W., Principles of Mathematical Analysis, McGraw-Hill, New York, 1964.
9. Trif, T., Probleme de calcul diferential si integral in R^n, Casa Cartii de Stiinta, Cluj-Napoca, 2003 	

VI. Thematic of didactic activities per weeks
Lectures

1. The system of real numbers [6, Chap. 1] [7, Chap. 1]
       1.1. Definitions of natural numbers, rational numbers, and real numbers  
       1.2 Exercises 
2. Sequences of real numbers  [6, Chap. 3] [7, Chap. 3]
       2.1. Intervals  
       2.2. Topology on the real axis 
       2.3. Sequences of real numbers, convergence 
       2.4. Exercises 
3. Completeness of the real axis and numerical series [6, Chap. 3] [7, Chap. 3]
       3.1. Completeness of the real axis  
       3.2. Numerical series, series with positive terms  
       3.3. Convergence criteria
       3.4. Alternating series 
       3.5. Exercises 
4. Function of real variable and real valued  [6, Chap. 4] [7, Chap. 4]
       4.1. Limits  
       4.2. Continuity  
       4.3. Monotonic functions
       4.4. Lipschitz functions, Banach fixed point theorem 
       4.5. Convex functions
       4.6. Exercises 
5. Differential calculus for real functions of a real variable [6, Chap. 5] [7, Chap. 5]
       5.1. Differential and derivatives of the first and higher order  
       5.2. Mean value theorems   
       5.3. Taylor formula   
6. Local extreme problems [6, Chap. 5] [7, Chap. 5]
       6.1. Local extreme problems  
       6.2. Exercises 
7. Integral calculus for real valued functions of a real variable [6, Chap. 5, 6] [7, Chap. 5, 6]
       7.1. Primitives   
       7.2. Definite integral
       7.3. The Leibniz-Newton formula    
       7.4. Exercises
8. The Darboux-Stieljes integral for real valued functions of a real variable [6, Chap.  6] [7, Chap.  6]
       8.1. The Darboux-Stieljes integral for real valued functions of a real variable       
       8.2. Exercises   
9. Vector valued functions of vector variables [6, Chap. 7] [7, Chap. 7] 
       9.1. The Euclidean space R^n    
       9.2. Elements of topology on R^n
       9.3. The completeness of R^n
       9.4. Limits
       9.5. Continuity
       9.6. The first order Frechet differential
       9.7. Exercises  
10. Partial derivatives [6, Chap. 7] [7, Chap. 7]
       10.1. Linear and quadratic functions   
       10.2. Partial derivatives of the first order and of higher order   
       10.3. Directional derivatives   
       10.4. Second order and higher order Frechet differentials    
       10.5. Schwarz theorem   
       10.6. Taylor formula 
       10.4. Exercises
11. Local extreme problems for functions of vector variables [6, Chap. 7] [7, Chap. 7] 
       11.1. The inverse and the implicit functions theorems
       11.2. Local extreme problems for functions of vector variables 
       11.3. Constrained extreme problems
       11.4. Exercises 
12. Double and triple integrals  [7, Chap. 8]   
       12.1. Double integrals on rectangles and on simple domains  
       12.2. Triple integrals on parallelepipeds and on simple domains
       12.3. Changes of variables  
       12.3. Exercises
13. Improper integrals and integrals depending on parameters [7, Chap. 6]  [7, Chap. 8]    
       13.1. Improper integrals  
       13.2. Integrals depending on parameters   
       13.3. Exercises 
14. Sequences and series of  functions  [7, Chap. 3, 4, 5, 6]  
       14.1. Sequences of  functions     
       14.2. Series of  functions  
       14.3. Exercises 

Seminars

1. The system of real numbers. Induction. Inequalities.
2. Sequences of real numbers. Recurrences.
3. Completeness of the real axis and numerical series. Number e. 
4. Function of real variable and real valued.  
5. Differential calculus for real functions of a real variable  
6. Local extreme problems. 
7. Primitives. Definite integral. The Leibniz-Newton formula    
8. Darboux-Stieljes integral for real valued functions of a real variable. BBP formulas.
9. Vector valued functions of vector variables.
10. Partial derivatives. 
11. Local extreme problems for functions of vector variables.
12. Double and triple integrals. 
13. Improper integrals and integrals depending on parameters. 
14. Sequences and series of  functions.  	

VII. Didactic methods used
Lectures, presentations, conversations, exercises, individual study, homework assignments.	

VIII. Assessment
The activity ends with a written final. The exam subjects have theoretical questions from all the studied topics, and one problem, among the problems studied at the course or the seminar. There is an evaluation of the overall seminar activity, and the documentations prepared for the programming projects will be evaluated. The final grade is the mean of the grades mentioned above. The final grade = Homework 30% + Seminar Activity 20%+Written Exam 50%.	

IX. Additional references
1. Megan, M., Bazele analizei matematice, I+II+III, Eurobit, Timisoara, 1997-1998.	

	Date:	2007, April 18							Course responsible,
Prof. dr. Marian Mureşan