Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Licenţă
Domeniul: Informatică
Programul de studii: Informatică - în limba engleză
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MMA0002	Mathematical Analysis	

Semester	Hours: C+S+L+P	Category	Status	
1	2+2+0+0	fundamental	compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
MURESAN Marian	Ph.D.	Prof.	Algebra, Mathematical Analysis and Geometry	*			

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

III. Course objectives
Getting know the algebraic and  topological structure of the Euclidean space IR^n and the basic notions and results concerning the single valued and multivariable differential and integral calculus.	

IV. Course contents
1. Real number system. 
2  Sequences of real numbers. The algebraic and topological structure of the real axis.
3. Completeness of the real axis. Series of real numbers.
4. Real functions of a real variable. Limits, continuity, monotonic functions, Lipschitz functions, Banach fixed point theorem, and convex functions.
5. Diferential calculus on the real axis: definitions, properties, mean value theorems, higher order derivatives, Taylor formula, and the study of functions variations.
6. Problemes on the existence of extremum points.
7. Primitives. Defined integrals, the Leibnitz-Newton formula.
8. The Darboux-Stieltjes formula.
9. Vector functions of vector argument.
10. Continuity, the Frechet differential.
11. Partial derivatives.
12. Second and higher order Frechet differential.
13. Existence of local extremum points. Inverse function theorem. Implicit functions.  
14. Duble and triple integrals. 	

V. Bibliography
1. ANDRICA D., DUCA I.D., PURDEA I., POP I.: Matematica de baza, Studium, Cluj-Napoca, 2002. 
2. BALAZS M., KOLUMBAN I.: Analiza matematica,Curs litografiat, Facultatea de Matematica, Univ. $Babes-Bolyai$. 
3. BRECKNER W. W.: Analiza matematica. Topologia spatiului R^n Cluj-Napoca, Universitatea, 1985. 
4. COBZAS S.: Analiza matematica (Calcul diferential), Presa Universitara Clujeana, Cluj-Napoca, 1998. 
5. MARUSCIAC I.: Analiza matematica. I, II, Cluj-Napoca, Universitatea $Babes-Bolyai$, 1980. 
6. MEGAN M.: Bazele analizei matematice,  Ed. BIT, Timisoara, vol I - III, 2000, 2001, 2002.
7. MURESAN, M.: Mathematical Analysis and Appications, Risoprint, Cluj-Napoca, 
8. MURESAN, M.:  A Concrete Approach to Classical Analysis, Springer, New York, 2009.
9. FIHTENHOLT G. M.: Curs de calcul diferential si integral, Vol. I, II. Bucuresti, Editura Tehnica, 1965. 
9. TRIF T.: Probleme de calcul diferential si integral in IR^n, Casa Cartii de Stiinta, Cluj-Napoca, 2003. 

VI. Thematic of didactic activities per weeks
Week 1.  Real number system. 

Week 2. Sequences of real numbers. The algebraic and topological structure of the real axis.

Week 3. . Completeness of the real axis. Series of real numbers.

Week 4. Real functions of a real variable. Limits, continuity, monotonic functions, Lipschitz functions, Banach fixed point theorem, and convex functions.

Week 5. Diferential calculus on the real axis: definitions, properties, mean value theorems, higher order derivatives, Taylor formula, and the study of functions variations.

Week 6. Problemes on the existence of extremum points.

Week 7. Primitives. Defined integrals, the Leibnitz-Newton formula.

Week 8. The Darboux-Stieltjes formula.

Week  9. Vector functions of vector argument.

Week 10. Continuity, the Frechet differential.

Week 11. Partial derivatives.

Week  12. Second and higher order Frechet differential.

Week  13. Existence of local extremum points. Inverse function theorem. Implicit functions.  

Week  14. Duble and triple integrals.	

VII. Didactic methods used
The main teaching methods used in the course are: exposition, discussions, individual study, home-works, quizzes. The lectures are based on the bibliography entries presented above: books, articles, web-sites. The concrete problems are discussed and solved during the seminar hours and at home.	

VIII. Assessment
 The students have to pass at a satisfactory level a midterm test. Its weight at the final grade is 20%.  The final grade is a weighted mean between the midterm test, weekly activity (20%), and the  written and oral exam (60%).	

IX. Additional references

		Dean,						Chair head,
   Prof. Tambulea Leon, Ph.D.			Conf. Breaz Simion, Ph.D.		

	Date,	25.09.2010				Course responsible,
							Prof. MURESAN Marian, Ph.D.