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.	Mathematics	*			

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. Differential calculus on the real axis: definitions, properties, mean value theorems, higher order derivatives, Taylor formula, and the study of functions variations.
6. Problems 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. Double and triple integrals. 	

V. Bibliography
1. Andras, Sz., Muresan, M., Mathematical Analysis and its Applications, EDP, Bucuresti, 2005,
  (Hungarian).
2. Breckner W. W., Mathematical Analysis. The Topology of the R^n Space, Universitatea din Cluj-Napoca, 1985 (Romanian).
3. Cobzas, S., Mathematical Analysis (Differential Calculus), Presa Universitara Clujeana, Cluj-Napoca, 1997 (Romanian).
4. Demidovitch, B., Requeil d’exercices et problemes d’analyse mathématique, Paris, 1965.
5. Muresan, M., Mathematics for Competitions, Cyprus Math. Soc., Nicosia, 2006. 
6. Muresan, M., A Concrete Approach to Classical Analysis, Springer, New York, 2009.
7. Rudin, W., Principles of Mathematical Analysis, 2nd Edition, McGraw-Hill, New York, 1964.
8. Trif, T., Differential and Integral Problems in R^n, Casa cartii de stiinta, Cluj-Napoca, 2003
    (Romanian). 	

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. Differential calculus on the real axis: definitions, properties, mean value theorems, higher order derivatives, Taylor formula, and the study of the variations of variations.
Week 6. Problems on the existence of extremum points.
Week 7. Primitives. Defined integrals, the Leibnitz-Newton formula.
Week 8. The Darboux-Stieltjes integral. Improper integral.
Week 9. Vector functions of vector argument.
Week 10. Continuity, the Frechet differential.
Week 11. Partial derivatives.
Week 12. Second and higher order Frechet differentials.
Week 13. Existence of local extremum points. Inverse function theorem. Implicit functions.  
Week 14. Double and triple integrals.	

VII. Didactic methods used

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. dr. Leon Tambulea				Prof. dr. Radu Precup	

	Date,							Course responsible,
 December, 15, 2011					Prof. MURESAN Marian, Ph.D.