syllabus
1. Information regarding the programme
1.1 Higher education institution 	Babes-Bolyai University, Cluj-Napoca	
1.2 Faculty	Mathematics and Computer Science	
1.3 Department	Computer Science	
1.4 Field of study	Mathematics	
1.5 Study cycle	Licence	
1.6 Study programme / Qualification 		

2. Information regarding the discipline 
2.1 Name of the discipline	Mathematical Analysis	
2.2 Course coordinator 	Prof. dr. Marian Muresan	
2.3 Seminar coordinator	Prof. dr. Marian Muresan	
2.4. Year of study	1	2.5 Semester	1	2.6. Type of evaluation	written	2.7 Type of discipline	compulsory	

3. Total estimated time (hours/semester of didactic activities) 
3.1 Hours per week 	4	Of which: 3.2 course	2	3.3 seminar/laboratory	2	
3.4 Total hours in the curriculum 	4	Of which: 3.5 course	2	3.6 seminar/laboratory	2	
Time allotment:	hours	
Learning using manual, course support, bibliography, course notes	2	
Additional documentation (in libraries, on electronic platforms, field documentation) 	2	
Preparation for seminars/labs, homework, papers, portfolios and essays	1	
Tutorship	1/2	
Evaluations	1/2	
Other activities: ..................		
3.7 Total individual study hours 	7	
3.8 Total hours per semester	84	
3.9 Number of ECTS credits	3	

4. Prerequisites (if necessary)
4.1. curriculum	•		
4.2. competencies	•		

5. Conditions (if necessary)
5.1. for the course	•		
5.2.  for the seminar /lab activities	20 exercises or problems weekly	

6. Specific competencies acquired 
Professional competencies	To realize a mathematical analysis exercise or problem and to solve it by at least one method.	
Transversal competencies	To do connections with other topics in mathematics and to use some mathematical packages.	

7. Objectives of the discipline (outcome of the acquired competencies)
7.1 General objective of the discipline
	•	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. 	
7.2 Specific objective of the discipline
	•		

8. Content
8.1 Course	Teaching methods	Remarks	
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			
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, Moscow, 1965 (French).
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.	
8.2 Seminar / laboratory	Teaching methods	Remarks	
1.	Recalling several notions and results from the high school. 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			
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 (French).
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.	

9. Corroborating the content of the discipline with the expectations of the epistemic community, professional associations and representative employers within the field of the program

•	 The content of the mathematical analysis course is designed to satisfy the expectations of several networks of professionals of recognised expertise. The results of our students validate this statement.

10. Evaluation
Type of activity	10.1 Evaluation criteria	10.2 Evaluation methods	10.3 Share in the grade (%)	
10.4 Course	 Exam	Written exam	3	
				
10.5 Seminar/lab activities	Exam 	Written exam	3	
	Midterm test (compulsory)	Midterm test and weekly activity	2 +1	
10.6 Minimum performance standards  5 	
•		

Date				Signature of course coordinator 	Signature of seminar coordinator
..........................		.........................................		............................................

Date of approval  					  Signature of the head of department	
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