Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate
SUBJECT

Code	Subject	
MMA0019	Mathematical Analysis 2 (Differential Calculus in R^n)	

Section	Sem.	Hours: C+S+L	Category	Type	
Mathematics	2	2+2+0	fundamental	compulsory	
Mathematics and Computer Science	2	2+2+0	fundamental	compulsory	

Teaching Staff in Charge	Assoc.Prof. TRIF Tiberiu Vasile, Ph.D., ttrif@math.ubbcluj.ro
Assoc.Prof. FINTA Zoltan, Ph.D., fzoltan@math.ubbcluj.ro	
Aims	Getting to know improper integrals, the topology of the Euclidean space R^n, and the differential calculus of functions of several real variables.	
Content	1. Improper integrals. Definition of improper integrals over different types of noncompact intervals. Improper integrals vs Riemann integrals. Computation of improper integrals: linearity, Leibniz-Newton formula, integration by parts, change of variable in improper integrals. Convergence tests for improper integrals.
2. Topology of the Euclidean space R^n. The Euclidian space R^n. Sequences in R^n. Compact sets in R^n. Limits of vector functions of vector variable. Continuity of vector functions of vector variable.
3. Differential calculus in R^n. Linear mappings, the norm of a linear mapping, bijective linear mappings. Derivative of a vector function of a real variable. Differentiability of vector functions of vector variable, directional derivatives, partial derivatives and their connection with differentiability. Operations with differentiable functions, differentiability of the inverse of a bijective function. Mean-value theorems for differentiable functions of vector variable. Functions of the class C^1. The inverse function theorem. Differentiable implicit functions. Optimization problems having equations as constraints. Second order partial derivatives, theorems of Schwarz and Young. The second order differential. Necessary and sufficient optimality conditions. Higher order partial derivatives, Taylor formula.	
Bibliography	l. BALÁZS M.: Matematikai analizis, Erdélyi Tankönyvtanács, Kolozsvár, 2000
2. BALÁZS M., KOLUMBÁN I.: Matematikai analizis, Dacia Könyvkiado, Kolozsvár-Napoca, 1978
3. BRECKNER W. W.: Analiza matematica. Topologia spatiului R^n. Universitatea din Cluj-Napoca, 1985
4. BROWDER A.: Mathematical Analysis. An Introduction, Springer-Verlag, New York, 1996
5. BUCUR G., CÂMPU E., G&Atilde;IN&Atilde; S.: Culegere de probleme de calcul diferential si integral, Vol. II, Editura Tehnica Bucuresti 1966. Vol. III, Editura Tehnic&atilde;, Bucuresti, 1967
6. COBZAS ST.: Analiz&atilde; matematic&atilde; (Calcul diferential), Presa Universitar&atilde; Clujean&atilde;, Cluj-Napoca, 1997
7. DEMIDOVICI B.P.: Culegere de probleme si exercitii de analiz&atilde; matematic&atilde;, Editura Tehnic&atilde;, Bucuresti, 1956
8. HEUSER H.: Lehrbuch der Analysis, Teil 1, 11. Auflage, B. G. Teubner, Stuttgart, 1994;
Teil 2, 9. Auflage, B. G. Teubner, Stuttgart, 1995
9. MEGAN M.: Bazele analizei matematice, Vol. I + Vol. II, Editura EUROBIT, Timisoara, 1997. Vol. III, Editura EUROBIT, Timisoara, 1998
10. RUDIN W.: Principles of Mathematical Analysis, 2nd Edition, McGraw-Hill, New York, 1964
11. WALTER W.: Analysis, I, II, Springer-Verlag, Berlin, 1990
12. TRIF T.: Probleme de calcul diferential si integral în R^n, Casa Cartii de Stiinta, Cluj-Napoca, 2003	
Assessment	A test paper during the semester (20%) + Exam (80%).