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

SUBJECT

Code
Subject
MMA0029 Mathematical Analysis 3 (Integral Calculus in R^n)
Section
Semester
Hours: C+S+L
Category
Type
Mathematics
3
2+2+0
speciality
compulsory
Mathematics and Computer Science
3
2+2+0
speciality
compulsory
Teaching Staff in Charge
Assoc.Prof. DIACONU Adrian, Ph.D.,  adiaconumath.ubbcluj.ro
Assoc.Prof. FINTA Zoltan, Ph.D.,  fzoltanmath.ubbcluj.ro
Assoc.Prof. BRECKNER Brigitte Erika, Ph.D.,  brigittemath.ubbcluj.ro
Aims
Getting to know the classical knowledges of integral calculus of functions of several real variables.
Content
I) Line integrals.
1) Vectorfuntions with values in Rn and with bundet variation. Vectorfuntions with the C1 class. The calculation of the total variation for a vectorfuntion with values in Rn and the C1 class.
2) Lines. The length and the lengt function for a lines and theirs expression in the case of the lines with the C1 class.
3) Integration on a line of a real function with n real variables. Calculation formulas in special situations.
4) Liniar differential forms and theirs integration. Calculation formulas in special situations.
5) The primitive an a vectorfuntion and the calculation of the integral of a liniar differential form on a line with the help of the primitive of the generating vectorfuntion.
6) Operations with lines. The conduct of the integral with regard these operations.
7) Theorems regarding the independece of line of the line integrals from a liniar differentil form. Necesary and sufficient conditions for the existence of the primitive of a vectorfunction.
8) Oriented arcs. The independece of the integral of the representative line of the arc. The integral on a oriented arc.
II) The Riemann integral of a real function with furthemore variables.
1) The Jordan measure of a set from Rn. Simple sets from R2 and R3 limited by the graphics of continuous functions.
2) The definition of the integrability and the integral of areal function with n real variables. The bounding of the integrabes functions.
3) The Darboux sums in the case of functions with furthemore variables. The theorem of Darboux in this case.
4) The Lebesgue criterion for the Riemann integrability.
5) The additivity with regard to the integration set of the integrability and of the integral.
6) The reduction of the integral for a function with mn variables to the iterate integrals. The Fubinii@s theorem. The calculation of the double and the triple integrals through the reduction of iterate integrals.
7) The connection between the integral of a set from R2 and the integral of his orieted border. The Green@s formula.
8) The formula of the changing of the variabels for a function with n variabels. The proof in the case of the double integral. Remarcabel cases for the double and the triple integral.
III) Surface integrals.
1) Surface parametrisations in R3. The measure of the surface determined through a parametrisation and the integral of a function defined on this surface with regard to the determined measure.
2) Differential forms with the second order and theirs integration on a surface parametrisation.
3) Oriented surface parts in R3. The integral of a differential form with the second order on an oriented surface part.
4) The theorems of Stokes and Gauss- Ostrogradski for the connection between integral types.
References
1. BALAZS M.: Analiza matematica, III si IV, Universitate, Cluj-Napoca, 1983,1984
2. BALAZS M., KOLUMBAN I.: Matematikai analizis, Dacia Konivkyado, Kolozsvar-Napoca, 1978
3. BOBOC N.: Analiza matematica, II, Universitate, Bucuresti, 1993
4. BUCUR G., CAMPU E., GAINA S.: Culegere de probleme de calcul diferential si integral, III, Editura tehnica, Bucuresti, 1967
5. COBZAS ST.: Analiza matematica (Calcul diferential), Presa universitara clujeana, Cluj-Napoca, 1997
6. COLOJOARA I.: Analiza matematica, Ed. did. si ped., Bucuresti, 1983
7. DEMIDOVICI B.P.: Culegere de probleme si exercitii de analiza matematica, Ed. tehnica, Bucuresti, 1956
8. SIKORSKI R.: Advanced Calculus, PWN-Polish Scientific Publishiers, Warsawa, 1969
9. WALTER W.: ANALYSIS I, Zweite Aufl. Berlin, Springer-Verlag,1990
10. ***: Analiza matematica, II, Ed. did. si pedag., Bucuresti, 1980
11. HEUSER H.: Lehrbuch der Analysis.Teil 2. 9. Auflage. Stuttgart: B. G. Teubner, 1995.
Assessment
Exam.
Links: Syllabus for all subjects
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