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

Code	Subject	
MMG0004	Geometry 3 (Curves and Surfaces)	

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	
Applied Mathematics	3	2+2+0	speciality	compulsory	

Teaching Staff in Charge	Lect. VACARETU Daniel, Ph.D., vacaretu@math.ubbcluj.ro
Prof. VARGA Csaba Gyorgy, Ph.D., csvarga@cs.ubbcluj.ro	
Aims	The course gives to the students the principal tools and also the necesary methods in the study of curves and surfaces with an accent on the intuitive side.	
Content	I. Local theory of curves.
1. Curves in euclidean space.Tangent line and normal plane.
2. Curves in general position. Osculating plane.
3. Frenet frame. Frenet formulas. Curvature and torsion.
4. Geometric iterpretation of curvature and torsion.
5. Contact beatween two curves.
6. Evolute and involute.
7. Envelope of a family of curves in plane.
II.Local theory of surfaces.
1. Surfaces in three-dimensional euclidean space.
2. Tangent plane and normal line of a surface.
3. First fundamental form of a surface.
4. Arc length , angle between two curves on surface .Area element.
5. Second fundamental form.
6. Normal curvature.
7. Asymptotic lines of a surface.
8. Principal curvatutres of a surface. Mean curvature and Gaussian curvature.
9. Theorema egregium.
10.Minimal surfaces and surfaces with constant curvature.
11.Darboux frame. Darboux formulas.
12.Geodesic curvature. Geodesic torsion.
13.Geodesic lines.
	
Bibliography	    1. BLAGA A. PAUL, Lectures on Classical Differential Geometry, Ed.   
                                        RISOPRINT, Cluj-Napoca, 2005
    2. DOBRESCU A., Curs de Geometrie Diferentţială, Ed. Didactică şi Pedagogică,   
                                Bucureşti, 1963
    3. ENGHIŞ P., ŢARINĂ M., Curs de Geometrie Diferenţială, Cluj-Napoca, 1985
    4. FINIKOV P.S., Curs de Geometrie Diferenţială, Ed. Tehnica, Bucureşti, 1954
    
    5. MURGULESCU E., col., Geometrie analitică şi diferenţială, Editura Didactică şi Pedagogică, Bucureşti, 1965.
    6. MURGULESCU E., col., Culegere de probleme de geometrie analitică şi diferenţială,     vol.1 Ed. Didactică şi Pedagogică, Bucureşti, 1971
    7. MURGULESCU E., col., Geometrie analitică in spatiu şi geometrie diferenţială,     
    Culegere de probleme, vol. 2    Ed. Didactică şi Pedagogică, Bucureşti. 
    8. PINTEA C., Geometrie, Presa Universitară Clujeană, 2001.
    9.TEODORESCU I.D., Geometrie Superioara, Ed. Didactică şi Pedagogică, Bucureşti ,   1970
    10.TEODORESCU I.D.,  TEODORESCU S.D., Culegere de probleme de Geometrie Superioara, Ed. Didactică şi Pedagogică, Bucureşti, 1975
	
Assessment	Final exam + 2 assessement tests + homework