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
Teaching Staff in Charge
Lect. VACARETU Daniel, Ph.D.,  vacaretumath.ubbcluj.ro
Lect. MEZEI Ildiko Ilonka,  ildiko.mezeimath.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.
References
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

Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject