### Fall 2002

Math345 : Differential Geometry I. A third year course.
Text book : Elementary Differential Geometry, by Barret O' Neil. Academic Press,1997.
Other Books: Differential Geometry of Curves and Surfaces, by Manfredo P. do Carmo. Prentice-Hall, New Jersey , 1976.
Content: In this period the first five chapters of " O'Neil " will be covered. See the "Course Syllabus" below
Course Schedule:
• Tuesday 13.40-15.30 , SAZ-19
• Fiday 10.40-12.30 , SAZ-19
Exams:
• (%20) First Midterm Exam: November 8 (During the Lecture hours)
• (%20) Second Midterm Exam: December 20 (During the Lecture hours)
• (%40) Final Exam :
• (%20) Homework:
• see homework assigments

Subjects Covered

September 23
• Tangent vectors
• Directional derivatives
• Curves
• Differential Forms
homework assigment Set 1

September 30
• Differential Forms
• Mappings

October 7
• Dot products and curves
• Frenet Formulas
• Space curves and covariant derivative

October 14

### Course Syllabus

famous curves

1. Sept.23 (Sections 1.2,1.3) Euclidean Space, Tangent vectors, Directional Derivatives.
2. Sept.30 (Sections 1.4.1.7) Curves in space, 1-forms, differential Forms, Mappings.
3. Oct.07 (Sections 2.1- 2.3) Dot product, curves , The Serret Frenet Formulas
4. Oct.14 (Sections 2.4-2.6 Covariant derivative, frame fields, connection forms (M.Ex.1)
5. Oct.21 (Sections 2.7,2.9) Regular Surfaces, Change of Parameters
6. Oct.28 (Sections 3.1-3.3) Isometries , the Tangent map, orientation.
7. Nov.4 (Sections 3.5-3.7) Euclidean Geometry, Congruence of curves.
8. Nov.11 (Sections 4.1-4.4 Surfaces in R^3 , Differentiable functions
and Tangent vectors, Differential forms on surfaces. (M.Ex.2)
9. Nov.08 (Sections 4.5-4.6) Mapping of surfaces, integration of forms.
10. Nov.15 (Sections 4.7-4.8) Topological properties of surfaces, Manifolds.
11. Nov.22. (Sections 5.1-5.3) Shape operator, Normal curvature, Gaussian curvature.
12. Nov.29 (Sections 5.4-5.5) Computational Techniques, the implicit example (M.Ex.3)
13. Dec.06 (Sections 5.6-5.7) Special curves and surfaces, surfaces of revolution.
14. Dec.13 Integrable Surfaces
15. Dec.20 Integrable surfaces
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