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### __ MATH 345 - DIFFERENTIAL GEOMETRY I__

### __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

varna2

__ Exams:__
(%20) First Midterm Exam ps file
: November 8 (SAZ- 04)
(%20) Second Midterm Exam ps file
: December 20 (18.00, Classroom SAZ-18)
(%40) Final Exam ps file
: January 13 (12.15 , Classroom SAZ-04)
(%20) Homework:
homework and midterm grades

** **__ Subjects Covered __

** September 23 **

Tangent vectors
Directional derivatives
Curves
Differential Forms

homework assignment Set 1

** September 30 **
Differential Forms
Mappings

** October 7 **

Dot products and curves
Frenet Formulas
Some special Space and Plane curves

** October 14 **
Arbitrary parametrization
Covariant dervative
Frame fields

Homework Set 2

** October 28 **
Isometries of R^3
Orientation

Homework Set 3

** November 4 **
Congruence of Curves
First Midterm

** 11 November **

Surfaces in R^3

** 18 November **

Patch Computations
Functions on surfaces
Tangent Space

IV th Homework Set

** 25 November **

Differential Forms on Surfaces
Mapping of Surfaces

** 02 December **

Mapping of Surfaces

** 09 December **

Integration of Forms on Surfaces

Topological Properties of Surfaces

** 16 December **

Abstract surfaces and Manifolds

Second Midterm Exam

** 23 December **
Manifolds
Some problems from surface theory

** 30 December **
Some problems from surface theory

__ 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 ** **

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. (M.Ex.1)

8. Nov.11 (Sections 4.1-4.4 Surfaces in R^3 , Differentiable functions

and Tangent vectors, Differential forms on surfaces.

9. Nov.18 (Sections 4.5-4.6) Mapping of surfaces, integration of forms.

10. Nov.25 (Sections 4.7-4.8) Topological properties of surfaces, Manifolds.

11. Dec.02. (Sections 5.1-5.3) Shape operator, Normal curvature, Gaussian
curvature.

12. Dec.09 (Sections 5.4-5.5) Computational Techniques, the implicit example

13. Dec.16 (Sections 5.6-5.7) Special curves and surfaces, surfaces
of revolution. (M.Ex.2)

14. Dec.23 Integrable Surfaces

15. Dec.30 Integrable surfaces
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