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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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    Last update September 2002


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