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MATH 345 - DIFFERENTIAL GEOMETRY I

Fall 2002,2018

Math345 : Differential Geometry I. A third year course.
Text Book: Differential Geometry of Curves and Surfaces, by Manfredo P. do Carmo. Prentice-Hall, New Jersey , 1976.
Other books : Elementary Differential Geometry, by Barret O' Neil. Academic Press,1997.
Elementary Differential Geometry, by Andrew Pressley, Springer Undergraduate Mathematics Series, 2012.
Content: In this period the first four chapters of " M. Do Carmo" will be covered. See the "Course Syllabus" below
Course Schedule:
  • Monday 15.40-17.30 , SA-Z02
  • Thursday 13.40-15.30 , SA-Z02
    varna2
    Exams:
  • (%20) First Midterm Exam ps file : October 22 (SA-Z02) (S
  • (%20) Second Midterm Exam ps file : November 26 (SA-Z02)
  • (%40) Final Exam ps file : January xx (SA-Zxx)
  • (%20) Homework:
  • homework and midterm grades
    Subjects Covered

    1. September 24
  • Parametrized Curves (1.2)
  • Regular Curves; Arc Length (1.3)
  • The Vector Product (1.4)
    2. October 1
  • The Local Theory of Curves (1.5)
  • The Local Canonical Form (1.6)
  • Global Properties of Plane Curves.2
    homework assignment Set 1
    3. October 8
  • Regular Surfaces
  • Change of Parameters;Difgerential Functions on Surfaces
    4. October 15
  • The Tangent Plane; the differentail of a map
  • The First Fundamenatl form
    5. October 22
  • Orientation of Surfaces
  • A Characterization of Compact Orientable Surfaces
  • First Midtrem Exam
    Homework Set 2
    6. October 29
  • The Gauss Map and its Fundamental Properties
    Homework Set 3
    7. November 5
  • The Gauss Map
    8. November 12
  • The Vector fields
    9. November 19
  • Ruled and Minimal Surfaces
    IV th Homework Set
    10. November 26
  • Differential Forms on Surfaces
  • Second Midterm Exam
    11. December 03
  • Isometries
    December 10
  • 12. The Gauss Theorem
    13. December 17
  • Parallel Transport
    14. December 20
  • The exponential Map

    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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