1. Lecture # 32 (Last) Dr. SOHAIL IQBAL
MTH352: Differential Geometry
For
 Master of Mathematics By
Dr. SOHAIL IQBAL
Assistant Professor
Department of Mathematics, CIIT Islamabad
MTH352: Differential Geometry

2. Last lecture
Contents:
Abstract Surfaces
Manifolds

3. Today’s lecture
Contents:
Geodesic Curves
Examples

4. MTH352: Differential Geometry
Geodesic Curves
MTH352: Differential Geometry

5. MTH352: Differential Geometry
Geodesic Curves
MTH352: Differential Geometry

6. MTH352: Differential Geometry
Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

11. MTH352: Differential Geometry
Examples
Geodesics on cylinders
Geodesics are helices on cylinders
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

13. MTH352: Differential Geometry
Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

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Examples
MTH352: Differential Geometry

26. Aim of the course: Main aim of the course is to:
Review of differential calculus.
Develop tools to study curves and surfaces in space.
Proper definition of surface. How to do calculus on surface.
A detailed study of geometry of surface.
A curved surface in space
A plane surface in space

27. MTH352: Differential Geometry
MTH352: Differential Geometry

28. Lecture 3 Contents: Directional derivatives
Definition
How to differentiate composite functions
(Chain rule)
How to compute directional derivatives
more efficiently
The main properties of directional derivatives
Operation of a vector field
Basic properties of operations of vector fields

29. MTH352: Differential Geometry
Lecture 4
MTH352: Differential Geometry

30. Lecture 5

31. MTH352: Differential Geometry
Lecture 6
MTH352: Differential Geometry

32. Lecture 7
Contents:
Introduction to Mappings
Tangent Maps

33. Lecture 8
Contents:
The Dot Product
Frames

34. Lecture 9 Contents: Formulas For The Dot Product The Attitude Matrix
Cross Product

35. Lecture 10 Contents: Speed Of A Curve Vector Fields On Curves
Differentiation of Vector Fields

36. MTH352: Differential Geometry
Lecture 11
Contents:
Curvature
Frenet Frame Field
Frenet Formulas
Unit-Speed Helix
MTH352: Differential Geometry

37. Lecture 12
Contents:
Frenet Approximation
Plane Curves

38. Lecture 13 Contents: Frenet Approximation Conclusion
Frenet Frame For Arbitrary Speed Curves
Velocity And Acceleration

39. Lecture 14 Contents: Frenet Apparatus For A Regular Curve
Computing Frenet Frame
The Spherical Image
Cylindrical Helix
Conclusion

40. Lecture 15 Contents: Cylindrical Helix Covariant Derivatives
Euclidean Coordinate Representation
Properties Of The Covariant Derivative
The Vector Field Of Covariant Derivatives

41. Lecture 16 Contents: From Curves to Space Frame Fields
Coordinate Functions

42. Lecture 17 Contents: Connection Form Connection Equations
How To Calculate Connection Forms

43. Lecture 18 Contents: Dual Forms Cartan Structural Equations
Structural Equations For Spherical Frame

44. MTH352: Differential Geometry
Lecture 19
MTH352: Differential Geometry

45. Lecture 20 Contents: Implicitly Defined Surfaces
Surfaces of Revolution
Properties Of Patches

46. Lecture 21 Contents: Parameter Curves on Surfaces Parametrizations
Torus
Ruled Surface

47. Lecture 22 Contents: Coordinate Expressions Curves on a Surface
Differentiable Functions

48. Lecture 23 Contents: Tangents Tangent Vector Fields
Gradient Vector Field

49. Lecture 24 Contents: Differential Forms Exterior Derivatives
Differential Forms On The Euclidean Plane
Closed And Exact Forms

50. Lecture 25 Contents: Mappings of Surfaces Tangent Maps of Mappings
Diffeomorphism

51. Lecture 26 Contents: Diffeomorphic Surfaces
Mapping of Differential Forms

52. Lecture 27

53. Lecture 28
Contents:
Stokes Theorem
Reparametrization

54. Lecture 29
Contents:
Connectedness
Compactness
Orientability

55. Lecture 30 Contents: Homotopy Simply Connectd Surfaces Poincare Lemma
Conditions of Orientability

56. Lecture 31
Contents:
Abstract Surfaces
Manifolds

57. Lecture 32
Contents:
Geodesic Curves
Examples

58. MTH352: Differential Geometry
End of the lecture
MTH352: Differential Geometry

59. MTH352: Differential Geometry
What’s Next
Final Examination
MTH352: Differential Geometry