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Mastering Differential Geometry: Geodesic Curves and Surface Analysis

Explore geodesic curves, directional derivatives, and surface mappings in this in-depth lecture series on differential geometry. Understand the calculus on surfaces, Frenet frame formulations, and covariant derivatives.

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Mastering Differential Geometry: Geodesic Curves and Surface Analysis

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  1. Lecture # 32 (Last) 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. Geodesic Curves MTH352: Differential Geometry

  5. Geodesic Curves MTH352: Differential Geometry

  6. Examples MTH352: Differential Geometry

  7. Examples MTH352: Differential Geometry

  8. Examples MTH352: Differential Geometry

  9. Examples MTH352: Differential Geometry

  10. Examples MTH352: Differential Geometry

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

  12. Examples MTH352: Differential Geometry

  13. Examples MTH352: Differential Geometry

  14. Examples MTH352: Differential Geometry

  15. Examples MTH352: Differential Geometry

  16. Examples MTH352: Differential Geometry

  17. Examples MTH352: Differential Geometry

  18. Examples MTH352: Differential Geometry

  19. Examples MTH352: Differential Geometry

  20. Examples MTH352: Differential Geometry

  21. Examples MTH352: Differential Geometry

  22. Examples MTH352: Differential Geometry

  23. Examples MTH352: Differential Geometry

  24. Examples MTH352: Differential Geometry

  25. 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

  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. Lecture 4 MTH352: Differential Geometry

  30. Lecture 5

  31. 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. 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. 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

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