Lecture # 32 (Last) Dr. SOHAIL IQBAL

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Presentation transcript:

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

Last lecture Contents: Abstract Surfaces Manifolds

Today’s lecture Contents: Geodesic Curves Examples

MTH352: Differential Geometry Geodesic Curves   MTH352: Differential Geometry

MTH352: Differential Geometry Geodesic Curves   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples MTH352: Differential Geometry

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

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

MTH352: Differential Geometry Examples   MTH352: Differential Geometry

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

MTH352: Differential Geometry   MTH352: Differential Geometry

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

MTH352: Differential Geometry Lecture 4   MTH352: Differential Geometry

Lecture 5  

MTH352: Differential Geometry Lecture 6   MTH352: Differential Geometry

Lecture 7 Contents: Introduction to Mappings Tangent Maps

Lecture 8 Contents: The Dot Product Frames

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

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

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

Lecture 12 Contents: Frenet Approximation Plane Curves

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

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

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

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

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

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

MTH352: Differential Geometry Lecture 19   MTH352: Differential Geometry

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

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

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

Lecture 23 Contents: Tangents Tangent Vector Fields Gradient Vector Field

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

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

Lecture 26 Contents: Diffeomorphic Surfaces Mapping of Differential Forms

Lecture 27  

Lecture 28 Contents: Stokes Theorem Reparametrization

Lecture 29 Contents: Connectedness Compactness Orientability

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

Lecture 31 Contents: Abstract Surfaces Manifolds

Lecture 32 Contents: Geodesic Curves Examples

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

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

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