lecture # 32 (last) mth352: differential geometry for master of mathematics by dr. sohail iqbal...

Lecture # 32 (Last)MTH352: Differential Geometry

For

Master of Mathematics

By

Dr. SOHAIL IQBALAssistant Professor

Department of Mathematics, CIIT Islamabad

1MTH352: Differential Geometry

Last lecture

Contents:

Abstract SurfacesManifolds

Contents:


Today’s lecture

Contents:

Geodesic CurvesExamples

Contents:


Geodesic Curves

MTH352: Differential Geometry 4

Geodesic Curves


Examples


Examples


Geodesics on cylindersGeodesics are helices on cylinders

Examples


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 plane surface in spaceA curved surface in space

26

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

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

Lecture 3

Lecture 4


Lecture 5

Lecture 6


Lecture 7

Contents:

Introduction to MappingsTangent Maps

Contents:

Introduction to MappingsTangent Maps

Lecture 8

Contents:

The Dot Product Frames

Contents:

The Dot Product Frames

Lecture 9

Contents:

Formulas For The Dot ProductThe Attitude MatrixCross Product

Contents:

Formulas For The Dot ProductThe Attitude MatrixCross Product

Lecture 10

Contents:

Speed Of A CurveVector Fields On CurvesDifferentiation of Vector Fields

Contents:

Speed Of A CurveVector Fields On CurvesDifferentiation of Vector Fields

Lecture 11

Contents:

CurvatureFrenet Frame FieldFrenet FormulasUnit-Speed Helix

Contents:

CurvatureFrenet Frame FieldFrenet FormulasUnit-Speed Helix


Lecture 12

Contents:

Frenet ApproximationPlane Curves

Contents:

Frenet ApproximationPlane Curves

Lecture 13

Contents:

Frenet ApproximationConclusionFrenet Frame For Arbitrary Speed CurvesVelocity And Acceleration

Contents:

Frenet ApproximationConclusionFrenet Frame For Arbitrary Speed CurvesVelocity And Acceleration

Lecture 14

Contents:

Frenet Apparatus For A Regular CurveComputing Frenet Frame The Spherical Image Cylindrical HelixConclusion

Contents:

Frenet Apparatus For A Regular CurveComputing Frenet Frame The Spherical Image Cylindrical HelixConclusion

Lecture 15

Contents:

Cylindrical Helix

Covariant DerivativesEuclidean Coordinate RepresentationProperties Of The Covariant DerivativeThe Vector Field Of Covariant Derivatives

Contents:

Cylindrical Helix

Covariant DerivativesEuclidean Coordinate RepresentationProperties Of The Covariant DerivativeThe Vector Field Of Covariant Derivatives

Lecture 16

Contents:

From Curves to SpaceFrame FieldsCoordinate Functions

Contents:

From Curves to SpaceFrame FieldsCoordinate Functions

Lecture 17

Contents:

Connection FormConnection EquationsHow To Calculate Connection Forms

Contents:

Connection FormConnection EquationsHow To Calculate Connection Forms

Lecture 18

Contents:

Dual FormsCartan Structural EquationsStructural Equations For Spherical Frame

Contents:

Dual FormsCartan Structural EquationsStructural Equations For Spherical Frame

Lecture 19


Lecture 20

Contents:

Implicitly Defined SurfacesSurfaces of RevolutionProperties Of Patches

Contents:

Implicitly Defined SurfacesSurfaces of RevolutionProperties Of Patches

Lecture 21

Contents:

Parameter Curves on SurfacesParametrizationsTorusRuled Surface

Contents:

Parameter Curves on SurfacesParametrizationsTorusRuled Surface

Lecture 22

Contents:

Coordinate ExpressionsCurves on a SurfaceDifferentiable Functions

Contents:

Coordinate ExpressionsCurves on a SurfaceDifferentiable Functions

Lecture 23

Contents:

Tangents Tangent Vector FieldsGradient Vector Field

Contents:

Tangents Tangent Vector FieldsGradient Vector Field

Lecture 24

Contents:

Differential FormsExterior DerivativesDifferential Forms On The Euclidean PlaneClosed And Exact Forms

Contents:

Differential FormsExterior DerivativesDifferential Forms On The Euclidean PlaneClosed And Exact Forms

Lecture 25

Contents:

Mappings of SurfacesTangent Maps of MappingsDiffeomorphism

Contents:

Mappings of SurfacesTangent Maps of MappingsDiffeomorphism

Lecture 26

Contents:

Diffeomorphic SurfacesMapping of Differential Forms

Contents:

Diffeomorphic SurfacesMapping of Differential Forms

Lecture 27

Lecture 28

Contents:

Stokes TheoremReparametrization

Contents:

Stokes TheoremReparametrization

Lecture 29

Contents:

ConnectednessCompactnessOrientability

Contents:

ConnectednessCompactnessOrientability

Lecture 30

Contents:

HomotopySimply Connectd SurfacesPoincare LemmaConditions of Orientability

Contents:

HomotopySimply Connectd SurfacesPoincare LemmaConditions of Orientability

Lecture 31

Contents:


Contents:


Lecture 32

Contents:


Contents:


End of the lecture


What’s Next


Final Examination

lecture # 32 (last) mth352: differential geometry for master of mathematics by dr. sohail iqbal...

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