Preface

The Point of this Book

Projects

Prerequisites

Book Features

Elliptic Functions and Maple Note

Thanks

For Users of Previous Editions

Maple 8 to 9

Note to Students

Chapter 1. The Geometry of Curves

1.1 Introduction

1.2 Arclength Parametrization

1.3 Frenet Formulas

1.4 Non-Unit Speed Curves

1.5 Some Implications of Curvature and Torsion

1.6 Green’s Theorem and the Isoperimetric Inequality

1.7 The Geometry of Curves and Maple

Chapter 2. Surfaces

2.1 Introduction

2.2 The Geometry of Surfaces

2.3 The Linear Algebra of Surfaces

2..4 Normal Curvature

2.5 Surfaces and Maple

Chapter 3. Curvatures

3.1 Introduction

3.2 Calculating Curvature

3.3 Surfaces of Revolution

3.4 A Formula for Gauss Curvature

3.5 Some Effects of Curvature(s)

3.6 Surfaces of Delaunay

3.7 Elliptic Functions, Maple and Geometry

3.8 Calculating Curvature with Maple

Chapter 4. Constant Mean Curvature Surfaces

4.1 Introduction

4.2 First Notions in Minimal Surfaces

4.3 Area Minimization

4.4 Constant Mean Curvature

4.5 Harmonic Functions

4.6 Complex Variables

4.7 Isothermal Coordinates

4.8 The Weierstrass-Enneper Representations

4.9 Maple and Minimal Surfaces

Chapter 5. Geodesics, Metrics and Isometries

5.1 Introduction

5.2 The Geodesic Equations and the Clairaut Relation

5.3 A Brief Digression on Completeness

5.4 Surfaces not in R3

5.5 Isometries and Conformal Maps

5.6 Geodesics and Maple

5.7 An Industrial Application

Chapter 6. Holonomy and the Gauss-Bonnet Theorem

6.1 Introduction

6.62 The Covariant Derivative Revisited

6..3 Parallel Vector Fields and Holonomy

6.4 Foucault's Pendulum

6.5 The Angle Excess Theorem

6.6 The Gauss-Bonnet Theorem

6.7 Applications of Gauss-Bonnet

6.8 Geodesic Polar Coordinates

6.9 Maple and Holonomy

Chapter 7. The Calculus of Variations and Geometry

7.1 The Euler-Lagrange Equations

7.2 Transversality and Natural Boundary Conditions

7.3 The Basic Examples

7.4 Higher-Order Problems

7.5 The Weierstrass E-Function

7.6 Problems with Constraints

7.7 Further Applications to Geometry and Mechanics

7.8 The Pontryagin maximum Principle

7.9 An Application to the Shape of a Balloon

7.10 The Caluclus of Variations and Maple

Chapter 8. A Glimpse at Higher Dimensions

8.1 Introduction

8.2 Manifolds

8.3 The Covariant Derivative

8.4 Christoffel Symbols

8.5 Curvatures

8.6 The Charming Doubleness

Appendix A. List of Examples

A.1 Examples in Chapter 1

A.2 Examples in Chapter 2

A.3 Examples in Chapter 3

A.4 Examples in Chapter 4

A.5 Examples in Chapter 5

A.6 Examples in Chapter 6

A.7 Examples in Chapter 7

A.8 Examples in Chapter 8

Appendix B. Hints and Solutions to Selected Problems

B.1. Chapter 1: The Geometry of Curves

B.2. Chapter 2: Surfaces

B.3. Chapter 3: Curvatures

B.4 Chapter 4: Constant Mean Curvature Surfaces

B.5 Chapter 5: Geodesics, Metrics and Isometries

B.6. Chapter 6: Holonomy and the Gauss-Bonnet Theorem

B.7. Chapter 7: The Calculus of Variations and Geometry

B.8. Chapter 8: A Glimpse of Higher Dimensions

Appendix C. Suggested Projects for Differential Geometry