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