
Book DetailsClassroom Resource MaterialsVolume: 30; 2007; 469 pp
Reprinted edition available: CLRM/59
Differential Geometry and Its Applications studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole. It mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. That mix of ideas offers students the opportunity to visualize concepts through the use of computer algebra systems such as Maple.
Differential Geometry and Its Applications emphasizes that this visualization goes hand in hand with understanding the mathematics behind the computer construction. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.

Table of Contents

cover

copyright page

title page

Contents

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

1 The Geometry of Curves

1.1 Introduction

1.2 Arclength Parametrization

1.3 Frenet Formulas

1.4 NonUnit 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

2 Surfaces

2.1 Introduction

Examples of Patches (or Parametrizations) on Surfaces

2.2 The Geometry of Surfaces

2.3 The Linear Algebra of Surfaces

2.4 Normal Curvature

2.5 Surfaces and Maple

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

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 WeierstrassEnneper Representations

4.9 Maple and Minimal Surfaces

4.9.1 Minimal Surface Plots

4.9.2 The Minimal Surface Equation

4.9.3 A Geometric Condition: Minimal Surfaces of Revolution

4.9.4 An Algebraic Condition

4.9.5 Maple and Area Minimization

4.9.6 Maple and the Weierstrass Enneper Representation

Special color pages

A geodesic on an ellipsoid

A closed geodesic on an unduloid

A nonclosed geodesic on an unduloid

Catalan’s surface

A perturbed Boy’s surface

Enneper’s surface

A helicoid

Henneberg’s surface

A planar lines of curvature surface

A twisted cylinder

Scherk’s fifth surface

Another view of the Bat

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 R^3

5.5 Isometries and Conformal Maps

5.6 Geodesics and Maple

5.6.1 Plotting Geodesics

5.6.2 Geodesics on the Cone

5.6.3 Geodesics on the Cylinder

5.6.4 Geodesics on the Unduloid

5.6.5 Geodesics on Surfaces not in R^3

5.6.6 Stereographic and Mercator Projections

5.7 An Industrial Application

6 Holonomy and the GaussBonnet Theorem

6.1 Introduction

6.2 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 GaussBonnet Theorem

6.7 Applications of GaussBonnet

6.8 Geodesic Polar Coordinates

6.9 Maple and Holonomy

7 The Calculus of Variations and Geometry

7.1 The EulerLagrange Equations

7.2 Transversality and Natural Boundary Conditions

7.3 The Basic Examples

7.4 HigherOrder Problems

7.4.1 A HigherOrder EulerLagrange Equation

7.4.2 HigherOrder Natural Boundary Conditions

7.5 The Weierstrass E Function

7.6 Problems with Constraints

7.6.1 Integral Constraints

7.6.2 Holonomic Constraints

7.6.3 Differential Equation 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 Calculus of Variations and Maple

7.10.1 Basic EulerLagrange Procedures

7.10.2 Buckling under Compression

7.10.3 The Double Pendulum

7.10.4 Constrained Particle Motion

7.10.5 Maple and the Mylar Balloon

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

Chapter 1: The Geometry of Curves

Chapter 2: Surfaces

Chapter 3: Curvatures

Chapter 4: Constant Mean Curvature Surfaces

Chapter 5: Geodesics, Metrics and Isometries

Chapter 6: Holonomy and the GaussBonnet Theorem

Chapter 7: The Calculus of Variations and Geometry

Chapter 8: A Glimpse at Higher Dimensions

Appendix C Suggested Projects for Differential Geometry

Project 1: Developable Surfaces

Project 2: The Gauss Map

Project 3: Minimal Surfaces and Area Minimization

Project 4: Unduloids

Project 5: The Shape of a Mylar Balloon

Bibliography

Index

About the Author


Reviews

... There is a good deal to like about this book: the writing is lucid, drawings and diagrams are plentiful and carefully done, and the author conveys a contagious sense of enthusiasm for his subject.
William J. Satzer, MAA Reviews


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 Book Details
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Reprinted edition available: CLRM/59
Differential Geometry and Its Applications studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole. It mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. That mix of ideas offers students the opportunity to visualize concepts through the use of computer algebra systems such as Maple.
Differential Geometry and Its Applications emphasizes that this visualization goes hand in hand with understanding the mathematics behind the computer construction. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.

cover

copyright page

title page

Contents

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

1 The Geometry of Curves

1.1 Introduction

1.2 Arclength Parametrization

1.3 Frenet Formulas

1.4 NonUnit 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

2 Surfaces

2.1 Introduction

Examples of Patches (or Parametrizations) on Surfaces

2.2 The Geometry of Surfaces

2.3 The Linear Algebra of Surfaces

2.4 Normal Curvature

2.5 Surfaces and Maple

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

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 WeierstrassEnneper Representations

4.9 Maple and Minimal Surfaces

4.9.1 Minimal Surface Plots

4.9.2 The Minimal Surface Equation

4.9.3 A Geometric Condition: Minimal Surfaces of Revolution

4.9.4 An Algebraic Condition

4.9.5 Maple and Area Minimization

4.9.6 Maple and the Weierstrass Enneper Representation

Special color pages

A geodesic on an ellipsoid

A closed geodesic on an unduloid

A nonclosed geodesic on an unduloid

Catalan’s surface

A perturbed Boy’s surface

Enneper’s surface

A helicoid

Henneberg’s surface

A planar lines of curvature surface

A twisted cylinder

Scherk’s fifth surface

Another view of the Bat

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 R^3

5.5 Isometries and Conformal Maps

5.6 Geodesics and Maple

5.6.1 Plotting Geodesics

5.6.2 Geodesics on the Cone

5.6.3 Geodesics on the Cylinder

5.6.4 Geodesics on the Unduloid

5.6.5 Geodesics on Surfaces not in R^3

5.6.6 Stereographic and Mercator Projections

5.7 An Industrial Application

6 Holonomy and the GaussBonnet Theorem

6.1 Introduction

6.2 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 GaussBonnet Theorem

6.7 Applications of GaussBonnet

6.8 Geodesic Polar Coordinates

6.9 Maple and Holonomy

7 The Calculus of Variations and Geometry

7.1 The EulerLagrange Equations

7.2 Transversality and Natural Boundary Conditions

7.3 The Basic Examples

7.4 HigherOrder Problems

7.4.1 A HigherOrder EulerLagrange Equation

7.4.2 HigherOrder Natural Boundary Conditions

7.5 The Weierstrass E Function

7.6 Problems with Constraints

7.6.1 Integral Constraints

7.6.2 Holonomic Constraints

7.6.3 Differential Equation 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 Calculus of Variations and Maple

7.10.1 Basic EulerLagrange Procedures

7.10.2 Buckling under Compression

7.10.3 The Double Pendulum

7.10.4 Constrained Particle Motion

7.10.5 Maple and the Mylar Balloon

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

Chapter 1: The Geometry of Curves

Chapter 2: Surfaces

Chapter 3: Curvatures

Chapter 4: Constant Mean Curvature Surfaces

Chapter 5: Geodesics, Metrics and Isometries

Chapter 6: Holonomy and the GaussBonnet Theorem

Chapter 7: The Calculus of Variations and Geometry

Chapter 8: A Glimpse at Higher Dimensions

Appendix C Suggested Projects for Differential Geometry

Project 1: Developable Surfaces

Project 2: The Gauss Map

Project 3: Minimal Surfaces and Area Minimization

Project 4: Unduloids

Project 5: The Shape of a Mylar Balloon

Bibliography

Index

About the Author

... There is a good deal to like about this book: the writing is lucid, drawings and diagrams are plentiful and carefully done, and the author conveys a contagious sense of enthusiasm for his subject.
William J. Satzer, MAA Reviews