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Differential Geometry and Its Applications
 
Differential Geometry and Its Applications
MAA Press: An Imprint of the American Mathematical Society
Now available in new edition: CLRM/59
Differential Geometry and Its Applications
Click above image for expanded view
Differential Geometry and Its Applications
MAA Press: An Imprint of the American Mathematical Society
Now available in new edition: CLRM/59
  • Book Details
     
     
    Classroom Resource Materials
    Volume: 302007; 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 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
    • 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 Weierstrass-Enneper 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 non-closed 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 Gauss-Bonnet 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 Gauss-Bonnet Theorem
    • 6.7 Applications of Gauss-Bonnet
    • 6.8 Geodesic Polar Coordinates
    • 6.9 Maple and Holonomy
    • 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
    • 7.4.1 A Higher-Order Euler-Lagrange Equation
    • 7.4.2 Higher-Order 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 Euler-Lagrange 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 Gauss-Bonnet 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 302007; 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.

  • 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 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
  • 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 Weierstrass-Enneper 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 non-closed 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 Gauss-Bonnet 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 Gauss-Bonnet Theorem
  • 6.7 Applications of Gauss-Bonnet
  • 6.8 Geodesic Polar Coordinates
  • 6.9 Maple and Holonomy
  • 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
  • 7.4.1 A Higher-Order Euler-Lagrange Equation
  • 7.4.2 Higher-Order 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 Euler-Lagrange 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 Gauss-Bonnet 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
Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
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