Differential Geometry and Its Applications.

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.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.