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| Preface | |
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| The Geometry of Curves | |
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| Introduction | |
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| Arclength Parametrization | |
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| Frenet Formulas | |
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| Nonunit Speed Curves | |
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| Some Implications of Curvature and Torsion | |
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| The Geometry of Curves and MAPLE | |
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| Surfaces | |
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| Introduction | |
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| The Geometry of Surfaces | |
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| The Linear Algebra of Surfaces | |
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| Normal Curvature | |
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| Plotting Surfaces in MAPLE | |
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| Curvature(s) | |
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| Introduction | |
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| Calculating Curvature | |
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| Surfaces of Revolution | |
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| A Formula for Gaussian Curvature | |
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| Some Effects of Curvature(s) | |
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| Surfaces of Delaunay | |
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| Calculating Curvature with MAPLE | |
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| Constant Mean Curvature Surfaces | |
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| Introduction | |
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| First Notions in Minimal Surfaces | |
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| Area Minimization | |
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| Constant Mean Curvature | |
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| Harmonic Functions | |
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| Geodesics, Metrics and Isometries | |
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| Introduction | |
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| The Geodesic Equations and the Clairaut Relation | |
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| A Brief Digression on Completeness | |
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| Surfaces not in R3 | |
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| Isometries and Conformal Maps | |
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| Geodesics and MAPLE | |
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| Holonomy and the Gauss-Bonnet Theorem | |
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| Introduction | |
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| The Covariant Derivative Revisited | |
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| Parallel Vector Fields and Holonomy | |
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| Foucault's Pendulum | |
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| The Angle Excess Theorem | |
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| The Gauss-Bonnet Theorem | |
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| Geodesic Polar Coordinates | |
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| Minimal Surfaces and Complex Variables | |
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| Complex Variables | |
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| Isothermal Coordinates | |
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| The Weierstrass-Enneper Representations | |
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| Bjurling's Problem | |
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| Minimal Surfaces which are not Area Minimizing | |
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| Minimal Surfaces and MAPLE | |
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| The Calculus of Variations and Geometry | |
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| The Euler-Lagrange Equations | |
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| The Basic Examples | |
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| The Weierstrass E-Function | |
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| Problems with Constraints | |
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| Further Applications to Geometry and Mechanics | |
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| The Pontryagin Maximum Principle | |
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| The Calculus of Variations and MAPLE | |
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| A Glimpse at Higher Dimensions | |
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| Introduction | |
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| Manifolds | |
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| The Covariant Derivative | |
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| Christoffel Symbols | |
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| Curvatures | |
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| The Charming Doubleness | |
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| List of Examples, Definitions and Remarks | |
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| Answers and Hints to Selected Exercises | |
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| References | |
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| Index | |