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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 | |