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| Preface | |
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| How to Use This Book | |
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| Surfaces and Straightness | |
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| When Do You Call a Straight Line? | |
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| How Do You Construct a Straight Line? | |
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| Local (and Infinitesimal) Straightness | |
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| Intrinsic Straight Lines on Cylinders | |
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| Geodesics on Cones | |
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| Is Shortest Always Straight? | |
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| Locally Isometric Surfaces | |
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| Local Coordinates for Cylinders and Cones | |
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| Geodesics in Local Coordinates | |
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| What Is Straight on a Sphere? | |
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| Intrinsic Curvature on a Sphere | |
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| Local Coordinates on a Sphere | |
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| Strakes, Augers, and Helicoids | |
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| Surfaces of Revolution | |
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| Hyperbolic Plane | |
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| Surface as Graph of a Function z=f(x,y) | |
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| Extrinsic Curves | |
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| Introduction | |
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| Give Examples of F.O.V.'s | |
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| Archimedian Property | |
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| Vectors and Affine Linear Space | |
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| Smoothness and Tangent Directions | |
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| Curvature of a Curve in Space | |
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| Curvature of the Graph of a Function | |
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| Osculating Circle | |
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| Strakes | |
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| When a Curve Does Not Lie in a Plane | |
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| Extrinsic Descriptions of Intrinsic Curvature | |
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| Smooth Surfaces and Tangent Planes | |
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| Extrinsic Curvature - Geodesics on Sphere | |
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| Intrinsic Curvature - Curves on Sphere | |
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| Intrinsic (Geodesic) Curvature | |
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| Geodesics on Surfacesthe Ribbon Test | |
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| Ruled Surfaces and the Converse of the Ribbon Test | |
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| Tangent Space, Metric, Directional Derivative | |
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| The Tangent Space | |
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| Mean Value Theorem: Curves, Surfaces | |
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| Natural Parametrizations of Curves | |
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| Riemannian Metric | |
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| Riemannian Metric in Local Coordinates on a Sphere | |
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| Riemannian Metric in Local Coordinates on a Strake | |
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| Vectors in Extrinsic Local Coordinates | |
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| Measuring Using the Riemannian Metric | |
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| Directional Derivatives | |
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| Directional Derivative in Local Coordinates | |
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| Differentiating a Metric | |
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| Expressing Normal Curvature | |
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| Geodesic Local Coordinates | |
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| Differential Operator | |
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| Metric in Geodesic Coordinates | |
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| Area, Parallel Transport, Intrinsic Curvature | |
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| The Area of a Triangle on a Sphere | |
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| Introducing Parallel Transport | |
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| The Holonomy of a Small Geodesic Triangle | |
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| Dissection of Polygons into Triangles | |
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| Gauss-Bonnet for Polygons on a Sphere | |
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| Parallel Fields and Intrinsic Curvature | |
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| Holonomy on Surfaces | |
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| Holonomy Explains Foucault's Pendulum | |
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| Intrinsic Curvature of a Surface | |
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| Gaussian Curvature Extrinsically Defined | |
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| Pep Talk to the Reader | |
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| Gaussian Curvature, Extrinsic Definition | |
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| Second Fundamental Form | |
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| The Gauss Map | |
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| Gauss-Bonnet and Intrinsic Curvature | |
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| Matrix of the Second Fundamental | |
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| Mean Curvature and Minimal Surfaces | |
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| Celebration of Our Hard Work | |
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| Applications of Gaussian Curvature | |
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| Gaussian Curvature in Local Coordinates | |
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| Curvature on Sphere, Strake, Catenoid | |
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| Circles, Polar Coordinates, and Curvature | |
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| Exponential Map and Shortest Is Straight | |
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| Ruled Surfaces and Ribbons | |
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| Surfaces with Constant Curvature | |
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| Curvature of the Hyperbolic Plane | |
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| Intrinsic Local Descriptions and Manifolds | |
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| Covariant Derivative and Connections | |
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| ManifoldsIntrinsic and Extrinsic | |
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| Christoffel Symbols, Intrinsic Descriptions | |
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| Intrinsic Curvature and Geodesics | |
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| Lie Brackets, Coordinate Vector Fields | |
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| Riemann Curvature Tensors | |
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| Calculation of Curvature Tensors in Local Coordinates | |
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| Intrinsic Calculations in Examples | |
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| Linear Algebraa Geometric Point of View | |
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| Where Do We Start? Geometric Affine Spaces | |
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| Vector Spaces | |
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| Inner ProductLengths and Angles | |
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| Linear Transformations and Operators | |
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| Areas, Cross Products, and Triple Products | |
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| Volumes, Orientation, and Determinants | |
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| Eigenvalues and Eigenvectors | |
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| Introduction to Tensors | |
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| Analysis from a Geometric Point of View | |
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| Smooth Functions | |
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| Invariance of Domain | |
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| Inverse Function Theorem | |
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| Implicit Function Theorem | |
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| Computer Scripts | |
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| Standard Functions | |
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| Strake | |
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| Surfaces of Revolution | |
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| Surfaces as Graph of a Function | |
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| Tangent Vectors to Curves | |
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| Curvature and Tangent Vectors | |
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| Osculating Planes | |
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| Osculating Circles | |
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| Frenet Frame | |
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| Tangent Planes to Surfaces | |
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| Curves on a Surface | |
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| Extrinisic Curvature Vectors | |
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| The Three Curvature Vectors | |
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| Ruled Surfaces | |
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| Non-dissectable Polyhedron | |
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| Sign of (Gaussian) Curvature | |
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| Mulitple Principle Directions | |
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| Gauss Map | |
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| Helicoid to Catenoid | |
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| Bibliography | |
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| Notation Index | |
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| Subject Index | |