Curves in the Plane Euclidean Spaces Curves in Space The Length of a Curve Curvature of Plane Curves Angle Functions First Examples of Plane Curves The Semicubical Parabola and Regularity 1.8 Exercises Notebook 1 Famous Plane Curves Cycloids Lemniscates of Bernoulli Cardioids The Catenary The Cissoid of Diocles The Tractrix Clothoids Pursuit Curves Exercises Notebook Alternative Ways of Plotting Curves Implicitly Defined Plane Curves The Folium of Descartes Cassinian Ovals Plane Curves in Polar Coordinates A Selection of Spirals Exercises Notebook 3 New Curves from Old Evolutes Iterated Evolutes Involutes Osculating Circles to Plane Curves Parallel Curves Pedal Curves Exercises Notebook 4 Determining a Plane Curve from its Curvature Euclidean Motions Isometries of the Plane Intrinsic Equations for Plane Curves Examples of Curves with Assigned Curvature Exercises Notebook 5 Global Properties of Plane Curves Total Signed Curvature Trochoid Curves The Rotation Index of a Closed Curve Convex Plane Curves The Four Vertex Theorem Curves of Constant Width Reuleaux Polygons and Involutes The Support Function of an Oval Exercises Notebook 6 Curves in Space The Vector Cross Product Curvature and Torsion of Unit-Speed Curves The Helix and Twisted Cubic Arbitrary-Speed Curves in R3 More Constructions of Space Curves Tubes and Tori Torus Knots Exercises Notebook 7 Construction of Space Curves The Fundamental Theorem of Space Curves Assigned Curvature and Torsion Contact Space Curves that Lie on a Sphere Curves of Constant Slope Loxodromes on Spheres 8.7 Exercises Notebook 8 Calculus on Euclidean Space Tangent Vectors to Rn Tangent Vectors as Directional Derivatives Tangent Maps or Differentials Vector Fields on R n Derivatives of Vector Fields Curves Revisited Exercises Notebook 9 Surfaces in Euclidean Space Patches in Rn Patches in R3 and the Local Gauss Map The Definition of a Regular Surface Examples of Surfaces Tangent Vectors and Surface Mappings Level Surfaces in R3 Exercises Notebook 10 Nonorientable Surfaces Orientability of Surfaces Surfaces by Identification The Mobius Strip The Klein Bottle Realizations of the Real Projective Plane Twisted Surfaces Exercises Notebook 11 Metrics on Surfaces The Intuitive Idea of Distance Isometries between Surfaces Distance and Conformal Maps The Intuitive Idea of Area Examples of Metrics Exercises Notebook 12 Shape and Curvature The Shape Operator Normal Curvature Calculation of the Shape Operator Gaussian and Mean Curvature More Curvature Calculations A Global Curvature Theorem Nonparametrically Defined Surfaces Exercises Notebook 13 Ruled Surfaces Definitions and Examples Curvature of a Ruled Surface Tangent Developables Noncylindrical Ruled Surfaces Exercises Notebook 14 Surfaces of Revolution and Constant Curvature Surfaces of Revolution Principal Curves Curvature of a Surface of Revolution Generalized Helicoids Surfaces of Constant Positive Curvature Surfaces of Constant Negative Curvature More Examples of Constant Curvature Exercises Notebook 15 A Selection of Minimal Surfaces Normal Variation Deformation from the Helicoid to the Catenoid Minimal Surfaces of More Examples of Minimal Surfaces Monge Patches and Scherk's Minimal Surface The Gauss Map of a Minimal Surface Isothermal Coordinates Exercises Notebook 16 Intrinsic Surface Geometry Intrinsic Formulas for the Gaussian Curvature Gauss's Theorema Egregium Christoffel Symbols Geodesic Curvature of Curves on Surfaces Geodesic Torsion and Frenet Formulas Exercises Notebook 17 Asymptotic Curves and Geodesics on Surfaces Asymptotic Curves Examples of Asymptotic Curves and Patches The Geodesic Equations First Examples of Geodesics Clairaut Patches Use of Clairaut Patches Exercises Notebook 18 Principal Curves and Umbilic Points The Differential Equation for Principal Curves Umbilic Points The Peterson-Mainardi-Codazzi Equations Hilbert's Lemma and Liebmann's Theorem Triply Orthogonal Systems of Surfaces Elliptic Coordinates Parabolic Coordinates and a General Construction Parallel Surfaces The Shape Operator of a Parallel Surface Exercises Notebook 19 Canal Surfaces and Cyclides of Dupin Surfaces Whose Focal Sets are 2-Dimensional Canal Surfaces Cyclides of Dupin via Focal Sets The Definition of Inversion Inversion of Surfaces Exercises Notebook 20 The Theory of Surfaces of Constant Negative Curvature Intrinsic Tchebyshef Patches Patches on Surfaces of Constant Negative Curvature The Sine-Gordon Equation Tchebyshef Patches on Surfaces of Revolution The Bianchi Transform Moving Frames on Surfaces in R3 Kuen's Surface as Bianchi Transform of the Pseudosphere The B.. acklund Transform Exercises Notebook 21 Minimal Surfaces via Complex Variables Isometric Deformations of Minimal Surfaces Complex Derivatives Minimal Curves Finding Conjugate Minimal Surfaces The Weierstrass Representation Minimal Surfaces via Bjorling's Formula Costa's Minimal Surface Exercises Notebook 22 Rotation and Animation using Quaternions Orthogonal Matrices Quaternion Algebra Unit Quaternions and Rotations Imaginary Quaternions and Rotations Rotation Curves Euler Angles Further Topics Exercises Notebook 23 Differentiable Manifolds The Definition of a Differentiable Manifold Differentiable Functions on Manifolds Tangent Vectors on Manifolds Induced Maps Vector Fields on Manifolds Tensor Fields Exercises Notebook 24 Riemannian Manifolds Covariant Derivatives Pseudo-Riemannian Metrics The Classical Treatment of Metrics The Christoffel Symbols in Riemannian Geometry The Riemann Curvature Tensor Exercises Notebook 25 Abstract Surfaces and their Geodesics Christoffel Symbols on Abstract Surfaces Examples of Abstract Metrics The Abstract Definition of Geodesic Curvature Geodesics on Abstract Surfaces The Exponential Map and the Gauss Lemma Length Minimizing Properties of Geodesics Exercises Notebook 26 The Gauss-Bonnet Theorem Turning Angles and Liouville's Theorem The Local Gauss-Bonnet Theorem An Area Bound A Generalization to More Complicated Regions The Topology of Surfaces The Global Gauss-Bonnet Theorem . Applications of the Gauss-Bonnet Theorem Exercises Notebook Bibliography Name Index Subject Index Notebook Index

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