Mod Differential Geom of Curves & Surfaces

Curves in the Plane: Euclidean Spaces. Curves in Rn. The Length of a Curve. Vector Fields along Curves. Curvature of Curves in the Plane. The Turning Angle. The Semicubical Parabola. Studying Curves in the Plane with Mathematica: Computing Curvature of Plane Curves. Computing Lengths of Curves. Filling Curves. Examples of Curves in Rn. Plotting Piecewise Defined Curves. Generalized Parabolas. Famous Plane Curves: Cycloids. Lemniscates of Bernoulli. Cardioids. The Cissoid of Diocles. The Tractrix. Clothoids. Alternate Methods for Plotting Plane Curves: Implicitly Defined Curves in R2. Cassinian Ovals. Plane Curves in Polar Coordinates. New Curves from Old: Evolutes. Iterated Evolutes. The Evolute of a Tractrix is a Catenary. Involutes. Tangent Lines and Normal Lines to Plane Curves. Osculating Circles to Plane Curves. Parallel Curves. Pedal Curves. Determining a Plane Curve form its Curvature: Euclidean Motions. Curves and Euclidean Motions. Intrinsic Equations for Plane Curves. Drawing Plane Curves with Assigned Curvature. Curves in Space: Preliminaries. Curvature and Torsion of Unit-Speed Curves in R3. Curvature and Torsion of Arbitrary-Speed Curves in R3. Computing Curvature and Torsion with Mathematica. The Helix and its Generalizations. Viviani's Curve. The Fundamental Theorem of Space Curves. Drawing Plane Curves with Assigned Curvature. Tubes and Knots: Tubes about Curves. Torus Knots. Calculus on Euclidean Space: Tangent Vectors to Rn. Tangent Vectors as Directional Derivatives. Tangent Maps. Vector Fields on Rn. Derivatives of Vector Fields on Rn. Curves Revisited. Surfaces in Euclidean Space: Local Surface in Rn. The Definition of a Surface in Rn. Tangent Vectors to Surfaces in Rn. Surface Mappings. Local Surfaces in R3. The Local Gauss Map. Examples of Local Surfaces: The Graph of a Function of Two Variables. The Sphere and the Ellipsoid. The Stereographic Ellipsoid. Tori. The Paraboloid. Sea Shells. Surfaces with Singularities. Nonorientable Surfaces: Orientability of Surfaces. Nonorientable Surfaces Described by Identifications. The Moebius Strip. The Klein Bottle. Realizations of the Real Projective Plane. Coloring Surfaces with Mathematica. Metrics on Surfaces: The Intuitive Idea of Distance on a Surface. The Intuitive Idea of Area on a Surface. A Program for Computing Metrics on a Surface. Examples of Metrics. Surfaces in 3-Dimensional Space: The Shape Operator. Normal Curvature. Calculation of the Shape Operator. Eigenvalues of the Shape Operator. The Gaussian and Mean Curvatures. The Three Fundamental Forms. Examples of Curvature Calculations by Hand. The Curvature of Nonparametriclly Defined Surfaces. Surfaces in 3-Dimensional Space via Mathematica: Programs for Computing the Shape Operator and Curvature. Examples of Curvature Calculations with Mathematica. The Gauss Map via Mathematica. Asymptotic Curves on Surfaces: Asymptotic Curves. Examples of Asymptotic Curves. Using Mathematica to Find Asymptotic Curves. Ruled Surfaces: Examples of Ruled Surfaces. Flat Ruled Surfaces. Noncylindrical Ruled Surfaces. Examples of Striction Curves of Ruled Surfaces. A Program for Ruled Surfaces. Developables. Surfaces of Revolution: Principal Curves. The Curvature of a Surface of Revolution. Generating a Surface of Revolution with Mathematica. The Catenoid. The Hyperboloid of Revolution. Surfaces of Constant Gaussian Curvature: The Elliptic Integral of the Second Kind. Surfaces of Revolution of Constant Positive Curvature. Surfaces of Revolution of Constant Negative Curvature. Kuen's Surface. Intrinsic Surface Geometry: Intrinsic Formulas for the Gaussian Curvature. Christoffel Symbols. The Mainardi-Codazzi Equations. Geodesic Curvature. Principal Curves and Umbilic Points: Principal Curves. Umbilic Points. Triply Orthogonal Systems of Surfaces. Elliptic Coordinates. Minimal Surfaces I: Normal Variation. Examples of Minimal Surfaces. Minimal Surfaces II: Isothermal Coordinates. Minimal Surfaces and Complex Function Theory. Finding Conjugate Minimal Surfaces. The Weierstrass Representation. The Weierstrass Patches Mathematica. Examples of Weierstrass Patches. Construction of Surfaces: Parallel Surfaces. The Shape Operator of a Parallel Surface. Pedal Surfaces. Generalized Helicoids. Twisted Surfaces. Differentiable Manifolds: The Definition of Differentiable Manifold. Differentiable Functions on Differentiable Manifolds. Tangent Vectors on Differentiable Manifolds. Induced Maps. Vector Fields on Differentiable Manifolds. Tensor Fields on Differentiable Manifolds. Riemannian Manifolds: Covariant Derivatives. Indefinite Riemannian Metrics. The Classical Treatment of Metrics. Abstract Surfaces: Metrics on Abstract Surfaces. Examples of Abstract Surfaces. Computing Curvature of Metrics on Abstract Surfaces. Orientability of an Abstract Surface. Geodesic Curvature for Abstract Surfaces. Geodesics on Surfaces: The Geodesic Equations. Clairaut Parameterizations. Examples of Clairaut Parameterization. Finding Geodesics Numerically with Mathematica. Appendices: General Programs. Plane Curves. Space Curves. Surfaces. Metrics. Bibliography. Index.