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Modern differential geometry of curves and surfaces with Mathematica.

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Bibliographic Details
Online Access: Full Text (via Taylor & Francis)
Main Author: Gray, Alfred, 1939-1998
Other Authors: Abbena, Elsa, Salamon, Simon
Format: eBook
Language:English
Published: Boca Raton, FL : Chapman & Hall CRC, 2006.
Edition:3rd ed. /
Series:Studies in advanced mathematics.
Subjects:
Mathematica (Computer file) > Textbooks.
Mathematica (Computer file)
Geometry, Differential > Data processing > Textbooks.
Curves on surfaces > Textbooks.
Curves on surfaces
Geometry, Differential > Data processing
Textbooks
Table of Contents:
  • 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
  • Exercises Notebook 1
  • Famous Plane Curves
  • Cycloids
  • Lemniscates of Bernoulli
  • Cardioids
  • The Catenary
  • The Cissoid of Diocles
  • The Tractrix
  • Clothoids
  • Pursuit Curves
  • Exercises Notebook 2
  • 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 UnitSpeed Curves
  • The Helix and Twisted Cubic
  • ArbitrarySpeed Curves in R3
  • More Constructions of Space Curves
  • Tubes and Tori
  • Torus Knots
  • Exercises Notebook 7
  • Construction of Space Curves
  • 1 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
  • Exercises Notebook 8
  • Calculus on Euclidean Space
  • Tangent Vectors to Rn
  • Tangent Vectors as Directional Derivatives
  • Tangent Maps or Differentials
  • Vector Fields on Rn
  • 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 11.1 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 Revolution
  • 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 PetersonMainardiCodazzi 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 2Dimensional
  • 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 Bj 'orling'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
  • PseudoRiemannian 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 27
  • Bibliography
  • Name Index Subject Index
  • Notebook Index.

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