Visual Differential Geometry and Forms : a Mathematical Drama in Five Acts /

An inviting, intuitive, and visual exploration of differential geometry and forms Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry . Using 235 hand-drawn diagrams, Needham deploys Newton...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Needham, Tristan (Autor)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Princeton : Princeton University Press, [2021]
Temas:
Geometry, Differential.
Differential forms.
Géométrie différentielle.
Formes différentielles.
MATHEMATICS > Geometry > Differential.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Intro
  • Contents
  • ACT I. The Nature of Space
  • 1. Euclidean and Non-Euclidean Geometry
  • 1.1 Euclidean and Hyperbolic Geometry
  • 1.2 Spherical Geometry
  • 1.3 The Angular Excess of a Spherical Triangle
  • 1.4 Intrinsic and Extrinsic Geometry of Curved Surfaces
  • 1.5 Constructing Geodesics via Their Straightness
  • 1.6 The Nature of Space
  • 2. Gaussian Curvature
  • 2.1 Introduction
  • 2.2 The Circumference and Area of a Circle
  • 2.3 The Local Gauss-Bonnet Theorem
  • 3. Exercises for Prologue and Act I
  • ACT II. The Metric
  • 4. Mapping Surfaces: The Metric
  • 4.1 Introduction
  • 4.2 The Projective Map of the Sphere
  • 4.3 The Metric of a General Surface
  • 4.4 The Metric Curvature Formula
  • 4.5 Conformal Maps
  • 4.6 Some Visual Complex Analysis
  • 4.7 The Conformal Stereographic Map of the Sphere
  • 4.8 Stereographic Formulas
  • 4.9 Stereographic Preservation of Circles
  • 5. The Pseudosphere and the Hyperbolic Plane
  • 5.1 Beltrami's Insight
  • 5.2 The Tractrix and the Pseudosphere
  • 5.3 A Conformal Map of the Pseudosphere
  • 5.4 The Beltrami-Poincaré Half-Plane
  • 5.5 Using Optics to Find the Geodesics
  • 5.6 The Angle of Parallelism
  • 5.7 The Beltrami-Poincaré Disc
  • 6. Isometries and Complex Numbers
  • 6.1 Introduction
  • 6.2 Möbius Transformations
  • 6.3 The Main Result
  • 6.4 Einstein's Spacetime Geometry
  • 6.5 Three-Dimensional Hyperbolic Geometry
  • 7. Exercises for Act II
  • ACT III. Curvature
  • 8. Curvature of Plane Curves
  • 8.1 Introduction
  • 8.2 The Circle of Curvature
  • 8.3 Newton's Curvature Formula
  • 8.4 Curvature as Rate of Turning
  • 8.5 Example: Newton's Tractrix
  • 9. Curves in 3-Space
  • 10. The Principal Curvatures of a Surface
  • 10.1 Euler's Curvature Formula
  • 10.2 Proof of Euler's Curvature Formula
  • 10.3 Surfaces of Revolution
  • 11. Geodesics and Geodesic Curvature
  • 11.1 Geodesic Curvature and Normal Curvature
  • 11.2 Meusnier's Theorem
  • 11.3 Geodesics are "Straight"
  • 11.4 Intrinsic Measurement of Geodesic Curvature
  • 11.5 A Simple Extrinsic Way to Measure Geodesic Curvature
  • 11.6 A New Explanation of the Sticky-Tape Construction of Geodesics
  • 11.7 Geodesics on Surfaces of Revolution
  • 11.7.1 Clairaut's Theorem on the Sphere
  • 11.7.2 Kepler's Second Law
  • 11.7.3 Newton's Geometrical Demonstration of Kepler's Second Law
  • 11.7.4 Dynamical Proof of Clairaut's Theorem
  • 11.7.5 Application: Geodesics in the Hyperbolic Plane (Revisited)
  • 12. The Extrinsic Curvature of a Surface
  • 12.1 Introduction
  • 12.2 The Spherical Map
  • 12.3 Extrinsic Curvature of Surfaces
  • 12.4 What Shapes Are Possible?
  • 13. Gauss's Theorema Egregium
  • 13.1 Introduction
  • 13.2 Gauss's Beautiful Theorem (1816)
  • 13.3 Gauss's Theorema Egregium (1827)
  • 14. The Curvature of a Spike
  • 14.1 Introduction
  • 14.2 Curvature of a Conical Spike
  • 14.3 The Intrinsic and Extrinsic Curvature of a Polyhedral Spike
  • 14.4 The Polyhedral Theorema Egregium
  • 15. The Shape Operator

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