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-Poincar�e Half-Plane -- 5.5 Using Optics to Find the Geodesics -- 5.6 The Angle of Parallelism -- 5.7 The Beltrami-Poincar�e Disc

6. Isometries and Complex Numbers -- 6.1 Introduction -- 6.2 M�obius 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

15.1 Directional Derivatives.

Online resource; title from digital title page (viewed on July 08, 2021).

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's geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner. Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss's famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n -manifold; and a detailed geometrical treatment of Einstein's field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell's equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan's method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book. Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught.

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