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Visual Differential Geometry and Forms A Mathematical Drama in Five Acts
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| 자료유형 | 단행본 |
|---|---|
| 서명/저자사항 | Visual Differential Geometry and Forms : A Mathematical Drama in Five Acts/ Tristan Needham. |
| 개인저자 | Needham, Tristan,author. |
| 발행사항 | Princeton: Princeton University Press, [2021]. |
| 형태사항 | 1 online resource (531 p.). |
| 기타형태 저록 | Print version: Needham, Tristan Visual Differential Geometry and Forms Princeton : Princeton University Press,c2021 9780691203690 |
| ISBN | 0691219893 9780691219899 |
| 일반주기 |
15.1 Directional Derivatives.
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| 내용주기 | 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 |
| 일반주제명 | Geometry, Differential. Differential forms. MATHEMATICS / Geometry / Differential |
| 언어 | 영어 |
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