| CONTENTS CHAPTER Ⅰ. INTRODUCTION 1. The nature of differential geometry 2. Directed line-segments. Vectors 3. Parallel and perpendicular vectors 4. Algebra of number triples 5. Applications to vectors. 6. The algebra of triples, continued 7. Applications to solid analytic geometry CHAPTER Ⅱ.SPACE CURVES 8. Parametric representation 9. Regular curves and regular parameters 10. Length of arc 11. The derived vectors 12. Tangent llne 13. Contact of the tangent line with the curve 14. Osculating plane 15. Trlhedral at a point 16. Curvature. Osculating circle 17. Torsion 18. Plane curves 19. The Frenet-Serret formulas 20. Singular points 21. Fundamental theorem 22. Cylindrical helices 23. Bertrand curves CHAPTER Ⅲ.CURVES AND SURFACES ASSOCIATED WITH A SPACE CURVE 24. Tangent surface of a space curve 25. Parametric representation of a surface 26. Envelopes 27. Developable surfaces 28. Rectifying developable 29. Polar developable. Osculating sphere 30. Involutes 31. Evolutes CHAPTER Ⅳ.FUNDAMENTALS OF THE THEORY OF SURFACES SECTION 32. Parametric representation 33. Linear element. First fundamental form 34. Directions at a point 35. Families and systems of curves 36. The directed normal. Second fundamental form 37. Classification of surfaces 38. Classification of points on a surface 39. Invariant properties of the fundamental forms CHAPTER Ⅴ. CURVATURE. IMPORTANT SYSTEMS OF CURVES 40. Curvature of a curve on the surface 41. Normal curvature 42. Euler's equation 43. Dupin's indicatrix of the normal curvature 44. Lines of curvature 45. Conjugate systems of curves 46. Asymptotic lines 47. Isometric systems CHAPTER Ⅵ. THE FUNDAMENTAL THEOREM 48. The formulas of Gauss 49. The equations of Gauss and Codazzi 50. Spherical representation CHAPTER Ⅶ. GEODESIC CURVATURE. GEODESICS 51. Geodesic curvature 52. Geodesics 53. Geodesic parallels 54. Differential equations of the geodesics 55. Bonnet's formula for geodesic curvature 56. Geodesic torsion 57. The trihedral of a curve on a surface CHAPTER Ⅷ. MAPPING OF SURFACES 58. Conformal, area-preserving, and isometric maps 59. The absolute properties of a surface. Applicability 60. Applicability of surfaces of constant curvature 61. Continuous deformations of surfaces of variable curvature. CHAPTER Ⅸ THE ABSOLUTE GEOMETRY OF ASURFACE CHAPTER Ⅹ SURFACES OF SPECIAL TYPE INDEX |
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