Book:C.E. Weatherburn/Differential Geometry of Three Dimensions/Volume I

Fourth Impression $1947$

Subject Matter

 * Differential Geometry

Contents

 * Preface to the Fourth Impression (University of W.A., Perth, Western Australia, 22 January, 1947.)


 * INTRODUCTION: VECTOR NOTATION AND FORMULAE
 * Sums, products, derivatives


 * CHAPTER I CURVES WITH TORSION
 * 1. Tangent
 * 2. Principal normal. Curvature
 * 3. Binormal. Torsion. Serret-Frenet formulae
 * 4. Locus of centre of curvature
 * EXAMPLES I
 * 5. Spherical curvature
 * 6. Locus of centre of spherical curvature
 * 7. Theorem: Curve determined by its intrinsic equations
 * 8 Helices
 * 9. Spherical indicatrix of tangent, etc.
 * 10. Involutes
 * 11. Evolutes
 * 12. Bertrand curves
 * EXAMPLES II


 * CHAPTER II: ENVELOPES. DEVELOPABLE SURFACES
 * 13. Surfaces
 * 14. Tangent plane. Normal


 * ONE-PARAMETER FAMILY OF SURFACES
 * 15. Envelope. Characteristics
 * 16. Edge of regression
 * 17. Developable surfaces


 * DEVELOPABLES ASSOCIATED WITH A CURVE
 * 18. Osculating developable
 * 19. Polar developable
 * 20. Rectifying developable


 * TWO-PARAMETER FAMILY OF SURFACES
 * 21. Envelope. Characteristic points
 * EXAMPLES III


 * CHAPTER III: CURVILINEAR COORDINATES ON A SURFACE. FUNDAMENTAL MAGNITUDES
 * 22. Curvilinear coordinates
 * 23. First order magnitudes
 * 24. Directions on a surface
 * 25. The normal
 * 26. Second order magnitudes
 * 27. Derivatives of $\mathbf n$
 * 28. Curvature of normal section. Meunier's theorem
 * EXAMPLES IV


 * CHAPTER IV: CURVES ON A SURFACE


 * LINES OF CURVATURE
 * 29. Principal directions and curvatures
 * 30. First and second curvatures
 * 31. Euler's theorem
 * 32. Dupin's indicatrix
 * 33. The surface $z = f (x, y)$
 * 34. Surface of revolution
 * EXAMPLES V


 * CONJUGATE SYSTEM
 * 35. Conjugate directions
 * 36. Conjugate systems


 * ASYMPTOTIC LINES
 * 37. Asymptotic lines
 * 38. Curvature and torsion


 * ISOMETRIC LINES
 * 39. Isometric parameters


 * NULL LINES
 * 40. Null lines, or minimal curves
 * EXAMPLES VI


 * CHAPTER V: THE EQUATIONS OF GAUSS AND OF CODAZZI
 * 41. Gauss's formulae for $\mathbf r_{11}$, $\mathbf r_{12}$, $\mathbf r_{22}$
 * 42. Gauss characteristic equation
 * 43. Mainardi-Codazzi relations
 * 44. Alternative expression. Bonnet's theorem
 * 45. Derivatives of the angle $\omega$
 * EXAMPLES VII


 * CHAPTER VI: GEODESICS AND GEODESIC PARALLELS


 * GEODESICS
 * 46. Geodesic property
 * 47. Equations of geodesics
 * 48. Surface of revolution
 * 49. Torsion of a geodesic


 * CURVES IN RELATION TO GEODESICS
 * 50. Bonnet's theorem
 * 51. Joachimsthal's theorems
 * 52. Vector curvature
 * 53. Geodesic curvature, $\kappa_g$
 * 54. Other formulae for $\kappa_g$
 * 55. Examples. Bonnet's formula


 * GEODESIC PARALLELS
 * 56. Geodesic parallels. Geodesic distance
 * 57. Geodesic polar coordinates
 * 58. Total second curvature of a geodesic triangle
 * 59. Theorem on geodesic parallels
 * 60. Geodesic ellipses and hyperbolas
 * 61. Liouville surfaces
 * EXAMPLES VIII


 * CHAPTER VII: QUADRIC SURFACES. RULED SURFACES


 * QUADRIC SURFACES
 * 62. Central quadrics. Curvilinear coordinates
 * 63. Fundamental magnitudes
 * 64. Geodesics. Liouville's equation
 * 65. Other properties. Joachimsthal's theorem
 * 66. Paraboloids
 * EXAMPLES IX


 * RULED SURFACES
 * 67. Skew surface or scroll
 * 68. Consecutive generators. Parameter of distribution
 * 69. Line of striction. Central point
 * 70. Fundamental magnitudes
 * 71. Tangent plane. Central plane
 * 72. Bonnet's theorem
 * 73. Asymptotic lines
 * EXAMPLES X


 * CHAPTER VIII EVOLUTE OR SURFACE OF CENTRES. PARALLEL SURFACES


 * SURFACE OF CENTRES
 * 74. Centro-surface. General properties
 * 75. Fundamental magnitudes
 * 76. Weingarten surfaces
 * 77. Lines of curvature
 * 78. Degenerate evolute


 * PARALLEL SURFACES
 * 79. Parallel surfaces
 * 80. Curvature
 * 81. Involutes of a surface


 * INVERSE SURFACES
 * 82. Inverse surface
 * 83. Curvature
 * EXAMPLES XI


 * CHAPTER IX: CONFORMAL AND SPHERICAL REPRESENTATIONS. MINIMAL SURFACES
 * CONFORMAL REPRESENTATION
 * 84. Conformal representation. Magnification
 * 85. Surface of revolution represented on a plane
 * 86. Surface of a sphere represented on a plane. Maps


 * SPHERICAL REPRESENTATION
 * 87. Spherical image. General properties
 * 88. Other properties
 * 89. Second order magnitudes
 * 90. Tangential coordinates


 * MINIMAL SURFACES
 * 91. Minimal surface. General properties
 * 92. Spherical image
 * 93. Differential equation in Cartesian coordinates
 * EXAMPLES XII


 * CHAPTER X: CONGRUENCES OF LINES
 * RECTILINEAR CONGRUENCES
 * 94. Congruence of straight lines. Surfaces of the congruence
 * 95. Limits. Principal planes
 * 96. Hamilton's formula
 * 97. Foci. Focal planes
 * 98. Parameter of distribution for a surface
 * 99. Mean ruled surfaces
 * 100. Normal congruence of straight lines
 * 101. Theorem of Malus and Dupin
 * 101. Isotropic congruence


 * CURVILINEAR CONGRUENCES
 * 103. Congruence of curves. Foci. Focal surface
 * 104. Surfaces of the congruence
 * 105. Normal congruence of curves
 * EXAMPLES XIII


 * CHAPTER XI: TRIPLY ORTHOGONAL SYSTEMS OF SURFACES
 * 106. Triply orthogonal systems
 * 107. Normals. Curvilinear coordinates
 * 108. Fundamental magnitudes
 * 109. Dupin's theorem. Curvature
 * 110. Second derivatives of $\mathbf r$. Derivatives of the unit normals
 * 111. Lamb's relations
 * 112. Theorems of Darboux
 * EXAMPLES XIV


 * CHAPTER XII: DIFFERENTIAL INVARIANTS FOR A SURFACE
 * 113. Point-functions for a surface
 * 114. Gradient of a scalar function
 * 115. Some applications
 * 116. Divergence of a vector
 * 117. Isometric parameters and curves
 * 118. Curl of a vector
 * 119. Vector functions (cont.)
 * 120. Formulae of expansion
 * 121. Geodesic curvature
 * EXAMPLES XV


 * TRANSFORMATION OF INTEGRALS
 * l22. Divergence theorem
 * 123. Other theorems
 * 124. Circulation theorem
 * EXAMPLES XVI


 * CONCLUSION: FURTHER RECENT ADVANCES
 * 125. Orthogonal systems of curves on a surface
 * 126. Family of curves on a surface
 * 127. Small deformation of a surface
 * 128. Oblique curvilinear coordinates in space
 * 129. Congruences of curves
 * EXAMPLES XVII
 * 130. Family of curves (continued)
 * 131. Family of surfaces


 * NOTE I. DIRECTIONS ON A SURFACE


 * NOTE II. ON THE CURVATURES OF A SURFACE


 * INDEX