Book:C.E. Weatherburn/Differential Geometry of Three Dimensions/Volume I
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C.E. Weatherburn: Differential Geometry of Three Dimensions, Volume $\text { I }$
Published $\text {1927}$, Cambridge at the University Press
Fourth Impression $1947$
Subject Matter
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 $\text I$ CURVES WITH TORSION
- 1. Tangent
- 2. Principal normal. Curvature
- 3. Binormal. Torsion. Serret-Frenet formulae
- 4. Locus of centre of curvature
- Examples $\text I$
- 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 $\text {II}$
- CHAPTER $\text {II}$: ENVELOPES. DEVELOPABLE SURFACES
- 13. Surfaces
- 14. Tangent plane. Normal
- ONE-PARAMETER FAMILY OF SURFACES
- 15. Envelope. Characteristics
- 16. Edge of regression
- 17. Developable surfaces
- ONE-PARAMETER FAMILY OF SURFACES
- DEVELOPABLES ASSOCIATED WITH A CURVE
- 18. Osculating developable
- 19. Polar developable
- 20. Rectifying developable
- DEVELOPABLES ASSOCIATED WITH A CURVE
- TWO-PARAMETER FAMILY OF SURFACES
- 21. Envelope. Characteristic points
- Examples $\text {III}$
- 21. Envelope. Characteristic points
- TWO-PARAMETER FAMILY OF SURFACES
- CHAPTER $\text {III}$: 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 $\text {IV}$
- CHAPTER $\text {IV}$: 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 $\text {V}$
- LINES OF CURVATURE
- CONJUGATE SYSTEM
- 35. Conjugate directions
- 36. Conjugate systems
- CONJUGATE SYSTEM
- ASYMPTOTIC LINES
- 37. Asymptotic lines
- 38. Curvature and torsion
- ASYMPTOTIC LINES
- ISOMETRIC LINES
- 39. Isometric parameters
- ISOMETRIC LINES
- NULL LINES
- 40. Null lines, or minimal curves
- Examples $\text {VI}$
- 40. Null lines, or minimal curves
- NULL LINES
- CHAPTER $\text {V}$: 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 $\text {VII}$
- CHAPTER $\text {VI}$: GEODESICS AND GEODESIC PARALLELS
- GEODESICS
- 46. Geodesic property
- 47. Equations of geodesics
- 48. Surface of revolution
- 49. Torsion of a geodesic
- GEODESICS
- 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
- CURVES IN RELATION TO GEODESICS
- 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 $\text {VIII}$
- GEODESIC PARALLELS
- CHAPTER $\text {VII}$: 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 $\text {IX}$
- QUADRIC SURFACES
- 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 $\text {X}$
- RULED SURFACES
- CHAPTER $\text {VIII}$: 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
- SURFACE OF CENTRES
- PARALLEL SURFACES
- 79. Parallel surfaces
- 80. Curvature
- 81. Involutes of a surface
- PARALLEL SURFACES
- INVERSE SURFACES
- 82. Inverse surface
- 83. Curvature
- Examples $\text {XI}$
- INVERSE SURFACES
- CHAPTER $\text {IX}$: 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
- CONFORMAL REPRESENTATION
- 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 $\text {XII}$
- MINIMAL SURFACES
- CHAPTER $\text {X}$: 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
- RECTILINEAR CONGRUENCES
- CURVILINEAR CONGRUENCES
- 103. Congruence of curves. Foci. Focal surface
- 104. Surfaces of the congruence
- 105. Normal congruence of curves
- Examples $\text {XIII}$
- CURVILINEAR CONGRUENCES
- CHAPTER $\text {XI}$: 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 $\text {XIV}$
- CHAPTER $\text {XII}$: 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 $\text {XV}$
- TRANSFORMATION OF INTEGRALS
- l22. Divergence theorem
- 123. Other theorems
- 124. Circulation theorem
- Examples $\text {XVI}$
- TRANSFORMATION OF INTEGRALS
- 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 $\text {XVII}$
- 130. Family of curves (continued)
- 131. Family of surfaces
- NOTE $\text {I}$. DIRECTIONS ON A SURFACE
- NOTE $\text {II}$. ON THE CURVATURES OF A SURFACE
- INDEX
Source work progress
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