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

## C.E. Weatherburn: Differential Geometry of Three Dimensions, Volume $\text { I }$

Published $\text {1927}$, Cambridge at the University Press

Fourth Impression $1947$

### 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
DEVELOPABLES ASSOCIATED WITH A CURVE
18. Osculating developable
19. Polar developable
20. Rectifying developable
TWO-PARAMETER FAMILY OF SURFACES
21. Envelope. Characteristic points
Examples $\text {III}$
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 $\text {V}$
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 $\text {VI}$
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
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 $\text {VIII}$
CHAPTER $\text {VII}$: QUADRIC SURFACES. RULED SURFACES
62. Central quadrics. Curvilinear coordinates
63. Fundamental magnitudes
64. Geodesics. Liouville's equation
65. Other properties. Joachimsthal's theorem
66. Paraboloids
Examples $\text {IX}$
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 $\text {X}$
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
PARALLEL SURFACES
79. Parallel surfaces
80. Curvature
81. Involutes of a surface
INVERSE SURFACES
82. Inverse surface
83. Curvature
Examples $\text {XI}$
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
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 $\text {XII}$
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
CURVILINEAR CONGRUENCES
103. Congruence of curves. Foci. Focal surface
104. Surfaces of the congruence
105. Normal congruence of curves
Examples $\text {XIII}$
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 $\text {XVI}$
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

Next