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Differential Geometry of Three Dimensions

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DIFFERENTIAL GEOMETRY OF THREE DIMENSIONS DIFFERENTIAL GEOMETRY OF THREE DIMENSIONS By G. E. "WEATHERBURN, M.A., D.Sc., LL.D. EMERITUS PROFESSOR 07 MATHEMATICS UNIVERSITY OF WESTERN AUSTRALIA. VOLUME I CAMBRIDGE AT THE UNIVERSITY PRESS 1955 V, PUBLISHED BY THB SYNDICS OF THE CAMBRIDGE UNIVERSITY PRESS London Office Bentiey House, N.W. I American Branch New York Agents for Canada,, India, and Pakistan' Maximilian First Edition 1927 Reprinted 1931 1939 1947 1955 First printed in Great Britain at The University Press, Cambridge Eeprmted by Spottwwoode, Sattantyne <b Co., Ltd , Colchester PEEFAOE TO THE FOURTH IMPRESSION present impression is substantially a reprint of the original THEwork. Since the book was first published a few errors have been corrected, and one or two paragraphs rewritten. Among the friends and correspondents who kindly drew my attention to desirable changes were Mr A S. Ramsey of Magdaler-e College, Cambridge, who suggested the revision of 5, and the late R J. A Barnard of Melbourne University, whose mfluence was partly responsible for my initial interest in the subject. The demand for the book, since its first appearance twenty years ago, has justified the writer's belief in the need for such a vectonal treatment. By the use of vector methods the presentation of the subject is both simplified and condensed, and students are encouraged to reason geometrically rather than analytically. At a later stage some of these students will proceed to the study of multidimensional differential geometry and the tensor calculus. It is highly desirable that the study of the geometry of Euclidean 3-space should thus come first, and this can be undertaken with most students at an earlier stage by vector methods than by the Ricci calculus. A student's appreciation of the more general case will undoubtedly be enhanced by an earlier acquaintance with differential geometry of three dimensions The more elementary parts of the subject are discussed in Chapters I-XI. The remainder of the book is devoted to differ- ential invariants for a surface and their applications. It will be apparent to the reader that these constitute a powerful weapon for analysing the geometrical properties of surfaces, and of systems of curves on a surface. The unit vector, n, normal to a surface at the current point, plays a prominent part m this discussion The first curvature of the surface :s the negative of the divergence of n; while the second curvature is expressible simply in terms of the divergence and the Laplacian of n with respect to the surface. CARNEGIE INSTITUTE TECHNni nr>v , ,, VI PEEFACB Extensive applications of these invariants to the geometry of surfaces are given in the second volume of this book. Applications to physical problems connected with curved surfaces have been given elsewhere* by the author. * 1. On differential invariants in geometry of surfaces, with some applications to mathematical physios Quarterly Journal of Mathematics, Yol 60, pp. 280-69 (Cambridge, 1925). 2 On small deformation of surfaces and of thin elastic shells. Ibid., Yol. 50, pp. 272-96 (1925). 8. On the motion of an extensible membrane in a given curved surface. Phil Yol 578-80 Mag , 23, pp (1037). 4 On transverse vibrations of cur\ed membranes. Phil Mag , Yol 28, pp 632- 84 (1989). C E. W. UNIVERSITY ov W A , PERTH, AUSTRALIA, 22 January, 1947. CONTENTS PAGE PEEFAOE v INTRODUCTION VECTOR NOTATION AND FORMULAE Sums, products, derivatives ... 1 CHAPTER I CURVES WITH TORSION ABT. 1. Tangent .10 2. Principal normal. Curvature 11 3. Binomial Torsion, Serret-Frenet formulae 13 4 Loous of centre of curvature . ... 17 EXAMPLES I . 18 5. Spherical curvature . 21 6. Locus of centre of spherical curvature . ... 23 7. Theorem Curve determined by its intrinsic equations 25 8. Helices ... 26 9 Spherical indioatrix of tangent, etc. .... 28 10. Involutes ... 30 11. Evolutes ... 32 12. Bertrand curves 34 EXAMPLES II 36 CHAPTER II ENVELOPES. DEVELOPABLE SURFACES 13. Surfaces 38 38 14. Tangent plane. Normal .... ... ONE-PARAMETER FAMILY OF SURFACES 40 16. Envelope Characteristics 42 16. Edge of regression 43 17. Developable surfaces DEVELOPABLES ASSOCIATED WITH A CURVE 45 18 Osculating developable 46 19. Polar developable 46 20. Rectifying developable ... TWO-PARAMETER FAMILY OF SURFACES 48 21. Envelope Characteristic points . .... EXAMPLES III B0 VU1 CONTENTS CHAPTER III CURVILINEAR COORDINATES ON A SURFACE FUNDAMENTAL MAGNITUDES ABT. 22. Curvilinear coordinates 51 23. First order magnitudes ....>... .53 24. Directions on a surface ... 55 The normal 25. ... 57 26 Second order magnitudes . 68 27 Derivatives of n 60 28. Curvature of normal section Mourner's theorem ... 61 EXAMPLES IV . ... 63 CHAPTER IV CURVES ON A SURFACE LINES OF CURVATURE 29. Principal directions and curvatures 66 30 First and second curvatures . 68 31. Eider's theorem . 72 32 Dupin's indicatiix . 74 33 The surface z=f (x, y} . 75 34. Surface of revolution . 77 EXAMPLES V ... 78 CONJUGATE SYSTEMS 35 Conjugate directions . 80 36 Conjugate systems ... 81 ASYMPTOTIC LINES 37. Asymptotic lines ... ... 83 38 Curvature and torsion . .... .84 ISOMETRIC LINES 39 Isometric parameters . .... 85 NOLL LINES 40 Null lines, or minimal curves . 87 EXAMPLES VI 88 CHAPTER V .THE EQUATIONS OF GAUSS AND OF CODAZZI 41. Gauss's formulae for Tin Tia, Tas 90 42. Gauss characteristic equation 93 43 Mainardi-Codazzi relations . ... 94 44. Alternative expression. Bonnet's theorem 95 45. Derivatives of the angle a 96 EXAMPLES VII . ... 96 CONTENTS iz CHAPTER VI GEODE&ICS AND GEODESIC PARALLELS GEODESIOS ART. PAaH 46 Geodesic property 99 47. of Equations geodesies . 100 ' 48. Surface of revolution 102 49. Torsion of a . geodesic . .... 103 CURVES IN RELATION TO GEODESIOS BO. Bonnet's theorem 105 51. Joaohimsthal's theorems . f 106 B2.< Vector curvature 108 53. Geodesic curvature, KO . 108 54. Other formulae for KO 110 55.' Examples. Bonnet's formula 112 GEODESIC PARALLELS 06. Geodesic parallels. Geodesic distance 113 57. Geodesic polar coordinates . .... 115 58. Total second curvature of a geodesic triangle . .116 59. Theorem on geodesic parallels 118 and . 60. Geodesic ellipses hyperbolas .... 119 81. Liouville surfaces 120 EXAMPLES VIII 121 CHAPTER VII QUADRIO SURFACES. RULED SURFACES QUADRIO SURFACES 62 Central quadrics. Curvilinear coordinates 124 63. Fundamental magnitudes 125 64. Geodesies Liouvillo's equation 127 65. Other properties. Joachimsthal's theorem 129 66. Paraboloids 131 EXAMPLES IX 133 RULED SURFACES 67. Skew surface or scroll ... ...... 135 68. Consecutive generators. Parameter of distribution . 136 69. Line of stnotion Central point .... 138 70. Fundamental magnitudes 139 71. Tangent plane. Central plane ... l40 72. Bonnet's theorem 143 73. Asymptotic lines 144 EXAMPLES X 144 X CONTENTS CHAPTER VIII EVOLTTTE OR SURFACE OF CENTRES. PARALLEL SURFACES SURFACE ov CENTRES AST. *AO 1*8 74. Centro-surface. General properties 1 B 1 76 Fundamental magnitudes 1&* 76 "Wemgarten surfaces 77. Lines of curvature 1&6 156 78 Degenerate evolute PARALLEL SURFACES 158 79 Parallel surfaces . ... ... 80 Curvature .159 81. Involutes of a surface 160 INVERSE SURFACES 162 82 Inverse surface . 83. Curvature 164 EXAMPLES XI 165 CHAPTER IX CONFORMAL AND SPHERICAL REPRESENTATIONS. MINIMAL SURFACES CONFORMAL REPRESENTATION 84 Conforms! representation Magnification 167 168 85 Surface of revolution represented on a plane . 8G Surface of a sphere represented on a plane. Maps . .170 SPHERICAL REPRESENTATION 87. Spherical image. General properties . 172 88. Other properties 173 89 Second order magnitudes 175 00. Tangential coordinates 175 MINIMAL SURFACES 91 Minimal surface. General properties 176 92 Spherical image . ....... 178 93. Differential equation in Cartesian coordinates . .179 EXAMPLES XII 181 CHAPTERS CONGRUENCES OF LINES RECTILINEAR CONGRUENCES 94. of lines. of Congruence straight Surfaces the congruence . .183 95 Limits. Principal planes 184 96 Hamilton's formula 187 97. Foci. Focal planes 189 CONTENTS Si ABT. PA0K 98. Parameter of distribution for a surface 192 99 Mean ruled surfaces 103 100. Normal congruence of straight luies 196 101 Theorem of Malus and Dupin 197 102. Isotropio congruence 198 CURVILINEAR CONGRUENCES 103 Congruence of curves. Foci Focal surface 199 104 Surfaces of the congruence 200 105 Normal congruence of curves . 202 EXAMPLES XIII ... 203 CHAPTER XI TRIPLY ORTHOGONAL SYSTEMS OF SURFACES 106. Triply orthogonal systems 207 107. Normals Curvilinear coordinates 207 108 Fundamental magnitudes .... ... 209 109. Dupin's theorem. Curvature ... 211 110 Second derivatives of r Derivatives of the unit normals . 213 111. Lamp's relations . 214 112. Theorems of Darboux 216 EXAMPLES XIV 218 CHAPTER XII DIFFERENTIAL INVARIANTS FOR A SURFACE 113. Point-functions for a surface ....... 220 114. Gradient of a scalar function 220 116 Some applications 223 116. Divergence of a vector . .... 225 117. Isometric parameters and curves 227 118 Curl of a vector 228 119 Vector functions (cont) . ... .... 230 120 Formulae of expansion 232 121. Geodesic curvature 233 EXAMPLES XV 236 TRANSFORMATION OP INTEGRALS 122. Divergence theorem 238 123. Other theorems 240 124 Circulation theorem 243 EXAMPLES XVI 244 Xli CONTENTS CONCLUSION FURTHER RECENT ADVANCES ABT. 125. of curves on a Orthogonal systems surface . 247 126. Family of curves on a surface 248 127 Small deformation of a surface 250 128. curvilinear coordinates in Obkque space . 251 129. Congruences of curves .... .... 252 EXAMVLES XVII . .254 130. Family of curves (continued) 258
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