Blaga P. Lectures on the differential geometry of - tiera.ru

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6 Contents 1.10.1 The geometrical meaning of the torsion . . . . . . . . . . . . . 53 1.10.2 Some further applications of the Frenet formulae . . . . . . . . 54 1.10.3 General helices. Lancret’s theorem . . . . . . . . . . . . . . . 56 1.10.4 Bertrand curves . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.11 The local behaviour of a parameterized curve around a biregular point . 62 1.12 The contact between a space curve and a plane . . . . . . . . . . . . . 64 1.13 The contact between a space curve and a sphere. The osculating sphere 66 1.14 Existence and uniqueness theorems for parameterized curves . . . . . . 68 1.14.1 The behaviour of the Frenet frame under a rigid motion . . . . . 68 1.14.2 The uniqueness theorem . . . . . . . . . . . . . . . . . . . . . 70 1.14.3 The existence theorem . . . . . . . . . . . . . . . . . . . . . . 72 2 Plane curves 75 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.2 Envelopes of plane curves . . . . . . . . . . . . . . . . . . . . . . . . 75 2.2.1 Curves given through an implicit equation . . . . . . . . . . . . 77 2.2.2 Families of curves depending on two parameters . . . . . . . . 79 2.2.3 Applications: the evolute of a plane curve . . . . . . . . . . . . 79 2.3 The curvature of a plane curve . . . . . . . . . . . . . . . . . . . . . . 81 2.3.1 The geometrical interpretation of the signed curvature . . . . . 84 2.4 The curvature center. The evolute and the involute of a plane curve . . . 86 2.5 The osculating circle of a curve . . . . . . . . . . . . . . . . . . . . . . 91 2.6 The existence and uniqueness theorem for plane curves . . . . . . . . . 92 3 The integration of the natural equations of a curve 95 3.1 The Riccati equation associated to the natural equations of a curve . . . 95 3.2 Examples for the integration of the natural equation of a plane curve . . 96 Problems 103 II Surfaces 113 4 General theory of surfaces 115 4.1 Parameterized surfaces (patches) . . . . . . . . . . . . . . . . . . . . . 115 4.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.2.1 Representations of surfaces . . . . . . . . . . . . . . . . . . . . 116 4.3 The equivalence of local parameterizations . . . . . . . . . . . . . . . . 119 4.4 Curves on a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Contents 7 4.5 The tangent vector space, the tangent plane and the normal to a surface . 123 4.6 The orientation of surfaces . . . . . . . . . . . . . . . . . . . . . . . . 127 4.7 Differentiable maps on a surface . . . . . . . . . . . . . . . . . . . . . 130 4.8 The differential of a smooth map between surfaces . . . . . . . . . . . 134 4.9 The spherical map and the shape operator of an oriented surface . . . . 136 4.10 The first fundamental form of a surface . . . . . . . . . . . . . . . . . 139 4.10.1 First applications . . . . . . . . . . . . . . . . . . . . . . . . . 140 The length of a segment of curve on a surface . . . . . . . . . . 140 The angle of two curves on a surface . . . . . . . . . . . . . . . 141 The area of a parameterized surface . . . . . . . . . . . . . . . 142 4.11 The matrix of the shape operator . . . . . . . . . . . . . . . . . . . . . 144 4.12 The second fundamental form of an oriented surface . . . . . . . . . . 146 4.13 The normal curvature. The Meusnier’s theorem . . . . . . . . . . . . . 148 4.14 Asymptotic directions and asymptotic lines on a surface . . . . . . . . . 150 4.15 The classification of points on a surface . . . . . . . . . . . . . . . . . 152 4.16 Principal directions and curvatures . . . . . . . . . . . . . . . . . . . . 156 4.16.1 The determination of the lines of curvature . . . . . . . . . . . 159 4.16.2 The computation of the curvatures of a surface . . . . . . . . . 161 4.17 The fundamental equations of a surface . . . . . . . . . . . . . . . . . 162 4.17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 4.17.2 The differentiation rules. Christoffel’s coefficients . . . . . . . . 162 Christoffel’s and Weingarten’s coefficients in curvature coordinates164 4.17.3 The Gauss’ and Codazzi-Mainardi’s equations for a surface . . 165 4.17.4 The fundamental theorem of surface theory . . . . . . . . . . . 167 4.18 The Gauss’ egregium theorem . . . . . . . . . . . . . . . . . . . . . . 172 4.19 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.19.2 The Darboux frame. The geodesic curvature and geodesic torsion 175 4.19.3 Geodesic lines . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Examples of geodesics . . . . . . . . . . . . . . . . . . . . . . 181 4.19.4 Liouville surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 182 5 Special classes of surfaces 185 5.1 Ruled surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 5.1.1 General ruled surfaces . . . . . . . . . . . . . . . . . . . . . . 185 The parameterization of a ruled surface . . . . . . . . . . . . . 186 The tangent plane and the first fundamental form of a ruled surface187 5.1.2 The Gaussian curvature of a ruled surface . . . . . . . . . . . . 189
Page 1: Lectures on the Di
Page 5: Contents Foreword 9 I Curves 11 1 S
Page 9 and 10: Foreword This book is based on the
Page 11: Part I Curves
Page 14 and 15: 14 Chapter 1. Space curves from the
Page 16 and 17: 16 Chapter 1. Space curves Definiti
Page 18 and 19: 18 Chapter 1. Space curves Figure 1
Page 20 and 21: 20 Chapter 1. Space curves (I, r) i
Page 22 and 23: 22 Chapter 1. Space curves Remark.
Page 24 and 25: 24 Chapter 1. Space curves and ρ
Page 26 and 27: 26 Chapter 1. Space curves given by
Page 28 and 29: 28 Chapter 1. Space curves • a sm
Page 30 and 31: 30 Chapter 1. Space curves Implicit
Page 34 and 35: 34 Chapter 1. Space curves Theorem
Page 36 and 37: 36 Chapter 1. Space curves Explicit
Page 38 and 39: 38 Chapter 1. Space curves From thi
Page 40 and 41: 40 Chapter 1. Space curves Theorem
Page 42 and 43: 42 Chapter 1. Space curves 1.7 The
Page 44 and 45: 44 Chapter 1. Space curves Remark.
Page 48 and 49: 48 Chapter 1. Space curves 1.9 Orie
Page 50 and 51: 50 Chapter 1. Space curves 1.10 The
Page 52 and 53: 52 Chapter 1. Space curves therefor
Page 54 and 55: 54 Chapter 1. Space curves Proposit
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56 Chapter 1. Space curves or 1 + r
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58 Chapter 1. Space curves or c0 ·
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60 Chapter 1. Space curves Let ω b
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62 Chapter 1. Space curves Corollar
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64 Chapter 1. Space curves vector r
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66 Chapter 1. Space curves a) If a
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68 Chapter 1. Space curves d) We sh
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70 Chapter 1. Space curves Then: τ
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72 Chapter 1. Space curves 1.14.3 T
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74 Chapter 1. Space curves
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76 Chapter 2. Plane curves Proof. I
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78 Chapter 2. Plane curves By subst
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80 Chapter 2. Plane curves The foll
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82 Chapter 2. Plane curves c) J(Jv)
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84 Chapter 2. Plane curves therefor
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86 Chapter 2. Plane curves whence,
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88 Chapter 2. Plane curves Figure 2
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90 Chapter 2. Plane curves whence w
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92 Chapter 2. Plane curves where A(
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94 Chapter 2. Plane curves
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96 Chapter 3. The integration of th
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98 Chapter 3. The integration of th
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100 Chapter 3. The integration of t
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104 Problems 6. A straight line seg
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106 Problems 19. Show that the Bern
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108 Problems 35. Find the envelopes
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110 Problems d) x 2 y 2 = (a 2 −
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4.1 Parameterized surfaces (patches
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4.2. Surfaces 117 Explicit represen
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4.3. The equivalence of local param
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4.3. The equivalence of local param
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4.5. The tangent vector space, the
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4.5. The tangent vector space, the
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4.6. The orientation of surfaces 12
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and 4.6. The orientation of surface
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4.7. Differentiable maps on a surfa
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4.7. Differentiable maps on a surfa
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where 4.8. The differential of a sm
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4.9. The spherical map and the shap
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4.10. The first fundamental form of
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4.11. The matrix of the shape opera
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4.12. The second fundamental form o
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4.13. The normal curvature. The Meu
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4.14. Asymptotic directions and asy
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4.15. The classification of points
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4.15. The classification of points
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4.16. Principal directions and curv
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4.17. The fundamental equations of
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4.18. The Gauss’ egregium theorem
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4.19 Geodesics 4.19.1 Introduction
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4.19. Geodesics 177 will denote it
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4.19. Geodesics 179 where, as we kn
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Examples of geodesics 4.19. Geodesi
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4.19. Geodesics 183 Here, for the f
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5.1 Ruled surfaces CHAPTER 5 Specia
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5.1. Ruled surfaces 187 Figure 5.2:
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5.1. Ruled surfaces 189 Proof. Sinc
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5.1. Ruled surfaces 191 surfaces, g
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5.1. Ruled surfaces 193 The three c
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5.1. Ruled surfaces 195 We can rega
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5.1. Ruled surfaces 197 5.1.5 Devel
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5.1. Ruled surfaces 199 • The str
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5.2. Minimal surfaces 201 Given a c
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5.2. Minimal surfaces 203 The follo
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5.2. Minimal surfaces 205 Proof. Le
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5.2.3 Ruled minimal surfaces 5.2. M
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5.2. Minimal surfaces 209 asymptoti
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5.2. Minimal surfaces 211 as γ is
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5.3. Surfaces of constant curvature
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5.3. Surfaces of constant curvature
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1. We consider, in the coordinate p
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Problems 219 12. Show that the norm
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Problems 221 33. Find the envelope,
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Problems 223 45. Find the equation
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Problems 225 58. Find the second fu
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Problems 227 67. Through a point M
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[1] Bär, C. - Elementare Different
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Bibliography 231 [29] Jellet, J.H.
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affine normal, 109 arc length, 19,
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182, 185, 187-189, 202, 205, 213, 2
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torus, 118, 119, 154, 191, 217, 219
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Blaga P. Lectures on the differential geometry of - tiera.ru

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