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

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36 Chapter 1. Space curves Explicit representation If we have a space curve given by the equations ⎧ ⎪⎨ y = f (x) ⎪⎩ z = g(x) then we can construct a parameterization ⎧ x = t ⎪⎨ y = f (t) ⎪⎩ z = g(t). For the derivatives we obtain immediately the expressions ⎧ x ⎪⎨ ′ = 1 y ′ = f ′ ⎪⎩ z = g ′ , which, when substituted into the equations (1.5.7), give X − x = Y − f (x) f ′ (x) , Z − g(x) = g ′ , (1.5.11) (x) while for the equation of the normal plane, after substituting the derivatives into the equation (1.5.9), we obtain For a plane curve given explicitly X − x + (Y − f (x)) f ′ (x) + (Z − g(x))g ′ (x) = 0. (1.5.12) y = f (x), we have the parametric representation ⎧ ⎪⎨ x = t ⎪⎩ y = f (t), and, thus, the equation of the tangent line is X − x = Y − f (x) f ′ (x) (1.5.13)
or, in a more familiar form, while for the normal we get ore Implicit representation 1.5. The tangent and the normal plane 37 Let us consider a curve given by the implicit equations ⎧ ⎪⎨ F(x, y) = 0 ⎪⎩ G(x, y) = 0. Let us suppose that, at a point (x0, y0, z0) Y − f (x) = f ′ (x)(X − x), (1.5.14) X − x + (Y − f (x)) f ′ (x) = 0 (1.5.15) Y − f (x) = − 1 f ′ (X − x). (1.5.16) (x) det � F ′ y F ′ z G ′ y G ′ z � � 0 Then, as we saw before, around this point, the curve can be represented as ⎧ ⎪⎨ y = f (x) ⎪⎩ z = g(x), i.e. the system (1.5.17) can be written as ⎧ ⎪⎨ F(x, f (x), g(x)) = 0 ⎪⎩ G(x, f (x), g(x)) = 0. By computing the derivatives with respect to x of F and G, we get the system ⎧ ⎪⎨ F ⎪⎩ ′ x + f ′ (x)F ′ y + g ′ (x)F ′ z = 0 G ′ x + f ′ (x)G ′ y + g ′ (x)G ′ z = 0, therefore ⎧ ⎪⎨ ⎪⎩ f ′ F ′ y + g ′ F ′ z = −F ′ x f ′ G ′ y + g ′ G ′ z = −G ′ x. (1.5.17) (1.5.18)
Page 1: Lectures on the Di
Page 5 and 6: Contents Foreword 9 I Curves 11 1 S
Page 7 and 8: Contents 7 4.5 The tangent vector 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 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
Page 56 and 57: 56 Chapter 1. Space curves or 1 + r
Page 58 and 59: 58 Chapter 1. Space curves or c0 ·
Page 60 and 61: 60 Chapter 1. Space curves Let ω b
Page 62 and 63: 62 Chapter 1. Space curves Corollar
Page 64 and 65: 64 Chapter 1. Space curves vector r
Page 66 and 67: 66 Chapter 1. Space curves a) If a
Page 68 and 69: 68 Chapter 1. Space curves d) We sh
Page 70 and 71: 70 Chapter 1. Space curves Then: τ
Page 72 and 73: 72 Chapter 1. Space curves 1.14.3 T
Page 74 and 75: 74 Chapter 1. Space curves
Page 76 and 77: 76 Chapter 2. Plane curves Proof. I
Page 78 and 79: 78 Chapter 2. Plane curves By subst
Page 80 and 81: 80 Chapter 2. Plane curves The foll
Page 82 and 83: 82 Chapter 2. Plane curves c) J(Jv)
Page 84 and 85: 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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