Lecture Notes for 120 - UCLA Department of Mathematics

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$Math 33B/1 S00 K. Solutions to Midterm Exam G1, Version B If you ...$

$Math 3C Homework 3 Solutions$

4.4. THE FIRST FUNDAMENTAL FORM 92 (b) Conclude that there is a surface of revolution with the same first fundamental form. (9) Assume a surface has a parametrization q (u, v) where apple guu g uv g vu g vv = apple r 2 0 0 r 2 where r (u) > 0 is only a function of u. Showthatthereisareparametrization u = u (s) such that the first fundamental form becomes apple apple gss g sv 1 0 = g vs g vv 0 r 2 (10) For a parametrized surface q (u, v) show that N ⇥ @q @u = g uu @q @v N ⇥ @q @v = g uv @q @v @q @u ⇥ @q @v @q @u ⇥ @q @v (11) If we have a parametrization where apple 1 0 [I] = 0 g vv g uv @q @u g vv @q @u then the coordinate function f (u, v) =u has ru = @q @u . (12) Show that it is always possible to find an orthogonal parametrization, i.e., g uv vanishes. (13) Show that if @g uu @v = @g vv @u = g uv =0 then we can reparametrize u and v separately, i.e., u = u (s) and v = v (t) , in such a way that we obtain Cartesian coordinates: (14) Show that if then and conclude that g ss = g tt =1, g st = 0 @ 2 q @u@v =0 q (u, v) =F (u)+G (v) @g uu @v = @g vv @u =0 Give an example where g uv 6=0.
4.5. SPECIAL MAPS AND PARAMETRIZATIONS 93 (15) Consider a unit speed curve ↵ (s) with non-vanishing curvature and the tube of radius R around it q (s, )=↵ (s)+R (N ↵ cos + B ↵ sin ) where T ↵ , N ↵ , B ↵ are the unit tangent, normal, and binormal to the curve. (a) Show that T ↵ and N ↵ sin + B ↵ cos are an orthonormal basis for the tangent space and that the normal to the tube is N = (N ↵ cos + B ↵ sin ). (b) Show that apple apple gss g s (1 appleR) 2 +(⌧R) 2 ⌧R = g ⌧R 2 R 2 g s 4.5. Special Maps and Parametrizations Definition 4.5.1. We call a map F : M 1 ! M 2 between surfaces an isometry if its differential preserves the first fundamental form I g1 (X, Y )=I g2 (DF (X) ,DF (Y )) We call the map area preserving if it preserves the areas of parallelograms spanned by vectors. We call the map conformal if it preserves angles between vectors. When the first surface is given as a parametrized surface these conditions can be quickly checked. Proposition 4.5.2. Let q : U ! M 1 be a parametrization and F : M 1 ! M 2 a map. If F q is also a parametrization, then the map is an isometry if and area preserving if and conformal if for some non-zero function . [I g1 ]=[I g2 ] det [I g1 ]=det[I g2 ] [I g1 ]= 2 [I g2 ] Proof. Note that it is not necessary to first check that F q is also a parametrization as that will be a consequence of any one of the three conditions if we define [I g2 ]= apple @F q @u @F q @v · @F q @u · @F q @u @F q @u @F q @v · @F q @v · @F q @v and observe that @F q @v , @F q @u are linearly independent if and only if the matrix [Ig2 ] has nonzero determinant. Next note that the chain rule implies that ✓ ◆ @q DF = @F q ✓ ◆ @q @u @u ,DF = @F q @v @v So all three conditions are necessarily true if the map is an isometry, area preserving, or conformal respectively. More generally, we see that ✓ DF (X) =DF X u @q ◆ @q + Xv = X u @F q + X v @F q @u @v @u @v
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Classical Differential Geometry Pet
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CONTENTS 4 Chapter 7. Other Topics
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1.1. CURVES 6 measurements are need
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1.1. CURVES 8 as we then get q (✓
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1.1. CURVES 10 these derivatives ar
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1.1. CURVES 12 (a) Show that if r (
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1.2. ARCLENGTH AND LINEAR MOTION 14
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1.3. CURVATURE 20 relates the side
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1.3. CURVATURE 22 of a that is norm
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1.3. CURVATURE 24 As the piece of t
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1.4. INTEGRAL CURVES 26 (14) Let q
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1.4. INTEGRAL CURVES 28 So at q 0 =
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CHAPTER 2 Planar Curves 2.1. Genera
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2.2. THE FUNDAMENTAL EQUATIONS 32 M
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2.2. THE FUNDAMENTAL EQUATIONS 34 (
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2.3. LENGTH AND AREA 36 (22) (Newto
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As ( 2.3. LENGTH AND AREA 38 sin
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2.3. LENGTH AND AREA 40 The choice
Page 42 and 43: 2.4. THE ROTATION INDEX 42 the hypo
Page 44 and 45: 2.4. THE ROTATION INDEX 44 where an
Page 46 and 47: 2.4. THE ROTATION INDEX 46 Exercise
Page 48 and 49: 2.5. TWO SURPRISING RESULTS 48 unit
Page 50 and 51: 2.6. CONVEX CURVES 50 Exercises. Po
Page 52 and 53: 2.6. CONVEX CURVES 52 Proof. Any cu
Page 54 and 55: 2.6. CONVEX CURVES 54 (9) Give an e
Page 56 and 57: 3.1. THE FUNDAMENTAL EQUATIONS 56 T
Page 58 and 59: 3.1. THE FUNDAMENTAL EQUATIONS 58 B
Page 60 and 61: 3.1. THE FUNDAMENTAL EQUATIONS 60 (
Page 62 and 63: 3.2. CHARACTERIZATIONS OF SPACE CUR
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Page 66 and 67: 3.3. CLOSED SPACE CURVES 66 (12) Pr
Page 68 and 69: 3.3. CLOSED SPACE CURVES 68 Thus th
Page 70 and 71: 3.3. CLOSED SPACE CURVES 70 (b) Sho
Page 72 and 73: CHAPTER 4 Basic Surface Theory 4.1.
Page 74 and 75: 4.1. SURFACES 74 Definition 4.1.6.
Page 76 and 77: 4.1. SURFACES 76 (6) Many classical
Page 78 and 79: 4.2. TANGENT SPACES AND MAPS 78 Thi
Page 80 and 81: 4.2. TANGENT SPACES AND MAPS 80 We
Page 82 and 83: 4.2. TANGENT SPACES AND MAPS 82 Def
Page 84 and 85: 4.3. THE ABSTRACT FRAMEWORK 84 Befo
Page 86 and 87: 4.3. THE ABSTRACT FRAMEWORK 86 in o
Page 88 and 89: 4.4. THE FIRST FUNDAMENTAL FORM 88
Page 90 and 91: 4.4. THE FIRST FUNDAMENTAL FORM 90
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Page 98 and 99: 4.6. THE GAUSS FORMULAS 98 and then
Page 100 and 101: CHAPTER 5 The Second Fundamental Fo
Page 102 and 103: 5.1. CURVES ON SURFACES 102 This sh
Page 104 and 105: 5.1. CURVES ON SURFACES 104 We also
Page 106 and 107: 5.1. CURVES ON SURFACES 106 Exercis
Page 108 and 109: 5.2. THE GAUSS AND WEINGARTEN MAPS
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Page 112 and 113: 5.3. THE GAUSS AND MEAN CURVATURES
Page 118 and 119: 5.4. PRINCIPAL CURVATURES 118 The f
Page 120 and 121: 5.4. PRINCIPAL CURVATURES 120 By le
Page 122 and 123: 5.5. RULED SURFACES 122 (16) Let q
Page 124 and 125: 5.5. RULED SURFACES 124 Definition
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Page 128 and 129: 5.5. RULED SURFACES 128 This shows
Page 130 and 131: 5.5. RULED SURFACES 130 (b) Show th
Page 132 and 133: 5.5. RULED SURFACES 132 (e) Show th
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Page 140 and 141: 6.3. ACCELERATION 140 Exercises. (1
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6.3. ACCELERATION 142 Alternately t
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6.4. THE GAUSS AND CODAZZI EQUATION
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6.5. LOCAL GAUSS-BONNET 150 (b) Use
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6.5. LOCAL GAUSS-BONNET 152 We furt
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6.5. LOCAL GAUSS-BONNET 154 clockwi
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6.5. LOCAL GAUSS-BONNET 156 where t
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7.1. GEODESICS 158 we must solve a
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7.2. UNPARAMETRIZED GEODESICS 160 T
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7.5. ISOMETRIES 162 since the v-cur
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7.6. THE UPPER HALF PLANE 164 Proof
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7.6. THE UPPER HALF PLANE 166 and t
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7.6. THE UPPER HALF PLANE 168 so so
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8.2. RIEMANNIAN GEOMETRY 170 The ke
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APPENDIX A Vector Calculus A.1. Vec
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A.2. GEOMETRY 174 the characteristi
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A.4. DIFFERENTIAL EQUATIONS 176 The
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A.4. DIFFERENTIAL EQUATIONS 178 Nex
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APPENDIX B Special Coordinate Repre
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This leaves us with finding L s s.
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B.3. MONGE PATCHES 184 So we immedi
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B.4. SURFACES GIVEN BY AN EQUATION
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B.6. CHEBYSHEV NETS 188 (1) Show th
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B.7. ISOTHERMAL COORDINATES 190 K =
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