Lecture Notes for 120 - UCLA Department of Mathematics

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

$Math 3C Homework 3 Solutions$

5.5. RULED SURFACES 128 This shows that X is in fact a planar circle or radius 1. For simplicity let us further assume that it is the unit circle in the (x, y)-plane, i.e., From the s-term we obtain showing that d2 c dv 2 is constant. Since dc dv ? dX dv and X (v) = (cos v, sin v, 0) ✓ X ⇥ dX dv 0 = d2 c dv 2 · ✓ = d2 c dv 2 · X ⇥ dX ◆ dv ✓ = d2 c dv 2 · X ⇥ dX ◆ dv 2 3 0 = d2 c dv 2 · 4 0 5 1 ◆ + d2 X ✓ dv 2 · X · X ⇥ dc ◆ dv ◆ ✓ X ⇥ dc dv also lies in the (x, y)-plane. In particular, 2 3 0 dc dv · 4 0 5 = h 1 we obtain ✓ ◆ dc dc dv = dv · X X + 2 dc dv ⇥ X = 4 0 0 h 3 2 4 0 0 h 3 5 5 ⇥ X = h dX dv This considerably simplifies the terms that are independent of s in the mean curvature equation ✓ ◆✓ ✓ dc dX 2 dv · X dv · X ⇥ dc ◆◆ ✓ = d2 c dv dv 2 · X ⇥ dc ◆ dv as we then obtain 2h dc dv · X = h d2 c dv 2 · dX dv = h dc dv · d2 X dv 2 = h dc dv · X When h =0the curve c also lies in the (x, y)-plane and the surface is planar. Otherwise dc dv · X =0which implies that 2 3 2 3 ✓ ◆ dc dc dv = dv · X X + 4 5 = 4 5 0 0 h 0 0 h
5.5. RULED SURFACES 129 and for a constant v 0 . The surface is then given by 2 c = 4 q (s, v) = 4 which shows explicitly that it is a helicoid. 3 0 0 5 hv + v 0 2 s cos v 3 s sin v 5 Exercises. (1) Let q (s) be a unit speed asymptotic line of a ruled surface q (u, v) = ↵ (v)+uX (v). Notethatu-curves are asymptotic lines. (a) Show that ⇤ =0 det ⇥ d↵ ¨q, X, dv + u dX dv (b) Assume for the remainder of the exercise that K < 0. Show that there is a unique asymptotic line through through every point that is not tangent to X. (c) Show that this asymptotic line can locally be reparametrized as where h du det dv = ↵ (v)+u (v) X (v) d↵ dX X, dv + u (v) dv , d 2 ↵ dv 2 2det ⇥ d↵ dv , X, dX dv + u (v) d2 X dv ⇤ 2 (2) Show that a generalized cylinder q (u, v) =↵ (v)+uX where X is a fixed unit vector admits a parametrization q (s, t) =c (t) +sX (t), where c is parametrized by arclength and lies a plane orthogonal to X. (3) Consider a parameterized surface q (u, v). ShowthattheGausscurvature vanishes if and only if @N @u , @N @v are linearly dependent everywhere. (4) Consider q (u, v) = u + v, u 2 +2uv, u 3 +3u 2 v (a) Determine when it defines a surface. (b) Show that the Gauss curvature vanishes. (c) What type of ruled surface is it? (5) Consider the Monge patch nX z = (ax + by) k + cx + dy + f k=2 (a) Show that the Gauss curvature vanishes. (b) Depending on the values of a and b determine the type of ruled surface. (6) Consider the equation xy =(z c) 2 (a) Determine when it defines a surface. i ⇤
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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
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2.4. THE ROTATION INDEX 42 the hypo
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2.4. THE ROTATION INDEX 44 where an
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2.4. THE ROTATION INDEX 46 Exercise
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2.5. TWO SURPRISING RESULTS 48 unit
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2.6. CONVEX CURVES 50 Exercises. Po
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2.6. CONVEX CURVES 52 Proof. Any cu
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2.6. CONVEX CURVES 54 (9) Give an e
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3.1. THE FUNDAMENTAL EQUATIONS 56 T
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3.1. THE FUNDAMENTAL EQUATIONS 58 B
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3.1. THE FUNDAMENTAL EQUATIONS 60 (
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3.2. CHARACTERIZATIONS OF SPACE CUR
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3.2. CHARACTERIZATIONS OF SPACE CUR
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3.3. CLOSED SPACE CURVES 66 (12) Pr
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3.3. CLOSED SPACE CURVES 68 Thus th
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3.3. CLOSED SPACE CURVES 70 (b) Sho
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CHAPTER 4 Basic Surface Theory 4.1.
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4.1. SURFACES 74 Definition 4.1.6.
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4.1. SURFACES 76 (6) Many classical
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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
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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
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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 140 and 141: 6.3. ACCELERATION 140 Exercises. (1
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Page 164 and 165: 7.6. THE UPPER HALF PLANE 164 Proof
Page 166 and 167: 7.6. THE UPPER HALF PLANE 166 and t
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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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Lecture Notes for 120 - UCLA Department of Mathematics

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