Lecture Notes for 120 - UCLA Department of Mathematics

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

$Math 3C Homework 3 Solutions$

3.1. THE FUNDAMENTAL EQUATIONS 56 Theorem 3.1.1. (Serret, 1851 and Frenet, 1852) If q (t) is a regular space curve with linearly independent velocity and acceleration, then or Moreover, dq dt dT dt dN dt dB dt d ⇥ ⇤ ds q T N B = dt dt apple = = ds dt T = apple ds dt N = apple ds dt T + ⌧ ds dt B = ⌧ ds dt N 2 ⇥ ⇤ q T N B 6 4 |v ⇥ a| |a (a · T) T| |v| 3 = |v| 2 ⌧ = det ⇥ v a j ⇤ |v ⇥ a| 2 = a (a · T) T N = |a (a · T) T| B = v ⇥ a |v ⇥ a| (v ⇥ a) · j |v ⇥ a| 2 0 0 0 0 1 0 apple 0 0 apple 0 ⌧ 0 0 ⌧ 0 Proof. The explicit formula for the N is our explicit formula for the principal normal. As T, N, B form an orthonormal basis we have ✓ ◆ ✓ ◆ ✓ ◆ dT dT dT dT dt = dt · T T + dt · N N + dt · B B Here dT dt · B = d dt dT dt · T = 1 d |T| 2 =0 2 dt dT dt · N = ds dT dt ds · N = ds dt apple ✓ ◆ v · B = a |v| |v| v d |v| |v| 2 dt ! · B =0 as B is perpendicular to T, N and thus also to v, a. The establishes Next note that dT dt = appleds dt N 3 7 5 This together with 0=B · dT dt = B · dB dt =0 dB dt · T
3.1. THE FUNDAMENTAL EQUATIONS 57 shows that dB dt = ✓ dB dt · N ◆ N However, we also have This implies Finally the equation 0= dB dt · N + B · dN dt = dB dt · N + ds dt ⌧ dN dt = dB dt = ⌧ ds dt N appleds dt T + ⌧ ds dt B is a direct consequence of the other two equations. The formula for the curvature follows from observing that ! dT ds · N = a v d |v| |v| |v| 2 · N ds = a |v| · = a (a · T) T |a (a · T) T| a · a (a · T) 2 |v||a (a · T) T| = |a|2 |v| 2 (a · v) 2 |v| 3 |a (a · T) T| |a (a · T) T| = |v| 2 q where |v||a (a · T) T| = |a| 2 |v| 2 (a · v) 2 . The formula for the binormal B now follows directly from the calculation T ⇥ N = 1 ✓ ◆ a (a · T) T |v| v ⇥ |a (a · T) T| = 1 ✓ ◆ |v| v ⇥ a |a (a · T) T| = v ⇥ a |v||a (a · T) T| = v ⇥ a |v ⇥ a| In the last equality recall that the denominators are the areas of the same parallelogram spanned by v and a. To establish the general formula for ⌧ we note (with more explanations to follow)
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Classical Differential Geometry Pet
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CONTENTS 4 Chapter 7. Other Topics
Page 6 and 7: 1.1. CURVES 6 measurements are need
Page 8 and 9: 1.1. CURVES 8 as we then get q (✓
Page 10 and 11: 1.1. CURVES 10 these derivatives ar
Page 12 and 13: 1.1. CURVES 12 (a) Show that if r (
Page 14 and 15: 1.2. ARCLENGTH AND LINEAR MOTION 14
Page 20 and 21: 1.3. CURVATURE 20 relates the side
Page 22 and 23: 1.3. CURVATURE 22 of a that is norm
Page 24 and 25: 1.3. CURVATURE 24 As the piece of t
Page 26 and 27: 1.4. INTEGRAL CURVES 26 (14) Let q
Page 28 and 29: 1.4. INTEGRAL CURVES 28 So at q 0 =
Page 30 and 31: CHAPTER 2 Planar Curves 2.1. Genera
Page 32 and 33: 2.2. THE FUNDAMENTAL EQUATIONS 32 M
Page 34 and 35: 2.2. THE FUNDAMENTAL EQUATIONS 34 (
Page 36 and 37: 2.3. LENGTH AND AREA 36 (22) (Newto
Page 38 and 39: As ( 2.3. LENGTH AND AREA 38 sin
Page 40 and 41: 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 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
Page 64 and 65: 3.2. CHARACTERIZATIONS OF SPACE CUR
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 94 and 95: 4.5. SPECIAL MAPS AND PARAMETRIZATI
Page 96 and 97: 4.5. SPECIAL MAPS AND PARAMETRIZATI
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
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5.1. CURVES ON SURFACES 106 Exercis
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5.2. THE GAUSS AND WEINGARTEN MAPS
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5.2. THE GAUSS AND WEINGARTEN MAPS
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5.3. THE GAUSS AND MEAN CURVATURES
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5.4. PRINCIPAL CURVATURES 118 The f
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5.4. PRINCIPAL CURVATURES 120 By le
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5.5. RULED SURFACES 122 (16) Let q
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5.5. RULED SURFACES 124 Definition
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i.e., d dv and X are parallel. In p
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5.5. RULED SURFACES 128 This shows
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5.5. RULED SURFACES 130 (b) Show th
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5.5. RULED SURFACES 132 (e) Show th
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6.1. CALCULATING CHRISTOFFEL SYMBOL
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6.1. CALCULATING CHRISTOFFEL SYMBOL
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6.2. GENERALIZED AND ABSTRACT SURFA
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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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