Lecture Notes for 120 - UCLA Department of Mathematics

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Contents Chapter 1. General Curve Theory 5 1.1. Curves 5 1.2. Arclength and Linear Motion 13 1.3. Curvature 20 1.4. Integral Curves 26 Chapter 2. Planar Curves 30 2.1. General Frames 30 2.2. The Fundamental Equations 31 2.3. Length and Area 36 2.4. The Rotation Index 42 2.5. Two Surprising Results 47 2.6. Convex Curves 50 Chapter 3. Space Curves 55 3.1. The Fundamental Equations 55 3.2. Characterizations <strong>of</strong> Space Curves 62 3.3. Closed Space Curves 66 Chapter 4. Basic Surface Theory 72 4.1. Surfaces 72 4.2. Tangent Spaces and Maps 77 4.3. The Abstract Framework 83 4.4. The First Fundamental Form 87 4.5. Special Maps and Parametrizations 93 4.6. The Gauss Formulas 97 Chapter 5. The Second Fundamental Form 100 5.1. Curves on Surfaces 100 5.2. The Gauss and Weingarten Maps and Equations 107 5.3. The Gauss and Mean Curvatures 111 5.4. Principal Curvatures 117 5.5. Ruled Surfaces 122 Chapter 6. Intrinsic Calculations on Surfaces 133 6.1. Calculating Christ<strong>of</strong>fel Symbols and The Gauss Curvature 133 6.2. Generalized and Abstract Surfaces 138 6.3. Acceleration 140 6.4. The Gauss and Codazzi Equations 143 6.5. Local Gauss-Bonnet 150 3
Page 1: Classical Differential Geometry Pet
Page 5 and 6: CHAPTER 1 General Curve Theory One
Page 7 and 8: 1.1. CURVES 7 to the differential e
Page 9 and 10: 1.1. CURVES 9 Example 1.1.5. A far
Page 11 and 12: 1.1. CURVES 11 Definition 1.1.13. W
Page 13 and 14: 1.2. ARCLENGTH AND LINEAR MOTION 13
Page 21 and 22: 1.3. CURVATURE 21 of how fast it ch
Page 23 and 24: 1.3. CURVATURE 23 Proof. The first
Page 25 and 26: 1.3. CURVATURE 25 Exercises. (1) Sh
Page 27 and 28: 1.4. INTEGRAL CURVES 27 The first o
Page 29 and 30: 1.4. INTEGRAL CURVES 29 (2) Conside
Page 31 and 32: 2.2. THE FUNDAMENTAL EQUATIONS 31 S
Page 33 and 34: 2.2. THE FUNDAMENTAL EQUATIONS 33 P
Page 35 and 36: 2.2. THE FUNDAMENTAL EQUATIONS 35 (
Page 37 and 38: 2.3. LENGTH AND AREA 37 We denote t
Page 39 and 40: 2.3. LENGTH AND AREA 39 can decide
Page 41 and 42: 2.3. LENGTH AND AREA 41 This tells
Page 43 and 44: 2.4. THE ROTATION INDEX 43 where th
Page 45 and 46: 2.4. THE ROTATION INDEX 45 It now f
Page 47 and 48: 2.5. TWO SURPRISING RESULTS 47 (7)
Page 49 and 50: 2.5. TWO SURPRISING RESULTS 49 Defi
Page 51 and 52: 2.6. CONVEX CURVES 51 Theorem 2.6.2
Page 53 and 54:
2.6. CONVEX CURVES 53 (c) Show that
Page 55 and 56:
CHAPTER 3 Space Curves 3.1. The Fun
Page 57 and 58:
3.1. THE FUNDAMENTAL EQUATIONS 57 s
Page 59 and 60:
3.1. THE FUNDAMENTAL EQUATIONS 59 P
Page 61 and 62:
3.1. THE FUNDAMENTAL EQUATIONS 61 (
Page 63 and 64:
3.2. CHARACTERIZATIONS OF SPACE CUR
Page 65 and 66:
3.2. CHARACTERIZATIONS OF SPACE CUR
Page 67 and 68:
3.3. CLOSED SPACE CURVES 67 Aregula
Page 69 and 70:
3.3. CLOSED SPACE CURVES 69 Theorem
Page 71 and 72:
3.3. CLOSED SPACE CURVES 71 (10) Sh
Page 73 and 74:
4.1. SURFACES 73 They are also know
Page 75 and 76:
4.1. SURFACES 75 Exercises. (1) A g
Page 77 and 78:
4.2. TANGENT SPACES AND MAPS 77 (10
Page 79 and 80:
4.2. TANGENT SPACES AND MAPS 79 Exa
Page 81 and 82:
4.2. TANGENT SPACES AND MAPS 81 In
Page 83 and 84:
4.3. THE ABSTRACT FRAMEWORK 83 (c)
Page 85 and 86:
4.3. THE ABSTRACT FRAMEWORK 85 Note
Page 87 and 88:
4.4. THE FIRST FUNDAMENTAL FORM 87
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
4.5. SPECIAL MAPS AND PARAMETRIZATI
Page 95 and 96:
4.5. SPECIAL MAPS AND PARAMETRIZATI
Page 97 and 98:
4.6. THE GAUSS FORMULAS 97 (c) Show
Page 99 and 100:
4.6. THE GAUSS FORMULAS 99 or or @
Page 101 and 102:
5.1. CURVES ON SURFACES 101 Taking
Page 103 and 104:
5.1. CURVES ON SURFACES 103 then q
Page 105 and 106:
5.1. CURVES ON SURFACES 105 thus yi
Page 107 and 108:
5.2. THE GAUSS AND WEINGARTEN MAPS
Page 109 and 110:
As dq dt 5.2. THE GAUSS AND WEINGAR
Page 111 and 112:
5.3. THE GAUSS AND MEAN CURVATURES
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
5.4. PRINCIPAL CURVATURES 117 (17)
Page 119 and 120:
5.4. PRINCIPAL CURVATURES 119 its p
Page 121 and 122:
5.4. PRINCIPAL CURVATURES 121 (5) C
Page 123 and 124:
5.5. RULED SURFACES 123 are best re
Page 125 and 126:
5.5. RULED SURFACES 125 The next re
Page 127 and 128:
5.5. RULED SURFACES 127 where g ss
Page 129 and 130:
5.5. RULED SURFACES 129 and for a c
Page 131 and 132:
5.5. RULED SURFACES 131 (b) Show th
Page 133 and 134:
CHAPTER 6 Intrinsic Calculations on
Page 135 and 136:
6.1. CALCULATING CHRISTOFFEL SYMBOL
Page 137 and 138:
6.1. CALCULATING CHRISTOFFEL SYMBOL
Page 139 and 140:
6.2. GENERALIZED AND ABSTRACT SURFA
Page 141 and 142:
6.3. ACCELERATION 141 becomes ¨q =
Page 143 and 144:
6.4. THE GAUSS AND CODAZZI EQUATION
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
6.5. LOCAL GAUSS-BONNET 151 where t
Page 153 and 154:
6.5. LOCAL GAUSS-BONNET 153 Proof.
Page 155 and 156:
6.5. LOCAL GAUSS-BONNET 155 an orth
Page 157 and 158:
CHAPTER 7 Other Topics We cover geo
Page 159 and 160:
7.2. UNPARAMETRIZED GEODESICS 159 (
Page 161 and 162:
7.4. CONSTANT GAUSS CURVATURE 161 T
Page 163 and 164:
7.5. ISOMETRIES 163 Basic examples
Page 165 and 166:
7.6. THE UPPER HALF PLANE 165 7.6.1
Page 167 and 168:
7.6. THE UPPER HALF PLANE 167 The g
Page 169 and 170:
CHAPTER 8 Global Theory 8.1. Global
Page 171 and 172:
8.2. RIEMANNIAN GEOMETRY 171 This e
Page 173 and 174:
A.1. VECTOR AND MATRIX NOTATION 173
Page 175 and 176:
A.3. DIFFERENTIATION AND INTEGRATIO
Page 177 and 178:
A.4. DIFFERENTIAL EQUATIONS 177 We
Page 179 and 180:
A.4. DIFFERENTIAL EQUATIONS 179 whe
Page 181 and 182:
B.2. SURFACES OF REVOLUTION 181 The
Page 183 and 184:
B.3. MONGE PATCHES 183 To find the
Page 185 and 186:
B.4. SURFACES GIVEN BY AN EQUATION
Page 187 and 188:
B.5. GEODESIC COORDINATES 187 To ca
Page 189 and 190:
B.7. ISOTHERMAL COORDINATES 189 (2)
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Lecture Notes for 120 - UCLA Department of Mathematics

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