Lecture Notes for 120 - UCLA Department of Mathematics

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2.3. LENGTH AND AREA 36 (22) (Newton, 1671, but the idea is much older for specific curves. Kepler considered it well-known.) Consider a regular planar curve q (t). For 3 consecutive values t ✏
2.3. LENGTH AND AREA 37 We denote the space of oriented lines by OL. To eliminate the choice of angle we identify OL with with points on the cylinder S 1 ⇥ [0, 1), where the first factor describes a unit direction in the plane and the second a distance. This eliminates the angle choice, but it still leaves us with having specified p as a distance to a specific point. The cylinder model helps us put a measure on OL so that we can integrate functions. This is simply done as a natural surface integral: ˆ l2OL ˆ f (l) dl = f (l (u, v)) @l @u ⇥ @l @v dudv If q :[a, b] ! R 2 is a curve denote by n q (l) the number of times the curve intersects the oriented line l. Theorem 2.3.1. (Cauchy-Crofton) The length of a curve can be computed via the formula L (q) = 1 ˆ n q (l) dl 4 l2OL Proof. Consider a regular curve q (t) :[a, b] ! R 2 . At a point q (t) of the curve consider all lines passing through this point. If we denote the direction of each of these oriented lines by (cos ✓, sin ✓) then the distance from this line to the origin is given by r (✓, t) =|( sin ✓, cos ✓) · q (t)| This gives us a parametrization of the set of lines that intersect the curve l (✓, t) = (cos ✓, sin ✓, p (✓, t)) However, different values of t correspond to the same line if that line intersects the curve in several points, i.e., n q (l) is the number of t values where l = l (✓, t). Thus the integral on the right hand side can be reinterpreted as ˆ 1 n q (l) dl = 1 ˆ dl (✓, t) 4 l2OL 4 (✓,t)2[0,2⇡]⇥[a,b] = 1 4 = 1 4 = 1 4 = 1 4 = 1 4 = 1 4 ˆ b ˆ 2⇡ a 0 ˆ b ˆ 2⇡ a 0 ˆ b ˆ 2⇡ a 0 ˆ b ˆ 2⇡ a 0 ˆ b ˆ 2⇡ a 0 ˆ L ˆ 2⇡ Now for a fixed s (or t) considertheintegral ˆ 2⇡ 0 0 0 @l @✓ ⇥ @l @t d✓dt 2 3 2 sin ✓ 4 cos ✓ 5 ⇥ 4 @r @✓ @r @t d✓dt 0 0 @r @t ( sin ✓, cos ✓) · dq dt 3 5 d✓dt d✓dt |( sin ✓, cos ✓) · T| ds dt d✓dt |( sin ✓, cos ✓) · T| d✓ |( sin ✓, cos ✓) · T| d✓ds
Page 1: Classical Differential Geometry Pet
Page 4 and 5: 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 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 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
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
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4.3. THE ABSTRACT FRAMEWORK 86 in o
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4.4. THE FIRST FUNDAMENTAL FORM 88
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4.5. SPECIAL MAPS AND PARAMETRIZATI
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4.5. SPECIAL MAPS AND PARAMETRIZATI
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4.6. THE GAUSS FORMULAS 98 and then
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CHAPTER 5 The Second Fundamental Fo
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5.1. CURVES ON SURFACES 102 This sh
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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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