Lecture Notes for 120 - UCLA Department of Mathematics

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

$Math 3C Homework 3 Solutions$

This leaves us with finding L s s. Since @q @s Thus B.2. SURFACES OF REVOLUTION 182 is a unit vector this is simply L s @N s = @s · @q @s = (h 00 cos µ, h 00 sin µ, r 00 ) · (r 0 cos µ, r 0 sin µ, h 0 ) = h 00 r 0 r 00 h 0 K = (h 00 r 0 r 00 h 0 ) h0 r H = h0 r + h00 r 0 r 00 h 0 In the case of cylinder, plane, and cone we note that K vanishes, but H only vanishes when it is a plane. This means that we have a selection of surfaces all with Cartesian coordinates with different H. We can in general simplify the Gauss curvature by noting that Thus yielding 1 = (r 0 ) 2 +(h 0 ) 2 ⇣ 0 = (r 0 ) 2 +(h 0 ) 2⌘ 0 =2r 0 r 00 +2h 0 h 00 ! K = r 00 (r0 ) 2 h 0 r 00 h 0 h 0 r ⇣ = r00 (r 0 ) 2 (h 0 ) 2⌘ r = = r 00 r @ 2 p @s 2 grr p grr This makes it particularly easy to calculate the Gauss curvature and also to construct examples with a given curvature function. It also shows that the Gauss curvature can be computed directly from the first fundamental form! For instance if we want K = 1, then we can just use r (s) =exp( s) for s>0 and then adjust h (s) for s 2 (0, 1) such that 1=(r 0 ) 2 +(h 0 ) 2 If we introduce a new parameter t =exp(s) > 1, then we obtain a new parametrization of the same surface q (t, µ) = q (ln (t) ,µ) = (exp( ln t) cos µ, exp ( ln t)sinµ, h (ln t)) ✓ 1 = t cos µ, 1 ◆ t sin µ, h (ln t)
B.3. MONGE PATCHES 183 To find the first fundamental form of this surface we have to calculate Thus d dt dh 1 h (ln t) = q ds t = 1 (r 0 ) 2 1 t = apple 1 1 I= t + 1 4 q 1 ( exp ( s)) 2 1 t = p 1 exp ( 2lnt) 1 r t 1 1 = 1 t 2 t 1 t 2 t 0 2 1 = 0 t 2 apple 1 t 2 0 0 1 t 2 This is exactly what the first fundamental form for the upper half plane looked like. But the domains for the two are quite different. What we have achieved is a local representation of part of the upper half plane. Exercises. (1) Show that geodesics on a surface of revolution satisfy Clairaut’s condition: r sin is constant, where is the angle the geodesic forms with the meridians. B.3. Monge Patches This is more complicated than the previous case, but that is only to be expected as all surfaces admit Monge patches. We consider q (u, v) =(u, v, f (u, v)) . Thus ✓ @q @u = 1, 0, @f ◆ , @u ✓ @q = 0, 1, @f ◆ @v @v N = r 1+ ⇣ @f @u , @f ⇣ @f @u @v , 1 ⌘ ⌘ 2 + ⇣ @f @v ⌘ 2
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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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4.2. TANGENT SPACES AND MAPS 78 Thi
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4.2. TANGENT SPACES AND MAPS 80 We
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4.2. TANGENT SPACES AND MAPS 82 Def
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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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Page 140 and 141: 6.3. ACCELERATION 140 Exercises. (1
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Page 160 and 161: 7.2. UNPARAMETRIZED GEODESICS 160 T
Page 162 and 163: 7.5. ISOMETRIES 162 since the v-cur
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
Page 168 and 169: 7.6. THE UPPER HALF PLANE 168 so so
Page 170 and 171: 8.2. RIEMANNIAN GEOMETRY 170 The ke
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Page 174 and 175: A.2. GEOMETRY 174 the characteristi
Page 176 and 177: A.4. DIFFERENTIAL EQUATIONS 176 The
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Page 184 and 185: B.3. MONGE PATCHES 184 So we immedi
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Page 188 and 189: B.6. CHEBYSHEV NETS 188 (1) Show th
Page 190: B.7. ISOTHERMAL COORDINATES 190 K =
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Lecture Notes for 120 - UCLA Department of Mathematics

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