Lecture Notes for 120 - UCLA Department of Mathematics

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

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6.5. LOCAL GAUSS-BONNET 152 We further have d 2 u ds 2 = d cos ✓ = sin ✓ d✓ ds ds , d 2 v ds 2 = d 1 r sin ✓ ds 1 dr = r 2 ds sin ✓ + 1 d✓ cos ✓ r ds = = ✓ 1 @r r 2 @u 1 @r r 2 @u du ds + @r @v dv ds ◆ sin ✓ + 1 d✓ cos ✓ r ds 1 @r cos ✓ sin ✓ + r 3 @v sin2 ✓ + 1 d✓ cos ✓ r ds And the Christoffel symbols are not hard to compute ✓ u dq ds , dq ◆ ✓ ◆ 2 u dv = vv ds ds = r @r 1 @u r 2 sin2 ✓ 1 @r = r @u sin2 ✓ Thus ✓ apple g = sin ✓ ✓ v dq ds , dq ◆ ds sin ✓ d✓ ds = d✓ ds + 1 @r r = d✓ ds + @r 1 @u r sin ✓ @u sin3 ✓ + 1 r = 2 v du dv uv ds ds + = 2 r @r @u v uv du dv ds ds + 1 @r r @v ✓ dv ds ◆ 2 ✓ dv ds ◆ 2 = 2 r 2 @r @u sin ✓ cos ✓ + 1 r 3 @r @v sin2 ✓ ◆ ✓ 1 @r 1 d✓ r @u sin2 ✓ + r cos ✓ cos ✓ r ds + 1 ◆ @r r 2 @u sin ✓ cos ✓ @r @u sin ✓ cos2 ✓ We can now prove the local Gauss-Bonnet theorem. It is stated in the way that Gauss and Bonnet proved it. Gauss considered regions bounded by geodesics thus eliminating the geodesic curvature, while Bonnet presented the version given below. Theorem 6.5.2. [Gauss-Bonnet] Assume as in the above Lemma that the parametrization gives a geodesic coordinate system. Let ✓ i be the exterior angles at the points where q has vertices, then ˆ ˆ L X KdA + apple g ds =2⇡ ✓i q(R) 0 ⇤
6.5. LOCAL GAUSS-BONNET 153 Proof. We have that ˆ q(R) @R KdA = = = ˆ R ˆ ˆ K p det [I]dudv R R @ 2 r @u 2 r rdudv @ 2 r @u 2 dudv The last integral can be turned into a line integral if we use Green’s theorem ˆ @ 2 r R @u ˆ@R 2 dudv = @r @u dv This line integral can now be recognized as one of the terms in the formula for the geodesic curvature ˆ ˆ @r L @u dv = @r dv @u ds ds Thus we obtain ˆ KdA + q(R) ˆ L Finally we must show that 0 apple g ds = = ˆ L 0 = = = 0 ˆ L 0 ˆ L 0 ˆ L ˆ 0 R ˆ L 0 @r 1 sin ✓ds @u r ✓ d✓ apple g apple g ds ds ˆ L 0 ◆ ds d✓ ds ds @ 2 ˆ r @u ˆ@R 2 dudv + @r L @u dv + d✓ 0 ds ds d✓ ds ds d✓ ds ds + X ✓ i =2⇡ For a planar simple closed curve this is a consequence of knowing that the rotation index must be 1 for such curves if they are parametrized to run counterclockwise. In this case we still know that the right hand side must be a multiple of 2⇡. The trick now is to compute the right hand side for each of the metrics apple 1 0 [I ✏ ]= 0 1 ✏ + ✏r 2 For each ✏ 2 [0, 1] this defines a metric on (a u ,b u ) ⇥ (a v ,b v ) and the rotation index for our curve has to be a multiple 2⇡. It is easy to see that the angles ✓ ✏ between the curve and the u-curves is continuous in ✏. Thus the right hand side also varies continuously. However, as it it is always a multiple of 2⇡ and is 2⇡ in case ✏ =0it follows that it is always 2⇡. ⇤ Clearly there are subtle things about the regions R we are allowed to use. Aside from the topological restriction on R there is also an orientation choice (counter
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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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Page 106 and 107: 5.1. CURVES ON SURFACES 106 Exercis
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Page 124 and 125: 5.5. RULED SURFACES 124 Definition
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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
Page 168 and 169: 7.6. THE UPPER HALF PLANE 168 so so
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Page 182 and 183: This leaves us with finding L s s.
Page 184 and 185: B.3. MONGE PATCHES 184 So we immedi
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