Lecture Notes for 120 - UCLA Department of Mathematics

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

$Math 3C Homework 3 Solutions$

B.6. CHEBYSHEV NETS 188 (1) Show that and uuu = 0 uvu = 0 = vuu vvv = r @r @v uvv = r @r @u = uuv = 0 vvu = r @r @u , u w 1w 2 = w 1w 2u v w 1w 2 = 1 r 2 w1w2v K = @ 2 r @u 2 r vuv B.6. Chebyshev Nets These correspond to a parametrization where the first fundamental form looks like: apple 1 c I = c 1 apple 1 cos ✓ = , cos ✓ 1 Real life interpretations that are generally brought up are fishnet stockings or nonstretchable cloth tailored to the contours of the body. The idea is to have a material where the fibers are not changed in length or stretched, but are allowed to change their mutual angles. Note that such parametrizations are characterized as having unit speed parameter curves. Exercises. (1) Show that any surface locally admits Chebyshev nets. Hint: Fix a point p = q (u 0 ,v 0 ) for a given parametrization and define new parameters ˆ u1 p s (u, v) = guu (x, v)dx t (u, v) = u ˆ 0 v1 v 0 p gvv (u, y)dy To show that these give a new parametrization near p show that and @s @v ! 0 as (u, v) ! (u 0,v 0 ) @t @u ! 0 as (u, v) ! (u 0,v 0 )
B.7. ISOTHERMAL COORDINATES 189 (2) Show that Chebyshev nets satisfy the following properties @ 2 q @u@v ? T pM uvw = uuu = vvv =0, uuv = @✓ @v sin ✓ vvu = @✓ @u sin ✓ @ 2 ✓ @u@v = K sin ✓ (3) Show that the geodesic curvature apple g of the u-coordinate curves in a Chebyshev net satisfy apple g = (4) (Hazzidakis) Show that p det [I] = sin ✓, and that integrating the Gauss curvature over a coordinate rectangle yields: ˆ K sin ✓dudv =2⇡ ↵ 1 ↵ 2 ↵ 3 ↵ 4 [a,b]⇥[c,d] @✓ @u where the angles ↵ i are the interior angles. B.7. Isothermal Coordinates These are also more generally known as conformally flat coordinates and have afirstfundamentalformthatlookslike: apple 2 0 I= 2 0 The proof that these always exist is called the local uniformization theorem. It is not asimpleresult,buttheimportanceofthesetypesofcoordinatesinthedevelopment of both classical and modern surface theory cannot be understated. There is also a global result which we will mention at a later point. Gauss was the first to work with such coordinates, and Riemann also heavily depended on their use. They have the properties that uuu = @ ln @u uvu = @ ln @v = vuu vvv = @ ln @v uvv = @ ln @u = vuv uuv = @ ln @v vvu = @ ln @u , w 3 w 1w 2 = 1 2 w1w2w3,
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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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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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Page 164 and 165: 7.6. THE UPPER HALF PLANE 164 Proof
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Page 174 and 175: A.2. GEOMETRY 174 the characteristi
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Page 184 and 185: B.3. MONGE PATCHES 184 So we immedi
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Lecture Notes for 120 - UCLA Department of Mathematics

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