Lecture Notes for 120 - UCLA Department of Mathematics

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1.1. CURVES 8 as we then get q (✓) =R (cos (✓ + ') , sin (✓ + ')) with ✓ being the angle to the x-axis. Yet a further scaling is possible as long as we exclude the origin apple 1 p x2 + y 2 apple dx dt dy dt This time the solutions are given by ✓ ✓ ◆ ✓ ◆◆ ✓ + ' ✓ + ' q (✓) =R cos , sin R R and we have to assume that R>0. Example 1.1.2. Consider = F (x, y) =x 2 y x y 2 = c When c 6= 0the solution set consists of two hyperbolas. They’ll be separated by the y-axis when c>0 and by the x-axis when c0 and in the third quadrant when a
1.1. CURVES 9 Example 1.1.5. A far more subtle problem is Newton’s inverse square law: d 2 q dt 2 = q |q| 3 = 1 q |q| 2 |q| Newton showed that the solutions are conic sections (intersections of planes and a cone) and can be lines, circles, ellipses, parabolas, or hyperbolas. Example 1.1.6. Finally we mention a less well known ancient example. This is the conchoid (conch-like) of Nicomedes. It is given by a quartic (degree 4) equation: x 2 + y 2 (y b) 2 R 2 y 2 =0 It is not clear why this curve looks like a conch. Descriptively it consists of two curves that are given as points (x, y) whose distance along radial lines to the line y = b is R. The radial line is simply the line that passes through the origin and (x, y). Sowearemeasuringthedistancefrom(x, y) to the intersection of this radial line with the line y = b. Asthatintersectionis ✓ x x y b ◆ 2 +(y b) 2 = R 2 ⇣ x y b, b ⌘ the condition is which after multiplying both sides by y 2 easily reduces to the above equation. The two parts of the curve correspond to points that are either above or below y = b. Notethatnopointony = b solves the equation as long as b 6= 0. A simpler formula appears if we use polar coordinates. The line y = b is described as and the point (x, y) by (x, y) =(b cot ✓, b) = b (cos ✓, sin ✓) sin ✓ ✓ ◆ b (x, y) = sin ✓ ± R (cos ✓, sin ✓) This gives us a natural parametrization of these curves. Another parametrization is obtained if we intersect the curve with the lines y = tx and use t as the parameter. This corresponds to t = tan ✓ in polar coordinates. Thus we obtain the parameterized form ✓ ◆ b (x, y) = t ± R p (1,t) 1+t 2 As we have seen, what we consider the same curve might have several different parametrizations. Definition 1.1.7. Two parametrized curves q (t) , q ⇤ (t ⇤ ) are reparametrizations of each other if it is possible to write t = t (t ⇤ ) as a function of t ⇤ and t ⇤ = t ⇤ (t) such that q (t) =q ⇤ (t ⇤ (t)) and q (t (t ⇤ )) = q ⇤ (t ⇤ ) If both of the functions t (t ⇤ ) and t ⇤ (t) are differentiable then it follows from the chain rule that dt dt ⇤ dt ⇤ dt =1 In particular, these derivatives never vanish and have the same sign. We shall almost exclusively consider such reparametrizations. In fact we shall usually assume that
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 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 =
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Page 32 and 33: 2.2. THE FUNDAMENTAL EQUATIONS 32 M
Page 34 and 35: 2.2. THE FUNDAMENTAL EQUATIONS 34 (
Page 36 and 37: 2.3. LENGTH AND AREA 36 (22) (Newto
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
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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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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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