776 CHAPTER 14 CALCULUS OF VECTOR-VALUED FUNCTIONS 46. Method for Computing N Let v(t) = Ir'(t)II. Show that v(t) r" (t) - v'(t) r (t) N (1 ) = 12 Iv(t)1" (t) - v'(t)r(t)ll Hint: N is the unit vector in the direction T'(t). Differentiate T(t) = r(t)/v(t) to show that v(f )r"(t) - v'(t)r'(t) is a positive multiple of T' (t). In Exercises 47-52, use Eq. (12) to find N at the point indicated. 47. (12, 13 ), 1 = 1 48. (t - sint, 1 - cost), t = it Su ed besh 50. (1-1, t, 12 ), 1 = -1 FIG 49. (+2 / 2, 13 1 3, 1 ), 1 = 1 the 51. (t, et , t), t= 0 52. (cosh t, sinh t, (2), t = 0 70 A 53. Let r(1) = (1, 3+3/2, 12 ). of (a) Find T, N, and B at the point corresponding to t = 1. x (b) Find the equation of the osculating plane at the point corresponding to by t = 1. 54. Let r(t) = (cost, sint, In(cost)). (a) Find T, N, and B at (1, 0, 0). (b) Find the equation of the osculating plane at (1, 0, 0). 55. Let r(t) = (t, 1 - t, t2). (a) Find the general formulas for T and N as functions of t. (b) Find the general formula for B as a function of t. (c) What can you conclude about the osculating planes of the curve based on your answer to b? 56. (a) What does it mean for a space curve to have a constant unit tan- gent vector T? (b) What does it mean for a space curve to have a constant normal vector N? (c) What does it mean for a space curve to have a constant binormal vector B? 57. Let f(x) = x2. Show that the center of the osculating circle at (xo, x 6) is given by (-4x8, 2 + 3x8) 58. Use Eq. (10) to find the center of curvature of r(t) = (12, 13) att = 1. In Exercises 59-68, find an equation of the osculating circle at the point indicated or indicate that none exists. 59. y = x2, x =1 60. y = x2, x=2 61. y = sinx, x = 62. y = sin x, x = n 63. y = et, x=0 064. y = Inx, x= 1 65. r(t) = (cost, sint), t = 66. r(t) = (12, 1 - 212), t = 2 67. r(t) = (1 - sint, 1 -2cost), t=x (0) is 1.10) = 12 68. r(t) = (cosh t, sinh t), t = 0 69. Figure 19 shows the graph of the half-ellipse y = +v2rx - px2, where r and p are positive constants. Show that the radius of curvature at the origin is equal to r. Hint: One way of proceeding is to write the ellipse in the form of Exercise 25 and apply Eq. (11).CUBIYOMUTOBUSAU 907. BY 10 SECTION 14.4 Curvature 777 fast 79. Show that both r'(t) and r"(t) lie in the osculating plane for a vector zing) function r(1). Hint: Differentiate r(t) = v(t)T(1). 80. Show that f ( x ) y (s ) = "(to ) + -N+ ((sinks)T - (cosics) N) is an arc length parametrization of the osculating circle at r(to). 81. Two vector-valued functions ri (s) and r2(s) are said to agree to order 2 at so if F1(so) = 12(50), 1(so) = 12(50), ri(so) = 12(50) it a Let r(s) be an arc length parametrization of a curve C, and let P be the terminal point of r(0). Let y (s) be the arc length parametrization of the osculating circle given in Exercise 80. Show that r(s) and y (s) agree to order 2 at s = 0 (in fact, the osculating circle is the unique circle that ap- proximates C to order 2 at P). in 82. Letr(t) = (x(t), y(t), z(t)) be a path with curvature k (t) and define the scaled path ri(t) = (xx(t), ay(t), Az(t)), where 1 0 is a constant. Prove that curvature varies inversely with the scale factor. That is, prove that the curvature Ki(t) of ri(t) is ki(t) = =k(t). This explains why the curva- al ture of a circle of radius R is proportional to 1/R (in fact, it is equal to 1/R). Hint: Use Eq. (3). 86. Follow steps (a)-(c) to prove that there is a number t called the torsion such that dB ds = -IN 15 dB IN (a) Show that TX ds ds and conclude that dB/ds is orthogonal to T. (b) Differentiate B . B = 1 with respect to s to show that dB/ds is orthog- onal to B. (c) Conclude that dB/ds is a multiple of N. 87. Show that if C is contained in a plane P, then B is a unit vector normal to P. Conclude that t = 0 for a plane curve. 88. Torsion means twisting. Is this an appropriate name for t? Ex- plain by interpreting t geometrically. 89. Use the identity ( max (b x c) = (a . c)b - (a . b)c to prove N X B = T, B X T = N 16 90. Follow steps (a)-(b) to prove dN ds = -KT + tB 17 (a) Show that dN/ds is orthogonal to N. Conclude that dN/ds lies in the plane spanned by T and B, and hence, dN/ds = aT + bB for some scalars a, b. (b) Use N . T = 0 to show that T . dN dT and compute a. Com- ds ds pute b similarly. Equations (15) and (17) together with dT/dt = KN are called the Frenet formulas. 91. Show that r' x r" is a multiple of B. Conclude that r x r B = 18 [rxrl