870 CHAPTER 13 Vector Functions 61. (a) Show that dB/ds is perpendicular to B. (b) Show that dB/ds is perpendicular to T. (c) Deduce from parts (a) and (b) that dB/ds-T(S)N for some number 7(s) called the torsion of the curve. (The torsion measures the degree of twisting of a curve.) (d) Show that for a plane curve the torsion is r(s) = 0. 62. The following formulas, called the Frenet-Serret formulas, are of fundamental importance in differential geometry: 1. dT/ds = KN 2. dN/ds=-KT + TB 3. dB/ds-TN (Formula 1 comes from Exercise 59 and Formula 3 comes from Exercise 61.) Use the fact that N = B x T to deduce Formula 2 from Formulas 1 and 3. 63. Use the Frenet-Serret formulas to prove each of the follow- ing. (Primes denote derivatives with respect to 1. Start as in the proof of Theorem 10.) (a) r"=s"T + K(s')²N (b) r' xr"= K(s')' B (c) r" [s" - K²(s')']T + [3ks's" + K'(s')² ]N + KT(S')³B (d) T= (r' xr").r" r'xr" ³ AM 64. Show that the circular helix r(t) = (a cos t, a sin t, bt), where a and b are positive constants, has constant curvature and constant torsion. [Use the result of Exercise 63(d).] 65. Use the formula in Exercise 63(d) to find the torsion of the curve r(t) = (1, ², ³). 66. Find the curvature and torsion of the curve x = sinh 1, y = cosh f, z = 1 at the point (0, 1, 0). 67. The DNA molecule has the shape of a double helix (see Figure 3 on page 850). The radius of each helix is about 10 angstroms (1 A = 10 cm). Each helix rises about 34 A during each complete turn, and there are about 2.9 x 10¹ complete turns. Estimate the length of each helix. 68. Let's consider the problem of designing a railroad track to make a smooth transition between sections of straight track Existing track along the negative x-axis is to be joined smoothly to a track along the line y = 1 for x ≥ 1. (a) Find a polynomial P = P(x) of degree 5 such that the function F defined by F(x) = P(x) 1 13.4 Motion in Space: Velocity and Acceleration if x ≤ 0 if 0