61 870CHAPTER 13 Vector Functions MOTO261. (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)Nfor 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 7 (s) = 0.smala lentorups barl62. The following formulas, called the Frenet-Serret formulas,are of fundamental importance in differential geometry:1. dT/ds = KN2. dN/ds = -KT + TB3. dB/ds=-TN04.12(Formula 1 comes from Exercise 59 and Formula 3 comesfrom Exercise 61.) Use the fact that N = B X T todeduce 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 t. Start as inthe proof of Theorem 10.)(a) r" = s'T + K(s')²N(d)(b) r' x r" = K (s')³ B(c) r" = [s" - K²(s')³]T + [3ks's" + K'(s')² ]N + KT(S"(r' xr") r"

61. (a) Show that dB/ds is perpendicular to B. di ba! (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 7 (s) = 0. T -

61. (a) Show that dB/ds is perpendicular to B. di ba! (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 7 (s) = 0. T -

Operations Research : Applications and Algorithms

4th Edition

ISBN:9780534380588

Author:Wayne L. Winston

Publisher:Wayne L. Winston

Chapter11: Nonlinear Programming

Section11.7: The Method Of Steepest Ascent

Problem 5P

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