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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 -

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
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870 CHAPTER 13 Vector Functions MOTO2 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 7 (s) = 0. smala lentor ups barl 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 04.12 (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 t. Start as in the 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"
Transcribed Image Text:870 CHAPTER 13 Vector Functions MOTO2 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 7 (s) = 0. smala lentor ups barl 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 04.12 (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 t. Start as in the 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"
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