CHAPTER 3 DIFFERENTIATION VECTORS OF A set of relations involving derivatives of the fundamental vectors and is Tˆ Nˆ Bˆ known collectively as the Frenet-Serret formulae given by dTˆ dNˆ dBˆ ,  kNˆ Bˆ ,  KTˆ ds ds ds Or in matrix form as K  0 dB ds and they obey the cyclic vector multiplication in an anticlockwise sense: Tˆ Nˆ Bˆ So that ; and Tˆ  Nˆ  Bˆ Nˆ  Bˆ  Tˆ Bˆ  Tˆ  Nˆ Clockwise move gives the negatives. Example 3.11 t  Prove that the curvature of the space curve is given numerically by  r r , where dots denote differentiation with respect to  K   r r 3  r We have d r r ds    dt ds dt   r Tˆ r d  d r 2 r  dt   T r T 2 d   2  r KNˆ r dTˆ dt  r Tˆ d  ds   r dt 2 2 3 3           r r T dt
CHAPTER 3 DIFFERENTIATION OF VECTORS A set of relations involving derivatives of the fundamental vectors Tˆ Nˆ and Bˆ is known collectively as the Frenet-Serret formulae given by dTˆ ds  kNˆ , dNˆ ds Bˆ KTˆ , dBˆ ds  Or in matrix form as K  0 dB ds and they obey the cyclic vector multiplication in an anticlockwise sense: Tˆ Nˆ Bˆ So that Tˆ  Nˆ  Bˆ ; Nˆ  Bˆ  Tˆ and Bˆ  Tˆ  Nˆ Clockwise move gives the negatives. Example 3.11 Prove that the curvature of the space curve r  r t  is given numerically by K  We have   r r  3 r , where dots denote differentiation with respect to  r  d r dt  d r ds ds dt   Tˆ r  r  d 2 r dt 2  T r  T  d dt  r  dTˆ ds 2 r  Tˆ d dt   r  KNˆ r 2      2    3  3 r r  T dt