CHAPTER 3 DIFFERENTIATION VECTORS OF dTˆ Then  kNˆ , where k is called the curvature of C at the specified point. The ds quality is called the radius of the curvature.  P 1k Example 3.9 Find , k and for the curve Tˆ Nˆ 3  sin 2tiˆ cos r tˆj We have d r 6 dt cos 2tiˆ sin 2tˆj   d r  36 cos 2t  36 sin 2t  6 dt 2 2 dt d r  Tˆ dt d r sin  2tiˆ  2tˆj Next d T 2 iˆ 2 ˆj sin 2t cos 2t   dt 3   dTˆ    KN ds dt dt 1 1 3 dTˆ  KNˆ  K  ds And K ds 2tiˆ 2 cos 2tˆj N   1 dTˆ 2 sin  Example 3.10 Find the curvature of the helix   cos  sin  rt a wtiˆ a wtˆj btkˆ We have d r dt  wtiˆ  awcos wtˆj  bkˆ d r dt sin2 cos2      a2w2 wt a2w2 wt b2 a2w2 b2 Tˆ 
CHAPTER 3 DIFFERENTIATION OF VECTORS Then dTˆ ds  kNˆ , where k is called the curvature of C at the specified point. The quality P  1k is called the radius of the curvature. Example 3.9 Find Tˆ , k and Nˆ for the curve r  sin 2tiˆ 3 cos tˆj We have d r dt d r dt  cos 2tiˆ  6 sin 2tˆj  36 cos 2 2t  36 sin 2 2t  6 Tˆ  d r dt d r dt  2tiˆ  sin 2tˆj Next d dt  2 sin 2t iˆ  2 cos 2t ˆj dTˆ ds  dt dt  1 3    KN And dTˆ ds  KNˆ  K  1 3 N  1 dTˆ K ds  2 sin 2tiˆ  2 cos 2tˆj Example 3.10 Find the curvature of the helix rt   a cos wtiˆ  a sin wtˆj  btkˆ We have d r dt d r dt  wtiˆ  awcos wtˆj  bkˆ  a2w2 sin2 wt  a2w2 cos2 wt  b2  a2w2  b2 Tˆ  T