CHAPTER 3 DIFFERENTIATION VECTORS OF ELEMENTS DIFFERENTIAL GEOMETRY OF C Differential Geometry involves a study of space curves and surfaces. If is a space curve defined by the function where s is arc length, then we have rs d r C which is a unit tangent vector to denoted by Tˆ . ds  Suppose  is an arbitrary representation of the curve C with the same r rt orientation as  , then we find r rs d r d r ds   r dt ds dt d d r the quantity r  dt  r dt r ds d r d r d r dt ds ds dt dt Example 3.8 Find the unit tangent vector along the helix  cos  sin  r a tiˆ a tˆj btkˆ a b where and are non-zero. Solution: d r dt sin cos  tiˆ  a tˆj  bkˆ    d r 2 dt ds  1 dt d r ds Thus dt dt Tˆ  Z S Definition: The rate at which changes with respect to is a Tˆ dTˆ Nˆ C measure of the curvature of C and is given by . ds d Tˆ direction of
CHAPTER 3 DIFFERENTIATION OF VECTORS ELEMENTS OF DIFFERENTIAL GEOMETRY Differential Geometry involves a study of space curves and surfaces. If C is a space curve defined by the function rs where s is arc length, then we have d r ds which is a unit tangent vector to C denoted by Tˆ . Suppose r  rt  is an arbitrary representation of the curve C with the same orientation as r  rs , then we find r  d r dt  d r ds ds dt the quantity ds dt d r dt d r ds  d r d r dt dt  r r d r dt ds Example 3.8 Find the unit tangent vector along the helix r  a cos tiˆ  a sin tˆj  btkˆ where a and b are non-zero. Solution: d r dt  sin tiˆ  a cos tˆj  bkˆ d r dt  ds dt    1 2 Thus Tˆ  d r dt ds dt Definition: The rate at which Tˆ changes with respect to S is a Z measure of the curvature of C and is given by dTˆ ds . C Nˆ direction of d Tˆ