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Classical differential geometry of space-curves  35

                        which can also be expressed as

                                                           x     y    z

                                                 τ = ρ  2    x      y     z       .        (2.25)

                                                         x       y      z

                        Exercise 2. If a curve is given in terms of a parameter t, show, making use of
                                                               ...
                        eqns (2.12), (2.13), and a similar equation for r ,that
                                                          ...           ...
                                                    ˙ r · (¨ r × r )  ˙ r · (¨ r × r )
                                                  2              2
                                             τ = ρ           = ρ           ,               (2.26)
                                                       ˙ s 6       (˙ r · ˙ r) 3
                        where ρ may be obtained from eqns (2.15) and (2.18).
                        Exercise 3. For the curve given by the parametric form
                                                               2
                                                                              3
                                                    3
                                         x = a(3t − t ),  y = 3at ,  z = a(3t + t ),
                                                                      2 −2
                                                            1 −1
                        where a is a constant, show that κ = τ = a  (1 + t )  .
                                                            3
                           2.3 Generalized co-ordinate approach

                        It may be instructive to derive the Serret-Frenet formulas using generalized co-ordi-
                        nates. In the process we introduce the concept of intrinsic differentiation. Given a set
                                                       3
                                                    2
                                                 1
                        of curvilinear co-ordinates x , x , x , a space-curve may be specified in terms of a
                        parameter t and the functions
                                                   i
                                                        i
                                                  x = x (t),  i = 1, 2, 3.
                          Arc-length is given in terms of the basic metric
                                                       2
                                                               i
                                                                  j
                                                     ds = g ij dx dx ,                     (2.27)
                        or in terms of the parameter t as

                                                         t     i   j
                                                             dx dx
                                                  s =     g ij      dt,                    (2.28)
                                                              dt dt
                                                       t 0
                        measured from some point on the curve where t = t 0 . Equation (2.27) can be re-
                        written as
                                                          i
                                                        dx dx j
                                                     g ij      = 1,                        (2.29)
                                                         ds ds
                        which according to eqn (1.55) may be interpreted as implying that the vector whose
                                                    i
                        contravariant components are dx /ds, i = 1, 2, 3, has magnitude unity. Of course, this
                        is just the unit tangent vector t to the curve; here we write
                                                             dx i
                                                         i
                                                        t =     .                          (2.30)
                                                             ds
                          For any vector field u we have by the Chain Rule
                                           du    ∂u dx j    i  dx j     dx j  i
                                              =         = u ,j   g i = u i,j  g
                                            ds   ∂x j  ds     ds         ds
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