Let γ : [ 0 , 2 π ] → R 3 be γ ( t ) = ( 4 t , cos ⁡ ( 3 t ) , sin ⁡ ( 3 t ) ) .Justify that γ is differentiable.

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Let   γ  :  [  0  ,  2  π  ]  →            R        3   be  γ  (  t  )  =  (  4  t  ,  cos  ⁡  (  3  t  )  ,  sin  ⁡  (  3  t  )  )  .Justify that   γ is differentiable.

drumette824ed · Accepted Answer

Step 1A vector valued function   f  :            R        n    →            R        m   is differentiable if its component functions       f    i    :            R        n    →      R   are differentiable. Here,   n  =  1 and   m  =  3 The component functions are,      f    1    (  t  )  =  4  t        f    2    (  t  )  =  cos  ⁡  (  3  t  )        f    3    (  t  )  =  sin  ⁡  (  3  t  )Hopefully it is clear that each of these component functions are differentiable. Then, we have that f is differentiable and its derivative is the Jacobian (matrix of partial derivatives),  ∇  f  =      (                                        ∂            t                                f            1                                                            ∂            t                                f            2                                                            ∂            t                                f            3                                )    =      (                            4                                      −          3          sin          ⁡          (          3          t          )                                      3          cos          ⁡          (          3          t          )                      )

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