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Grad of composite function

  1. Aug 18, 2009 #1
    1. The problem statement, all variables and given/known data

    http://img21.imageshack.us/img21/8175/46521897.jpg [Broken]

    3. The attempt at a solution

    I think I have a starting point, but I'm not 100% sure
    Basically I thought of just computing grad(f(α(t)) · dα/dt and showing its equal to zero.

    Am I on the right track, or shall I try another approach?
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Aug 19, 2009 #2

    lanedance

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    Homework Helper

    hey tnp, i think you might want to start by taking the gradient of both sides of the original equation: grad(f(α(t)) = grad(c)

    as the prove is an "if & only if", you might have to think about whether this proves both directions of the theorem, ie. "if" and "only if"
     
  4. Aug 19, 2009 #3
    \cdotok, say I were to take grad of lhs, I need to apply chain rule since f(α(t))

    so..

    df/dt = (df/dα)(dα/dt)

    I recognise (dα/dt) as [tex]\dot{\alpha}[/tex], which leads me to

    df/dt = (df/dα)([tex]\dot{\alpha}[/tex])

    Also, since [tex]\alpha[/tex] has components [tex]\alpha[/tex]1, [tex]\alpha[/tex]2, [tex]\alpha[/tex]3, ......, [tex]\alpha[/tex]n+1

    df/dα = ([tex]\partial[/tex]f/d[tex]\alpha[/tex]1, [tex]\partial[/tex]f/d[tex]\alpha[/tex]2, ....,[tex]\partial[/tex]f/d[tex]\alpha[/tex]n+1) which I recognise is [tex]\nabla[/tex]f(α(t)),

    this df/dt = [tex]\nabla[/tex]f(α(t)) [tex]\cdot[/tex] [tex]\dot{\alpha}[/tex] which is what I wanted.

    I hope i'm correct up to here and it isn't too messy to show with the latex...o:)

    But as you said before, the question states, if and only if, which means I have to show both ways. Puzzled as to how to do the reverse way...
     
  5. Aug 20, 2009 #4
    can anybody help please? :P
     
  6. Aug 20, 2009 #5
    Could you possibly provide me the name of the textbook?

    Regards.
     
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