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Dot Product Involving Path of a Curve

  1. Feb 13, 2013 #1
    1. The problem statement, all variables and given/known data
    Let ##\gamma(t)## be a path describing a level curve of ##f : \mathbb{R}^2 \to \mathbb{R}##. Show, for all ##t##, that ##( \nabla f ) (\gamma(t))## is orthogonal to ##\gamma ' (t)##


    2. Relevant equations
    ##\gamma(t) = ((x(t), y(t))##
    ##\gamma ' (t) = F(\gamma(t))##
    ##F = \nabla f = \left( \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}\right)## [this is called a gradient field]

    None of these were given for the question at hand but I think they might be useful.


    3. The attempt at a solution
    If ##x(t)## and ##y(t)## are the parameters for ##\gamma(t)##, then ##( \nabla f ) (\gamma(t)) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)##

    But both ##\nabla f (\gamma(t))## and ##\gamma ' (t)## are equal to ##F(\gamma(t))##, so the dot product is

    ##F(\gamma(t)) \cdot F(\gamma(t)) = ||F(\gamma(t))||^2##

    At this point, I'm stuck. I don't think ##||F(\gamma(t))||^2## would be 0 for any ##t## let alone for any function ##f : \mathbb{R}^2 \to \mathbb{R}##. Did I make a bad assumption/simplification somewhere?
     
  2. jcsd
  3. Feb 13, 2013 #2

    HallsofIvy

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    Note that [itex]f(\lambda(t))[/itex] is a constant for all t and use the chain rule.
     
  4. Feb 13, 2013 #3
    EDIT: I see now, thank you for the help!
     
    Last edited: Feb 13, 2013
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