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Gradient operation proof

  1. Nov 25, 2012 #1
    1. The problem statement, all variables and given/known data
    Show that the operation of taking the gradient of a function has the given property. Assume u and v are differentiable functions of x and y and that a and b are constants.

    Operation: (∇(u))n = n*un-1*∇u

    2. Relevant equations

    The gradient vector of f is <∂f/∂x,∂f/∂y>, where f is a function of x and y (in other words f(x,y)).

    3. The attempt at a solution

    I tried proof by induction, but I have a lot of gaps.

    Base, n=1: ∇u = n*u0∇u. But what is u0 if u is a function?

    Assume n = k: ∇uk = n*uk-1∇u

    For k=k+1, ∇uk+1 = n*uk∇u.

    Isn't proof by induction for sums though? I can't seem to identify the sum here.


    Any help would be greatly appreciated!
     
  2. jcsd
  3. Nov 25, 2012 #2

    Dick

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    You have got to mean ## \nabla (u^n)=n u^{n-1} \nabla u ##, ## (\nabla(u))^n ## doesn't mean anything. Just use the chain rule for partial derivatives.
     
  4. Dec 1, 2012 #3
    Oh I see. Thanks!
     
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