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Identity by using gradient

  1. Mar 15, 2015 #1
    1. The problem statement, all variables and given/known data
    Let us consider three scalar fields u(x), v(x), and w(x). Show that they have a relationship such that f(u, v, w) = 0 if and only if

    (∇u) × (∇v) · (∇w) = 0.

    2. Relevant equations


    3. The attempt at a solution
    I could think nothing but just writing the components of (∇u) × (∇v) · (∇w) = 0. How could i prove this?
     
    Last edited: Mar 15, 2015
  2. jcsd
  3. Mar 15, 2015 #2
    1. The problem statement, all variables and given/known data
    Let us consider three scalar fields u(x), v(x), and w(x).
    Show that they have a relationship such that f(u, v, w) = 0 if and only if
    (∇u) × (∇v) · (∇w) = 0.

    2. Relevant equations


    3. The attempt at a solution
    I can do nothing but just writing components of (∇u) × (∇v) · (∇w). How can I prove this identity?
     
  4. Mar 15, 2015 #3

    Ray Vickson

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    If ##\vec{A} = \nabla u##, ##\vec{B} = \nabla v## and ##\vec{C} = \nabla w##, what does the given condition say about the directions of the vectors ##\vec{A}, \vec{B}, \vec{C}##?
     
  5. Mar 15, 2015 #4
    Scalar triple product means det(a,b,c) or volume of a parallelepiped. Is it a key to solve this problem?
     
  6. Mar 15, 2015 #5
    When you wrote the components out, what did you get?
     
  7. Mar 15, 2015 #6
    I wrote
    %C0%B9%BF%A2.JPG
    by using levi-civita symbol
     
  8. Mar 16, 2015 #7

    Dick

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    One way is easy. If f(u,v,w)=0 then to show the triple product is zero you just have to show that the gradients of u, v and w are linearly dependent. Use the chain rule. The other way is harder, you need to have some sort of integrability theorem like Frobenius. I'm a little fuzzy on that. What have you got?
     
    Last edited: Mar 16, 2015
  9. Mar 16, 2015 #8

    mfb

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    Staff: Mentor

    Sorry, am I missing something? There is always such a function f(u,v,w) - just define it to be zero everywhere. Are we looking for a non-trivial (!) linear function f?
     
  10. Mar 16, 2015 #9

    Dick

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    Good point. I'm guessing the problem should state that f has at least one nonzero partial derivative at points where you want to show the triple product is zero. Just a guess.
     
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