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Vector cross product identity proof

  1. Feb 2, 2009 #1
    1. The problem statement, all variables and given/known data


    [tex] \bigtriangledown\times\\(v\times w)= v(\bigtriangledown\cdot w) - w(\bigtriangledown\cdot v)+ (v\cdot\bigtriangledown)w - (w\cdot\bigtriangledown) v [/tex]


    I've tried expanding left side and get
    [v1(dw2/dy+dw3/dz)-w1(dv2/dy+dv3/dz)]i +
    [v2(dw3/dz+dw1/dx)-w2(dv3/dz+dv1/dx)]j +
    [v3(dw1/dx+dw2/dy)-w3(dv2/dy+dv1/dx)]k

    is this right way to start it? another way to attack it?
     
  2. jcsd
  3. Feb 2, 2009 #2
    You can try the following: First note that [tex] \nabla [/tex] is a differential operator, so its action on the product is the sum of actions on each factor

    [tex] \bigtriangledown\times\\(v\times w)= \bigtriangledown\times\\(V\times w) + \bigtriangledown\times\\(v\times W) [/tex]

    (I used capital letters to denote the factor on which [tex] \nabla [/tex] acts in each term. The other factor can be treated as a constant). Then use the vector identity

    [tex] a \times( b \times c) = b(a \cdot c) - c (a \cdot b) [/tex]
     
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