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Vector triple product causing a contradiction in this proof

  1. Jul 6, 2015 #1
    1. The problem statement, all variables and given/known data

    Prove the following identity

    [tex]\nabla (\vec{F}\cdot \vec{G}) = (\vec{F}\cdot \nabla)\vec{G} + (\vec{G}\cdot \nabla)\vec{F} + \vec{F} \times (\nabla \times \vec{G}) + \vec{G}\times (\nabla \times \vec{F})[/tex]

    2. Relevant equations

    vector triple product

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

    3. The attempt at a solution

    The first thing I wanted to do was investigate what expanding according to the vector triple product would do to the original statement I am trying to prove. This happens:

    [tex]\nabla (\vec{F}\cdot \vec{G}) = (\vec{F}\cdot \nabla)\vec{G} + (\vec{G}\cdot \nabla)\vec{F} + \nabla (\vec{F}\cdot \vec{G}) - (\vec{F}\cdot \nabla)\vec{G} + \nabla (\vec{F}\cdot \vec{G}) - (\vec{G}\cdot \nabla)\vec{F} = 2\nabla (\vec{F}\cdot \vec{G}) [/tex]

    What's happening here? Is it not valid to use the vector differential operator in an expansion of the vector triple product? Why not?
     
  2. jcsd
  3. Jul 6, 2015 #2
    It may be because the vector operator does not commute in the dot product like ordinary vectors.
     
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