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curl of a cross product watch

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    struggling on how to show curl(U x V) in terms of directional derivative and divergence using suffix notation/Levi-Civita tensor.

    tried to type what I got so far in to latex but it failed and dont have the effort to take a picture and upload but all ive done so far is essentially expand the curl and cross product in to its suffix notations.

    something like
    epsilon(ijk)*delta(j) (espilon(ijk)*U(j)*V(k))
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    (Original post by Kim-Jong-Illest)
    struggling on how to show curl(U x V) in terms of directional derivative and divergence using suffix notation/Levi-Civita tensor.

    tried to type what I got so far in to latex but it failed and dont have the effort to take a picture and upload but all ive done so far is essentially expand the curl and cross product in to its suffix notations.

    something like
    epsilon(ijk)*delta(j) (espilon(ijk)*U(j)*V(k))
    The problem you've got yourself into is that you're using the same set of indices for the two different cross products. Try to attack this by breaking down the expression into simpler pieces.

     (U \times V)_{i} = \epsilon_{ijk} U_{j} V_{k}

    So you've got the ith component of the vector cross product using the first Levi-Civita symbol. Now we have

     (\nabla \times W)_m = \epsilon_{mpq} \frac{\partial W_{p}}{\partial x_{q}}

    Take W to be U \times V, substitute and use some well known identities,
 
 
 
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