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Proving vector identities using Cartesian tensor notation

  1. Jun 6, 2010 #1
    1. The problem statement, all variables and given/known data
    1. Establish the vector identity
    [tex]

    \nabla . (B[/tex] x [tex]A) = (\nabla[/tex] x [tex]A).B - A.(\nabla[/tex] x [tex]B)
    [/tex]

    2. Calculate the partial derivative with respect to [tex]x_{k}[/tex] of the quadratic form [tex]A_{rs}x_{r}x_{s}[/tex] with the [tex]A_{rs}[/tex] all constant, i.e. calculate [tex]A_{rs}x_{r}x_{s,k}[/tex]


    2. Relevant equations



    3. The attempt at a solution
    1.
    [tex]

    \nabla . (B[/tex] x [tex]A) = \epsilon_{ijk}A_{j}B_{k,i}
    [/tex]

    Now I don't know what to do next.

    2.

    [tex]
    A_{rs}x_{r}x_{s,k} = A_{rs}\partial_{k}(x_{r}x_{s}) = A_{rs}(x_{r}\partial_k x_{s} + x_{s}\partial_k x_{r})[/tex]

    I have no idea if this is right or not.

    I'm pretty good at proving vector identities (and Cartesian tensor notation in general), but I get lost when partial derivatives/nablas are involved. Any tips would be greatly appreciated!!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 6, 2010 #2

    vela

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    Your notation is unclear and misleading. When you write [itex]\nabla\cdot(B\times A) = \epsilon_{ijk}A_j B_{k,i}[/itex], it looks like you're only differentiating Bk. Also, you got the indices wrong since you have BxA, not AxB. On the other hand, you should have [itex]\nabla\cdot(A \times B)[/itex] on the LHS anyway.

    Instead, you should write

    [tex]\nabla\cdot(A \times B) = \partial_i (\epsilon_{ijk} A_j B_k) = \epsilon_{ijk} \partial_i (A_j B_k)[/tex]

    Use the product rule to differentiate and then convert back to vector notation.


    In the second problem, use the fact that the x's are independent, so [itex]\partial_i x_j = 0[/itex] if [itex]i \ne j[/itex] and [itex]\partial_i x_j = 1[/itex] if [itex]i = j[/itex], i.e. [itex]\partial_i x_j = \delta_{ij}[/itex].
     
    Last edited: Jun 6, 2010
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