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Laplacian of dyad

  1. Oct 3, 2016 #1
    1. The problem statement, all variables and given/known data

    Given the dyad formed by two arbitrary position vector fields, u and v, use indicial notation in Cartesian coordinates to prove:

    $$\nabla^2 ({\vec u \vec v}) = \vec v \nabla^2 {\vec u} + \vec u \nabla^2 {\vec v} + 2\nabla {\vec u} \cdot {(\nabla \vec v)}^T
    $$


    2. Relevant equations

    Per my professor's notes, the Laplacian of a dyad (also a tensor) is given as:
    $$
    \nabla^2 {\mathbf {S}} = \nabla \cdot {S_{ij,k} \mathbf{e_{i}e_{j}e_{k}}} = S_{ij,kk} \mathbf{e_{i}e_{j}}
    $$


    3. The attempt at a solution

    $$
    \nabla^2 {\mathbf {uv}} = (u_{i}v_{j})_{,kk} = u_{i,kk}v_{j} + u_{i}v_{j,kk} \\
    u_{i,kk}v_{j} + u_{i}v_{j,kk} = \vec v \nabla^2 {\vec u} + \vec u \nabla^2 {\vec v}
    $$


    I don't know where the following terms come from:
    $$
    2\nabla {\vec u} \cdot {(\nabla \vec v)}^T
    $$

    Does anyone have any suggestions? I feel that I am missing a step or something.
     
  2. jcsd
  3. Oct 3, 2016 #2

    fresh_42

    Staff: Mentor

    I'm not good with coordinates, so I leave this up to you.
    But ##\nabla (\mathbf{u}\mathbf{v}) = \mathbf{u}(\nabla \mathbf{v}) + (\nabla \mathbf{u})\mathbf{v}## and the next differentiation gives you the third term (twice).
     
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