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Identity for laplacian of a vector dotted with a vector

  1. Oct 6, 2009 #1
    1. The problem statement, all variables and given/known data

    I have [itex]$\int \nabla^2 \vec{u} \cdot \vec{v} dV$[/itex] where u and v are velocities integrated over a volume. I want to perform integration by parts so that the derivative orders are the same. This is the Galerkin method.

    2. Relevant equations



    3. The attempt at a solution

    I have found identities involving [itex]$\nabla \vec{u}$[/itex] and [itex]$\nabla \vec{v}$[/itex] as a tensor scalar product and I have tried to work out a product rule:
    [itex]$\nabla \cdot (\vec{v} \cdot \nabla \vec{u}) = \nabla \vec{u} : \nabla \vec{v} = \nabla^2 \vec{u} \cdot \vec{v}$[/itex].

    I am having trouble figuring out if this is correct. I know i have scalars on the right hand side. On the left hand side I have the divergence of a vector dotted with a tensor, which I think will lead to a scalar.

    Any help is most appreciated.

    Thank you,
    dakg
     
    Last edited: Oct 6, 2009
  2. jcsd
  3. Oct 6, 2009 #2
    Is it Green's First Identity that I need? Does it hold for vectors?
     
    Last edited: Oct 6, 2009
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