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Math Help - Vector/Tensor Proof

  1. #1
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    Vector/Tensor Proof

    I'm trying to prove what is shown in figure 1 where V is a vector and T is a tensor. However if I arbitrarily assign values to the vectors and tensors and try to prove it by performing the operation, I end up calculating that all three terms are exactly the same which obviously cannot be. Input anyone?
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    Quote Originally Posted by Jensen View Post
    I'm trying to prove what is shown in figure 1 where V is a vector and T is a tensor. However if I arbitrarily assign values to the vectors and tensors and try to prove it by performing the operation, I end up calculating that all three terms are exactly the same which obviously cannot be. Input anyone?
    I'm lost on something here. How can you calculate \nabla \cdot ( V \cdot T^T )? V \cdot T^T is a scalar. You can't take the divergence of a scalar.

    If this is supposed to be a gradient operation instead of a divergence, notice that
    \nabla ( V \cdot T^T ) = V \cdot ( \nabla T^T ) + T \cdot ( \nabla V )
    is an identity. You can easily show this by working it out component by component.
    \partial _i (V_j T_j) = V_i \partial _j T_j + T_i \partial _j V_j
    using the summation convention.

    -Dan
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    As I understand it, the dot product of a vector and a tensor is a vector so you could take the divergence of what is shown in the parenthesis.
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Jensen View Post
    As I understand it, the dot product of a vector and a tensor is a vector so you could take the divergence of what is shown in the parenthesis.
    Hmmmm.... I see what you are saying for a higher dimensional case, and I'll admit that I didn't think it through, but vectors with the correct transformation properties are tensors also and you can't do the problem when T is a vector.

    Otherwise I think you can make the same argument that I did, just in a slightly different form. Now the T_j are a set of vectors is the only difference.

    -Dan
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