Thread: 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?

Originally Posted by Jensen

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 ? 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

is an identity. You can easily show this by working it out component by component.

using the summation convention.

-Dan

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.

Originally Posted by Jensen

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 are a set of vectors is the only difference.

-Dan