What is the dot product of tensors?

[sugarmolecule]

Hello,

I was trying to follow a proof that uses the dot
product of two rank 2 tensors, as in A dot B.

How is this dot product calculated?

A is 3x3, Aij, and B is 3x3, Bij, each a rank 2 tensor.

Any help is greatly appreciated.

Thanks!

sugarmolecule

 

[Fredrik]

I've never heard of a dot product of tensors. Can you give us more details? Tip: If this is from a book, check if it's available at books.google.com. You might even be able to show us the specific page where you found this.

 

[sugarmolecule]

Hi,

I found this reference online that lists a potential intepretation:

www.math.mtu.edu/~feigl/courses/CFD-script/tensor-review.pdf

It lists the dot product of two rank-2 tensors U, V in 3-space as:

U_ikV_kj

Does that look right?

Thanks,

sugarmolecule

 

[maze]

nevermind i was thinking of something else.

 

[Fredrik]

Hi,

I found this reference online that lists a potential intepretation:

www.math.mtu.edu/~feigl/courses/CFD-script/tensor-review.pdf

It lists the dot product of two rank-2 tensors U, V in 3-space as:

U_ikV_kj

Does that look right?

Thanks,

sugarmolecule

I suspected that. I didn't know that anyone uses term "dot product" about rank 2 tensors, but if they do, it's logical that they mean precisely that. I don't see a reason to call it a dot product though. To me, that's just the definition of matrix multiplication, and if we insist on thinking of U and V as tensors, then the operation would usually be described as a ''contraction" of two indices of the rank 4 tensor that you get when you take what your text calls the "dyadic product" of U and V.

 

[dakg]

There is a single dot tensor product and a double dot scalar product of two tensors.

It is explained here:
http://www.math.mtu.edu/~feigl/courses/CFD-script/tensor-review.pdf
(sorry same link as above)

I find it used in computational fluid dynamics problems.