Writing 3rd Order Tensor Symmetric Part in Tensor Form

[mikeeey]

Can some one write for me the Symmetric part of a third order tensor (as a tensor form)

Thanks .

 

[HallsofIvy]

What do you mean by "symmetric" for a third order tensor? A second order tensor would be represented (in a given coordinate system) by a 3 by 3 matrix (or 4 by 4 if you are counting time) and it would be symmetric if and only if A_{ij}= A_{ji}. But a third order tensor would be represented by a "3 by 3 by 3" array, A_{ijk}. And then we can have several different kinds of "symmetry":
A_{ijk}= A_{jik}, or A_{ijk}= A_{ikj}, or A_{ijk}= A_{kji}. You could even have a kind of symmetry by "rotating" the indices: A_{ijk}= A_{kij}= A_{jki}.

 

[mikeeey]

Ok i will explain .
T_{[abc]} = \frac{1}{6} \big( T_{abc} -T_{acb} + T_{bca} -T_{bac} + T_{cab} -T_{cba} \big)
this is the anti-symmetric part of a third order tensor, i want to write me the symmetric part of a third order tensor

 

[mikeeey]

T_{[abc]} = \frac{1}{6} \big( T_{abc} -T_{acb} + T_{bca} -T_{bac} + T_{cab} -T_{cba} \big)

 

[lpetrich]

The symmetric part is like the antisymmetric part, but with all signs +. Its symmetry should be easy to verify.

 

[mikeeey]

Thanks you very much . Lperrich.