- #1

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**1. Problem statement:**

Assume that u is a vector and A is a 2nd-order tensor. Derive a transformation rule for a 3rd order tensor Z

_{ijk}such that the relation u

_{i}= Z

_{ijk}A

_{jk}remains valid after a coordinate rotation.

## Homework Equations

:[/B]

Transformation rule for 3rd order tensors: Z'

_{ijk}= C

_{il}C

_{jm}C

_{kn}Z

_{lmn}. Transformation rule of 2nd order tensors: A'

_{jk}= C

_{jm}C

_{kn}A

_{mn}. Transformation rule for 1st order tensors: u'

_{i}= C

_{il}u

_{l}.

**3. My attempt:**

To begin with, I am confused as to the wording of this question. I assume that it means: come up with an expression for Z'

_{ijk}such that the relation u'

_{i}= Z'

_{ijk}A'

_{jk}holds, but if I am wrong, I would appreciate an explanation of what we are trying to do! If I am correct, then I don't see why the normal transformation rule for third order tensors does not work here. I have:

Z'

_{ijk}A'

_{jk}= C

_{il}C

_{jm}C

_{kn}Z

_{lmn}C

_{jm}C

_{kn}A

_{mn}= C

_{il}Z

_{lmn}A

_{mn}= u'

_{i}

I think I've done something very wrong here, but I am unfamiliar with tensors and I don't know how to go about fixing it. Help would be much appreciated, thank you!