# Homework Help: Transformation Properties of a tensor

1. Feb 16, 2014

### CAF123

1. The problem statement, all variables and given/known data
$D_{ijk}$ is an array with $3^3$ elements, which is not known to represent a tensor. If for every symmetric tensor represented by $a_{jk}$ $$b_i = D_{ijk}a_{jk},$$ represents a vector, what can be said about the transformation properties under rotations of the coordinate axes of $D_{ijk}$ and $D_{ijk} + D_{ikj}$?

2. Relevant equations
Transformation between vector components in frame S' and components in frame S, where the rotation matrix $L$ takes $S \rightarrow S'$ is $b_i' = l_{ij}b_j$

Quotient Theorem to identify tensors.

3. The attempt at a solution
$D_{ijk}$ may be represented as a matrix of size 3 x 3. Can I interpret the coordinate axes to mean, for example, the standard Cartesian axes?

$$b_i' = D_{ijk}' a_{jk}' = l_{ij}b_j = l_{ij}D_{jlm}a_{lm} = l_{ij} D_{jlm}a_{ml}\,\,\,\,(1)$$ using that $a_{lm}$ is a symmetric tensor.

Since $a_{jk}$ is a second rank tensor, it obeys the transformation law; $a_{jk}' = l_{j\alpha}l_{k \beta} a_{\alpha \beta}$. Since the rotation matrix $L$ satisfies $LL^{T} = I$, we have that $l_{\beta k} l_{\alpha j}a_{jk}' = a_{\alpha \beta}$ so then subbing into (1) gives$$l_{ij}D_{jlm} a_{ml} = l_{ij}D_{jlm} l_{lk}l_{mj}a_{jk}'$$
which is equal to $$l_{ij} (l_{jm})^{T} D_{jlm} l_{lk} a_{jk}' = D_{ijl}l_{lk} a_{jk}' = b_i'$$ using $LL^T = I$. I am not really sure if this result helps or not. I think the aim of the question is to show using the transformation properties of a second rank symmetric tensor and a vector that $D_{ijk}$ is also a tensor, but I am not sure.

2. Feb 22, 2014

### CAF123

Can anyone give me a hint on how to progress?
Thanks.