CAF123
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Homework Statement
##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}##?
Homework 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.
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.