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In three dimensions, the general rank-two isotropic tensor is $a{\delta }_{ij}$ and the general rank-three tensor invariant under proper rotations is $b{ϵ}_{ijk}$. Here $\delta$ is the Kronecker delta and $ϵ$ the Levi-Civita symbol. The latter is orientation-sensitive: an ordinary third-rank tensor invariant under all orthogonal transformations is zero instead. This distinction does not change the stress conclusion.

For an isotropic material, the constitutive tensor must consequently have $A_{ijk}=b{ϵ}_{ijk}$ in the proper-rotation convention. Hence ${\sigma }_{ij}=b{ϵ}_{ijk}{B}_k=-{\sigma }_{ji}$. Mechanical stress in the model is also symmetric, so ${\sigma }_{ij}=-{\sigma }_{ij}$ and every component vanishes. Thus nonzero stress linear in the field requires anisotropic material coefficients. This is the isotropic linear magnetostriction vanishes restriction, not a claim that all higher-order magnetostriction is impossible in isotropic materials.