= Quadratic minor null Lagrangian
{title2=$K_{k\gamma}\epsilon_{\alpha\beta\gamma}\epsilon_{ijk}u_{i,\alpha}u_{j,\beta}$}
For constant $K$, this quadratic combination of two-by-two gradient minors is a <divergence>. Indeed,
$$
K_{k\gamma}\epsilon_{\alpha\beta\gamma}\epsilon_{ijk}\overline{u_{i,\alpha}}u_{j,\beta}=\partial_\alpha\left(K_{k\gamma}\epsilon_{\alpha\beta\gamma}\epsilon_{ijk}\overline{u_i}u_{j,\beta}\right),
$$
because the term containing $u_{j,\beta\alpha}$ vanishes by antisymmetry of the <Levi-Civita symbol>. Its volume integral vanishes for homogeneous displacement boundary data. It also vanishes pointwise on <rank-one matrices>, since $\nu_\alpha\nu_\beta$ is symmetric. Consequently one can strengthen an elastic energy density by such a term without changing fixed-boundary incremental energies or <acoustic tensors>.
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