= Planar nematic divergence-elasticity identity
{title2=$|\nabla\cdot Q|^2=\lambda_0^2|\nabla\chi|^2$}
For a planar <nematic director> $\mathbf n=(\cos\chi,\sin\chi)$ and constant amplitude $\lambda_0$, the <nematic order parameter> is
$$
Q=\frac{\lambda_0}2\begin{pmatrix}\cos2\chi&\sin2\chi\\\sin2\chi&-\cos2\chi\end{pmatrix}.
$$
Differentiation gives $\nabla\cdot Q=\lambda_0(-\sin2\chi\,\chi_x+\cos2\chi\,\chi_y,\cos2\chi\,\chi_x+\sin2\chi\,\chi_y)$. The coefficient matrix is orthogonal, proving the identity. Hence $K|\nabla\cdot Q|^2/2$ is the angle-gradient energy with effective modulus $K\lambda_0^2$. The full tensor-gradient norm instead obeys $\sum_{ijk}(\partial_kQ_{ij})^2=2\lambda_0^2|\nabla\chi|^2$ in this normalization; the two local densities must not be confused.
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