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The bulk potential gives fluctuations in $\lambda$ a nonzero restoring force, whereas slowly rotating the nematic director costs only gradients. At lengths much larger than the amplitude correlation length, it is therefore the leading gradient expansion approximation to set $\lambda ={\lambda }_0$ while retaining $\mathbf{n}(\mathbf{r})$.

DifferentiatinggivesSubstitution into the elastic term yields

The anchoring conditions are $\theta (0)=0$ modulo $\pi$ and $\theta (L)=\pi /2$ modulo $\pi$, since the nematic director identifies angles differing by $\pi$. Their difference can therefore be $m\pi /2$ for any odd integer $m$. The Euler-Lagrange equation ofis ${\theta }^{\prime \prime }=0$, so every stationary solution has the formIts free energy per unit length in the $y$ direction isThe smallest possible $m^2$ is one, giving exactly the two degenerate global minima $m=1$ and $m=-1$.

For a planar nematic director $\mathbf{n}=(\cos \chi ,\sin \chi )$ and constant amplitude ${\lambda }_0$, the nematic order parameter isDifferentiation gives $\nabla \cdot Q={\lambda }_0(-\sin 2\chi {\chi }_x+\cos 2\chi {\chi }_y,\cos 2\chi {\chi }_x+\sin 2\chi {\chi }_y)$. 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 ${\sum }_{ijk}({\partial }_k{Q}_{ij}{)}^2=2{\lambda }_0^2|\nabla \chi {|}^2$ in this normalization; the two local densities must not be confused.

In simple shear flow, a persistent polar order parameter with $|\xi |<1$ has no fixed angle and rotates continuously. Integrating $\dot{\theta }=g(\xi \cos 2\theta -1)/2$ through $2\pi$ gives the displayed period. An unoriented nematic director repeats after half that time. At $\xi =0$, the magnitude relaxes independently while the angular speed is $-g/2$.

A uniaxial nematic order parameter has one distinguished nematic director and equal eigenvalues in its perpendicular plane. In the displayed normalization the eigenvalues are $\lambda ,-\lambda /2,-\lambda /2$, giving $\mathrm{Tr}{Q}^2=3{\lambda }^2/2$ and $\mathrm{Tr}{Q}^3=3{\lambda }^3/4$. Positive and negative $\lambda$ describe prolate and oblate orientational anisotropy, respectively; reversing the nematic director leaves the tensor unchanged.