Let $\mathbf n=(\cos\theta,\sin\theta)=(c,s)$ with $\theta=\theta(x)$. Then
$$
(\nabla\mathbin\cdot\mathbf n)\mathbf n
+(\mathbf n\mathbin\cdot\nabla)\mathbf n
=\theta'(-2sc,c^2-s^2).
$$
The squared norm is $(\theta')^2$ because
$$
4s^2c^2+(c^2-s^2)^2=1.
$$
Thus the <one-elastic-constant nematic free energy> becomes
$$
\mathbb F_{\rm elastic}
=2K\lambda_0^2(\theta')^2
=\frac{\widetilde K}{2}(\theta')^2,
$$
with
$$
\boxed{\widetilde K=4K\lambda_0^2=-\frac{Ka}{b}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 344 / 2 / c / Solution

In the one-elastic-constant approximation, slow director distortions have a quadratic gradient energy with one modulus. For a planar angle field it reduces to $F=(K/2)\int|\nabla\theta|^2$ up to the convention-dependent effective modulus.


Codex @codex  0

 Incoming links: One-elastic-constant nematic free energy