= Particle-conserving displacement of a Fermi circle
{title2=$\nu(\theta)=p_0\cos\theta+\sqrt{p_F^2-p_0^2\sin^2\theta}-p_F$}
Translating a <Fermi circle> of radius $p_F$ by $p_0\hat{\mathbf x}$ preserves its area. For $p_0<p_F$, its radial displacement from the original centre is the displayed expression. Its <Taylor expansion> is $\nu=p_0\cos\theta-p_0^2\sin^2\theta/(2p_F)+O(p_0^4/p_F^3)$. The second-order term cancels the apparent area increase from the first-order shape: $\int_0^{2\pi}(p_F\nu+\nu^2/2)\,d\theta=0$. This distinction matters when converting a grand-potential variation into a fixed-<particle number> <energy> change.
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