 First-order free swimming of a planar filament

For $y=ϵg(x,t)$ on $0\le x\le L$, uniform local drag and force/torque balance give $U_1=0$, $V_1=-4⟨g_t{⟩}_x+6⟨x{g}_t{⟩}_x/L$, and ${\Omega }_1=6⟨g_t{⟩}_x/L-12⟨x{g}_t{⟩}_x/L^2$. The tangent can be replaced by the undeformed direction at this order because the velocity is already first order.