Least-squares projection of filament deformation velocity
= Least-squares projection of filament deformation velocity
{title2=$\int_0^L(g_t+V_1+\Omega_1x)\binom{1}{x}\,dx=0$}
At first order, the transverse drag depends on $g_t+V_1+\Omega_1x$. Its zero force and torque conditions make this residual orthogonal to $1$ and $x$. Thus $-V_1-\Omega_1x$ is the <linear least-squares projection> of the prescribed transverse deformation velocity onto the span of $1$ and $x$. This recasts a force-balance calculation as a projection.