= Propulsive force of a periodic planar filament
{title2=$F_x=\frac{\xi_\perp-\xi_\parallel}{T}\int_0^T\int_0^\lambda y_x y_t\,dx\,dt$}
For a small-amplitude planar filament without mean axial drift, <resistive-force theory> gives the leading time-averaged force on the filament as $F_x=(\xi_\perp-\xi_\parallel)T^{-1}\int_0^T\int_0^\lambda y_x y_t\,dx\,dt$. The <parallel and perpendicular drag coefficients of a slender filament> must differ to obtain propulsion at this order. A right-going wave gives negative axial force; a left-going wave gives positive axial force.
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