$w=4$ and $L=6.25$ ($k_F=1$ and $k_B=0.4$): in-plane piece $a(x)$ (heavy solid curve); out-of-plane piece $b(x)$ (dashed curve); adiabatic prediction (light solid curve); second term in Eq. (\[lt\]) scaled to match the maximum of $b(x)$ (light solid curve). []{data-label="fig:shortwalltorque"}](fig6.eps){width="8cm"}

${\bf N}_{\rm st}(x)$ differs from ${\bf N}_{\rm ad}(x)$ because gradients in the gradient field induce single electron spin moments to precess around ${\bf B}_{\rm eff}(x)$ rather to align perfectly with it. Fig. \[fig:shortwalltorque\] shows $a(x)$ and $b(x)$ as calculated for a typical short domain wall. The associated torques lie in the plane of the magnetization and perpendicular to that plane, respectively. These non-adiabatic contributions to the torque are both oscillatory functions of position that do not go immediately to zero when the the magnetization becomes uniform. In other words, $a(x)$ and $b(x)$ are generically non-local functions of the magnetization ${\bf
M}(x)$. The positive-valued function that falls to zero at the edges of the domain wall (light solid curve in Fig. \[fig:shortwalltorque\]) is the second function in Eq. (\[lt\]) with $c_2$ chosen to match $b(x)$ at their common maximum. Evidently, the proposed torque function Eq. (\[lt\]) gives at best a qualitative account of the out-of-plane non-adiabatic torque.

A convenient measure of the degree 