  We derive the gradient expansion for the exchange energy of a spin-polarized
electron gas by perturbing the uniformly spin polarized state and thus inducing
a small non-collinearity that is slowly varying in space. We show that the
exchange-energy contribution due to the induced longitudinal gradient of the
spin polarization to the exchange energy differs from the contribution due to
the transverse gradient. The difference is present at any non-vanishing spin
polarization and becomes larger with increasing spin polarization. We argue
that improved generalized gradient approximations of Spin-Density-Functional
Theory must account for the difference between the longitudinal and transverse
spin stiffness.
