  We derive a closed form expression for the quantum corrections to the kinetic
energy density (KED) in the Thomas-Fermi (TF) limit of a linear potential model
system in three dimensions (the Airy gas). The universality of the expression
is tested numerically in a number of three dimensional model systems: (i)
jellium surfaces, (ii) hydrogen-like potentials, (iii) systems confined by an
harmonic potential in one and (iv) all three dimensions, and (v) a system with
a cosine potential (the Mathieu gas). Our results confirm that the usual
gradient expansion of extended Thomas-Fermi theory (ETF) does not describe the
quantum oscillations for systems that incorporate surface regions where the
electron density drops off to zero. We find that the correction derived from
the Airy gas is universally applicable to relevant spatial regions of systems
of type (i), (ii), and (iv), but somewhat surprisingly not (iii). We discuss
possible implications of our findings to the development of functionals for the
kinetic energy density.
