  We formulate $K^+$-nucleus quasielastic scattering in a manner which closely
parallels standard treatments of $e^-$-nucleus quasielastic scattering. For
$K^+$ scattering, new responses involving scalar contributions appear in
addition to the Coulomb (or longitudinal) and transverse $(e,e')$ responses
which are of vector character. We compute these responses using both nuclear
matter and finite nucleus versions of the Relativistic Hartree Approximation to
Quantum Hadrodynamics including RPA correlations. Overall agreement with
measured $(e,e')$ responses and new $K^+$ quasielastic scattering data for
$^{40}$Ca at $|\qs|=500$ MeV/c is good. Strong RPA quenching is essential for
agreement with the Coulomb response. This quenching is notably less for the
$K^+$ cross section even though the new scalar contributions are even more
strongly quenched than the vector contributions. We show that this
``differential quenching'' alters sensitive cancellations in the expression for
the $K^+$ cross section so that it is reduced much less than the individual
responses. We emphasize the role of the purely relativistic distinction between
vector and scalar contributions in obtaining an accurate and consistent
description of the $(e,e')$ and $K^+$ data within the framework of our nuclear
structure model.
