  This paper is divided into two parts. In the first one the von Weizs\"acker
term is introduced to the Magnetic TF theory and the resulting MTFW functional
is mathematically analyzed. In particular, it is shown that the von
Weizs\"acker term produces the Scott correction up to magnetic fields of order
$B \ll Z^2$, in accordance with a result of V. Ivrii on the quantum mechanical
ground state energy. The second part is dedicated to gradient corrections for
semiclassical theories of atoms restricted to electrons in the lowest Landau
band. We consider modifications of the Thomas-Fermi theory for strong magnetic
fields (STF), i.e. for $B \ll Z^3$. The main modification consists in replacing
the integration over the variables perpendicular to the field by an expansion
in angular momentum eigenfunctions in the lowest Landau band. This leads to a
functional (DSTF) depending on a sequence of one-dimensional densities. For a
one-dimensional Fermi gas the analogue of a Weizs\"acker correction has a
negative sign and we discuss the corresponding modification of the DSTF
functional.
