  We present a microscopic theory for superconductivity in a magnetic field
based on a local approximation approach. We derive an expression for free
energy density $F$ as a function of temperature $T$ and vector potential {\bf
a}, and two basic equations of the theory: the first is an implicit solution
for energy gap parameter amplitude $|\Delta_{\bf k}|$ as a function of wave
vector {\bf k}, temperature $T$ and vector potential {\bf a}; and the second is
a London-like relation between electrical current density {\bf j} and vector
potential {\bf a}, with an ``effective superconducting electron density'' $n_s$
that is both $T$- and {\bf a}-dependent. The two equations allow determination
of spatial variations of {\bf a} and $|\Delta_{\bf k}|$ in a superconductor for
given temperature $T$, applied magnetic field ${\bf H}_a$ and sample geometry.
The theory shows the existence of a ``partly-paired state,'' in which paired
electrons (having $|\Delta_{\bf k}|>0$) and de-paired electrons (having
$|\Delta_{\bf k}|=0$) co-exist. Such a ``partly-paired state'' exists even at
T=0 when $H_a$ is above a threshold for a given sample, giving rise to a
non-vanishing Knight shift at T=0 for $H_a$ above the threshold. We expect the
theory to be valid for highly-local superconductors for all temperatures and
magnetic fields below the superconducting transition. In the low-field limit,
the theory reduces to the local-limit result of BCS. As examples, we apply the
theory to the case of a semi-infinite superconductor in an applied magnetic
field ${\bf H}_a$ parallel to the surface of the superconductor and the case of
an isolated vortex in an infinite superconductor, and determine, in each case,
spatial variations of quantities such as {\bf a} and $|\Delta_{\bf k}|$. We
also calculate...
