  We formulate the problem of the formation of magnetically supercritical cores
in magnetically subcritical parent molecular clouds, and the subsequent
collapse of the cores to high densities, past the detachment of ions from
magnetic field lines and into the opaque regime. We employ the six-fluid MHD
equations, accounting for the effects of grains (negative, positive and
neutral) including their inelastic collisions with other species. We do not
assume that the magnetic flux is frozen in any of the charged species. We
derive a generalized Ohm's law that explicitly distinguishes between flux
advection (and the associated process of ambipolar diffusion) and Ohmic
dissipation, in order to assess the contribution of each mechanism to the
increase of the mass-to-flux ratio of the central parts of a collapsing core
and possibly to the resolution of the magnetic flux problem of star formation.
We show how our formulation is related to and can be transformed into the
traditional, directional formulation of the generalized Ohm's law, and we
derive formulae for the perpendicular, parallel and Hall conductivities
entering the latter, which include, for the first time, the effect of inelastic
collisions between grains. In addition, we present a general (valid in any
geometry) solution for the velocities of charged species as functions of the
velocity of the neutrals and of the effective flux velocity (which can in turn
be calculated from the dynamics of the system and Faraday's law). The last two
sets of formulae can be adapted for use in any general non-ideal MHD code to
study phenomena beyond star formation in magnetic clouds. The results,
including a detailed parameter study, are presented in two accompanying papers.
