  We develop a 3+1+1 covariant formalism with cosmological and astrophysical
applications. First we give the evolution and constraint equations both on the
brane and off-brane in terms of 3-space covariant kinematical,
gravito-electro-magnetic (Weyl) and matter variables. We discuss the junction
conditions across the brane in terms of the new variables. Then we establish a
closure condition for the equations on the brane. We also establish the
connection of this formalism with isotropic and anisotropic cosmological
brane-worlds. Finally we derive a new brane solution in the framework of our
formalism: the tidal charged Taub-NUT-(A)dS brane, which obeys the closure
condition.
