  We have made the first map of CO(J=3-2) emission covering the disk of the
edge-on galaxy, NGC~4631, which is known for its spectacular gaseous halo. The
strongest emission, which we model with a Gaussian ring,occurs within a radius
of 5 kpc. Weaker disk emission is detected out to radii of 12 kpc, the most
extensive molecular component yet seen in this galaxy. From comparisons with
infrared data, we find that CO(J=3-2) emission more closely follows the hot
dust component, rather than the cold dust,consistent with it being a good
tracer of star formation. The first maps of $R_{3-2/1-0}$, H$_2$ mass surface
density and SFE have been made for the inner 2.4 kpc radius region. Only 20\%
of the SF occurs in this region and excitation conditions are typical of galaxy
disks, rather than of central starbursts. The SFE suggests long gas consumption
timescales ($>$ $10^9$ yr).
  The velocity field is dominated by a steeply rising rotation curve in the
region of the central molecular ring followed by a flatter curve in the disk. A
very steep gradient in the rotation curve is observed at the nucleus, providing
the first evidence for a central concentration of mass:
M$_{dyn}\,=\,5\,\times\,10^7$ M$_\odot$ within a radius of 282 pc. The velocity
field shows anomalous features indicating the presence of molecular outflows;
one of them is associated with a previously observed CO(J=1-0) expanding shell.
Consistent with these outflows is the presence of a thick ($z$ up to $1.4$ kpc)
CO(J=3-2) disk. We suggest that the interaction between NGC~4631 and its
companion(s) has agitated the disk and also initiated star formation which was
likely higher in the past than it is now. These may be necessary conditions for
seeing prominent halos.
