  Starting from a system of planar electrons in a strong magnetic field normal
to the plane, interacting with perturbing electromagnetic fields, an effective
Lagrangian for the fermions in the lowest Landau level (L.L.L.) has been
derived. By choosing a suitable background electrostatic potential, an
incompressible droplet of these electrons is constructed. The gauge invariant
effective Lagrangian for the electrons in the L.L.L. is shown to split
naturally into a $1+1$ dimensional Lagrangian for the electrons on the surface
of the droplet and into a $2+1$ dimensional gauge-field Lagrangian representing
the contribution of the interior of the droplet. Upon bosonization, the former
represents the surface vibrations of the droplet. Individually neither of these
two actions is gauge invariant, but it is shown that the gauge dependence from
the two pieces cancels out. This demonstrates that the edge degrees of freedom
are essential for maintaining gauge invariance.
