  We introduce a method-of-lines formulation of the closest point method, a
numerical technique for solving partial differential equations (PDEs) defined
on surfaces. This is an embedding method, which uses an implicit representation
of the surface in a band containing the surface. We define a modified equation
in the band, obtained in a straightforward way from the original evolution PDE,
and show that the solutions of this equation are consistent with those of the
surface equation. The resulting system can then be solved with standard
implicit or explicit time-stepping schemes, and the solutions in the band can
be restricted to the surface. Our derivation generalizes existing formulations
of the closest point method and is amenable to standard convergence analysis.
