Using Hodge theory for a surface, we obtain a new representation for solutions to the time harmonic Maxwell equations in both bounded and unbounded domains of R^3. This representation has no spurious resonances, does not suffer from low frequency breakdown, and reduces standard boundary value problems to Fredholm equations of second kind. There are many interactions between the topology of the boundary and aspects of this representation. This representation also provides a method to find force free, Beltrami fields, and the spectra for a family of self adjoint boundary problems for the curl operator parametrized by Lagrangian subspaces if the first cohomology group of the boundary.