Relative to the choice of a projective renormalization, a generalization of the
Gauss map permits a Jacobian matrix, J, of rank N1 to be defined for any
given vector field of dimension N. The characteristic polynomial of such a
three dimensional matrix is always a cubic. If three universal coordinate
functions are defined in terms of the similarity invariants of the Jacobian
matrix, then the characteristic polynomial equation, in terms of these universal
variables, defines a universal surface which has the topological features of
the Gibbs surface for a Van der Waals gas. The method permits the evaluation of
those thermodynamic properties which correspond to the limiting spinodal line of
single phase stability, the binodal line of mixed phase stability, and the
critical point, for any dynamical process constructed from a dynamical system.
The method establishes a theoretical foundation, in terms of projective
geometry, for the chemical law of corresponding states. All differentiable three
dimensional systems can be mapped to the Van der Waals gas.
