  Because only two variables are needed to characterize a simple thermodynamic
system in equilibrium, any such system is constrained on a 2D manifold. Of
particular interest are the exact 1-forms on the cotangent space of that
manifold, since the integral of exact 1-forms is path-independent, a crucial
property satisfied by state variables such as the internal energy dE and the
entropy dS. Our prior work shows that given an appropriate language of vector
calculus, a machine can re-discover the Maxwell equations and the
incompressible Navier-Stokes equations from simulation. We speculate that we
can enhance this language by including differential forms. In this paper, we
use the example of classical thermodynamics to show that there exists a simple
algorithm to automate the process of finding exact 1-forms on a thermodynamic
manifold. Since entropy appears in various fields of science in different
guises, a potential extension of this work is to use the machinery developed in
this paper to re-discover the expressions for entropy from data in fields other
than classical thermodynamics.
