# Introduction to differential forms in thermodynamics

There are two articles by S.G. Rajeev: Quantization of Contact Manifolds and Thermodynamics and A Hamilton-Jacobi Formalism for Thermodynamics in which he reviews the formulation of thermodynamics in terms of contact geometry and explains a number of examples such as van der Waals gases and the thermodynamics of black holes in this picture.

Contact geometry is intended primarily to applications of mechanical systems with time varying Hamiltonians by adding time to the phase space coordinates. The dimension of contact manifolds is thus odd. Contact geometry is formulated in terms of a basic one form, the contact one form:

$$ \alpha = dq^0 -p_i dq^i$$

($q^0$ is the time coordinate). The key observation in Rajeev's formulation is that one can identify the contact structure with the first law:

$$ \alpha = dU -TdS + PdV$$

I'm afraid that from the aesthetic side, there is not too much differential geometry to discover in (equilibrium) thermodynamics (at least on an undergrad level and if you don't want to bother with the conceptual question how to properly define the idea of heat for the most abstract situations). I suppose any book on thermodynamics has some sections, which makes use of the mathematical properties, which come from holding on parameter constant and so on.

So I suggest that starting with the axioms and the potentials, you involve yourself with the following basic statements, which make "heavy use" of the formalism:

Maxwell relations

Gibbs-Duhem equation

Gibbs–Helmholtz equation

(The articles all contain the derivations too)

Chapter 7 of my online book Classical and quantum mechanics via Lie algebras derives in 17 pages (pp. 161-177) the main concepts of equilibrium thermodynamics in a physically elementary and mathematically rigorous form. Differential forms appear on p.167 where reversible transformations are defined, and are applied on p.168 to the Gibbs-Duhem equation and the first law of thermodynamics.

Note that Chapter 7 is completely self-contained can be read independent from the earlier chapters.