I've studied differential geometry just enough to be confident with differential forms. Now I want to see application of this formalism in thermodynamics.
I'm looking for a small reference, to learn familiar concepts of (equilibrium?) thermodynamics formulated through differential forms.
Once again, it shouldn't be a complete book, a chapter at max, or an article.
UPD Although I've accepted David's answer, have a look at the Nick's one and my comment on it.
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:
(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.
Professor Hannay wrote a very interesting article "Carnot and the fields formulation of elementary thermodynamics," Am. J. Phys. 74 2, February 2006, pp134-140. Putting aside the rather strange and unusual notation he shows that how Carnot's efficiency formula can be written using differential forms, specifically with wedge product. Rewritten in a more conventional form than is in the article Carnot's efficiency equation is written as $$ \frac {1}{T} \tilde q \wedge \tilde dT = \tilde d \tilde w ,$$ where the ~ denotes a differential form, $\tilde d$ is the exterior derivative, $\tilde q$ and $\tilde w$ are the heat and work 1-forms. Hannay also writes the 1st and 2nd laws as: $$ \tilde d \tilde q + \tilde d \tilde w = 0$$ and $$ \tilde d \left( \frac {\tilde q}{T}\right) =0 $$
So there is use for higher order forms than just 1-forms in thermostatics.
