I've been working on some notes on thermodynamics. The fun bits are:
a) Using differential equations for functions of time. This lets
you give a very explicit treatment of Carnot's theorem on efficiency.
b) Then using differential forms. This lets you prove the theorems
of thermodynamics, like Maxwell's relation and the Clausius-Clapeyron
theorem, very straightforwardly. They also make deriving thermodynamic
identities much easier (essentially in the same way Jacobian methods
do).
c) Including differential equations for two chambers coupled together.
This shows in detail how heat conduction affects the frequency of
oscillations, a simplified version of how the speed of sound depends
on heat conduction.
d) Presenting Gibbs' way of looking at various thermodynamic diagrams,
which ties in with differential forms.
e) Looking at Gibbs' method for finding equilibria and deriving the
stability conditions using the second derivative test for maxima
when we have constraints (using bordered Hessians).
f) Including a very simple example of irreversible thermodynamics,
just by including quadratic terms in the work and heat (a little
like including air resistance in mechanics). This lets you study
the increase of entropy very explicitly and without handwaving.
My aim has been to state equations and theorems clearly enough that
we can check they are in fact true for concrete examples - and to
include these examples. I have also tried to illustrate analytical
results with diagrams. Lastly I mean to include historical examples.
