The Maxwell relations are some very general identities about the partial derivatives of functions of several variables. You don’t need to understand anything about thermodyamics to understand them, but they’re used a lot in that subject, so discussions of them tend to use notation from that subject.

Last time I went through a standard way to derive these relations for a function of two variables. Now I want to give a better derivation, which I found here:

• David J. Ritchie, A simple method for deriving Maxwell’s relations, *American Journal of Physics* **36** (1958), 760–760.

This paper is just one page long, and I can’t really improve on it, but I’ll work through the ideas in more detail. It again covers only the case of a function of two variables, and I won’t try to go beyond that case now—maybe later.

So, remember the setup. We have a smooth function on the plane

We call the coordinates on the plane and and we give the partial derivatives of funny names:

None of these funny names or the minus sign has any effect on the actual math involved; they’re just traditional in thermodynamics. So, mathematicians, please forgive me! If I ever generalize to the *n*-variable case, I’ll switch to more reasonable notation.

We instantly get this:

and since we get

so

Believe it or not, this simple relation contains all four of Maxwell’s relations within it!

To see this, note that both sides are smooth 2-forms on the plane. Now, the space of 2-forms at any one point of the plane is a 1-dimensional vector space. So, we can divide any 2-form at a point by any nonzero 2-form at that point and get a real number.

In particular, suppose and are functions on the plane such that at some point. Then we can divide both sides of the above equation by and get

at this point. We can now get the four Maxwell relations simply by making different choices of and We’ll choose them to be either or The argument will only work if so I’ll always assume that. The argument works the same way each time so I’ll go faster after the first time.

### The first relation

Take and and substitute them into the above equation. We get

or

Next we use a little fact about differential forms and partial derivatives to simplify both sides:

and similarly

If you were scarred as a youth when plausible-looking manipulations with partial derivatives turned out to be unjustified, you might be worried about this—and rightly so! Later I’ll show how to justify the kind of ‘cancellation’ we’re doing here. But anyway, it instantly gives us the first Maxwell relation:

### The second relation

This time we take Substituting this into our general formula

we get

and doing the same sort of ‘cancellation’ as last time, this gives the second Maxwell relation:

### The third relation

This time we take Substituting this into our general formula

we get

which gives the third Maxwell relation:

### The fourth relation

This time we take Substituting this into our general formula

we get

or

giving the fourth Maxwell relation:

You can check that other choices of and don’t give additional relations of the same form.

### Determinants

So, we’ve see that all four Maxwell relations follow quickly from the equation

if we can do ‘cancellations’ in expressions like this:

when one of the functions equals one of the functions This works whenever Let’s see why!

First of all, by the inverse function theorem, if at some point in the plane, the functions and serve as coordinates in some neighborhood of that point. In this case we have

Yes: the ratio of 2-forms is just the Jacobian of the map sending to This is clear if you know that 2-forms are ‘area elements’ and the Jacobian is a ratio of area elements. But you can also prove it by a quick calculation:

and thus

so the ratio is the desired determinant.

How does this help? Well, take

and now suppose that either or equals either or For example, suppose Then we can do a ‘cancellation’ like this:

or to make it clear that the partial derivatives are being done in the coordinate system:

This justifies all our calculations earlier.

### Conclusions

So, we’ve seen that all four Maxwell relations are unified in a much simpler equation:

which follows instantly from

This is a big step forward compared to the proof I gave last time, which, at least as I presented it, required cleverly guessing a bunch of auxiliary functions—even though these auxiliary functions turn out to be incredibly important in their own right.

So, we should not stop here: we should think hard about the physical and mathematical meaning of the equation

And Ritchie does this in his paper! But I will talk about that next time.

• Part 1: a proof of Maxwell’s relations using commuting partial derivatives.

• Part 2: a proof of Maxwell’s relations using 2-forms.

• Part 3: the physical meaning of Maxwell’s relations, and their formulation in terms of symplectic geometry.

For how Maxwell’s relations are connected to Hamilton’s equations, see this post:

Ironically, ‘this’ ―writing partial derivatives as ratios of top-rank exterior forms― is the safe way to do only

justifiedmanipulations. The curly ∂ is a warning that something funny is going on and you should not trust the notation. (Unfortunately, you can’t always trust a straight d either.)Of course, not all situations involve simple hydrostatic pressure; some require a full stress tensor and instead of one has something like (where hydrostatic pressure is the special case ). Notation tends to vary between linear elasticity theory, fluid dynamics, and other fields where the stress and deformation tensors appear.

We can take as independent any pair of variables: . The first two choices do not mix the variables on the two sides of the equation and simply yield . Geometrically this means the area elements in the PV and TS planes are equal. Suppose, for example, we choose as the independent pair of variables. Then and . Substituting in we get , whence , which is the fourth relation. The other three relations can be derived in an analogous way using the other three pairs.

The paper by Ritchie that John cited, stresses the importance of this, giving it this physical interpretation: that the work done by the system is equal to the heat absorbed by the system, when undergoing any cyclic process.

Yes! I was going to talk about the physical and mathematical interpretation next time. This is a very nice physical interpretation.

Maxwell Relations are commonly known as a set of four partial differential equations between four thermodynamic quantities or potentials: pressure (P), volume (V), temperature (T), and entropy (S).

[….]