Maxwell relations — derivation from each potential: Foundations

This page assumes you have seen none of the notation on the parent page. We build each symbol from nothing, anchor it to a picture, and only then use it. Read top to bottom; nothing here depends on anything below it.

1. A function of two variables — the "landscape"

Before any physics, we need the idea of a quantity that depends on two knobs at once.

Figure — Maxwell relations — derivation from each potential

The whole topic lives on pictures like this. Later, the "height" $z$ will be an energy and the floor-axes $(x, y)$ will be physical quantities. But the geometry is exactly this hill.

Recall Why do we need

two variables and not one? Because a gas needs two numbers to pin down its state ::: e.g. its volume and its temperature. One number is not enough to say what the gas is doing.

2. Slope in one direction — the partial derivative

3. The total change — the differential $d z$

Now: if I take one small step that is a bit East and a bit North at the same time, how much does the height change in total?

We give the two slopes short names, $M$ and $N$ : $d z = M d x + N d y, M \equiv (\frac{\partial z}{\partial x})_{y}, N \equiv (\frac{\partial z}{\partial y})_{x} .$

4. The crossed-slopes-agree fact (Schwarz / equality of mixed partials)

This is the engine of the entire topic.

This one line is every Maxwell relation. Once we know what plays the role of $z, x, y, M, N$ in thermodynamics, we just plug in.

Recall What single condition must a function satisfy for this to work?

It must be smooth (a genuine height function / state function) ::: so that its value depends only on where you are, not on the path you took to get there.

5. State function vs. path — why energy qualifies

The four thermodynamic potentials $U, H, F, G$ are state functions (heights on landscapes), so their differentials are exact and Section 4 applies to each. Heat $δ Q$ and work $δ W$ are not, which is why the parent note must convert them via $T d S$ and $P d V$ before proceeding.

6. The physical axes — meet $T, S, P, V$

Now we swap the abstract $(x, y, z)$ for real physical quantities. Four symbols carry the whole subject.

$S$ and $V$ are hard-or-easy to measure in opposite ways: you can easily read $P, V, T$ off gauges, but you cannot stick a probe in and read $S$ . That asymmetry is the whole reason Maxwell relations are useful — they trade an unmeasurable slope (like $\partial S / \partial P$ ) for a measurable one (like $\partial V / \partial T$ ).

7. The master differential $d U = T d S - P d V$

Now the abstract " $d z = M d x + N d y$ " becomes physical.

Because $U$ is a state function (Section 5), the crossed-slopes fact (Section 4) applies to it — and to $H, F, G$ once we build them.

8. Legendre transform — changing which axis you stand on

The parent note builds $H, F, G$ from $U$ by "adding $P V$ " or "subtracting $T S$ ." Here is the picture.

Definition "Natural variables"

The pair of axes for which a potential's differential comes out with clean coefficients ( $T, S, P, V$ and nothing messy). For $U$ they are $(S, V)$ . The transforms give:

potential	is $U$ with...	natural axes
$U$	—	$(S, V)$
$H = U + P V$	$V$ traded for $P$	$(S, P)$
$F = U - T S$	$S$ traded for $T$	$(T, V)$
$G = U - T S + P V$	both traded	$(T, P)$

Full machinery lives in Thermodynamic potentials & Legendre transforms. For this page you only need: each potential is another smooth hill, drawn over a different pair of axes, and Section 4 applies to every one.

How the foundations feed the topic

Read it as: the calculus (left branch) gives the engine; the physics (right branch) supplies the hills; together they produce the four Maxwell relations.

Equipment checklist

Test yourself — if any reveal surprises you, reread that section before the main note.

What does the curly $\partial$ warn you about, that a straight $d$ does not? ::: That other variables are being held fixed; it's a slope in one frozen direction.
What does the subscript in $(\partial z / \partial x)_{y}$ tell you? ::: Exactly which variable is held constant — and it is part of the quantity, not optional.
In $d z = M d x + N d y$ , what is $M$ ? ::: The slope of the height in the $x$ -direction, $(\partial z / \partial x)_{y}$ — the number multiplying $d x$ .
State the equality of mixed partials in one line. ::: $(\partial M / \partial y)_{x} = (\partial N / \partial x)_{y}$ — crossed slopes agree.
Why does this equality hold for $U, H, F, G$ but not for heat $δ Q$ ? ::: $U, H, F, G$ are state functions (heights); heat is a path function, not the differential of any hill.
Name the two conjugate pairs. ::: $(T, S)$ and $(P, V)$ .
Why is the term $- P d V$ negative? ::: Expansion does work on the surroundings, draining internal energy, so $U$ drops as $V$ rises.
What does a Legendre transform change, and what does it preserve? ::: It swaps which variable is the natural axis (e.g. $V \to P$ ); it preserves all the physical information.

Connections

Parent topic — where these foundations get used
Equality of mixed partial derivatives (Schwarz theorem) — the engine (Section 4)
First and Second Laws of Thermodynamics — source of $d U = T d S - P d V$ (Section 7)
Thermodynamic potentials & Legendre transforms — building $H, F, G$ (Section 8)