Criterion for an exact differential

Mixed-partial equality criterion for when a differential form is the differential of a state function; used to justify Maxwell relations and identify thermodynamic potentials.

Criterion for an exact differential

Statement

Let $D\subset\mathbb R^2$ be an open, simply connected domain and let $M,N\in C^1(D)$ . Consider the $1$ -form

\omega = M(x,y)\,dx + N(x,y)\,dy.

Then the following are equivalent:

$\omega$ is exact: there exists a scalar potential $F\in C^2(D)$ such that $dF=\omega$ (i.e. $\partial_x F=M$ and $\partial_y F=N$ ).
The mixed partials satisfy $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}\qquad\text{everywhere on }D.$

Equivalently, line integrals of $\omega$ are path-independent on $D$ .

Multivariable version (thermodynamic coordinates)

For $\omega=\sum_{i=1}^n M_i(x)\,dx_i$ with $M_i\in C^1(D)\subset\mathbb R^n$ on a simply connected domain, $\omega$ is exact iff

\frac{\partial M_i}{\partial x_j}=\frac{\partial M_j}{\partial x_i}\qquad\text{for all }i,j.

Key hypotheses

Coefficients $M,N$ are continuously differentiable (at least $C^1$ ).
The domain $D$ is simply connected (no “holes”), ensuring that closedness implies exactness in this setting.

Key conclusions

A differential expression is the differential of a single-valued state function precisely when the corresponding cross-derivative compatibility holds.
In thermodynamics, this criterion underlies:
- identification of state functions like internal energy and entropy ;
- derivations of Maxwell relations (see Maxwell relations theorem and Maxwell from mixed partials ).

Proof idea / significance

“Exact $\Rightarrow$ mixed partials equal” follows immediately from Clairaut’s theorem: if $M=\partial_x F$ and $N=\partial_y F$ with $F\in C^2$ , then $\partial_y M=\partial_{yx}F=\partial_{xy}F=\partial_x N$ .
“Mixed partials equal $\Rightarrow$ exact” can be shown by defining
$F(x,y)=\int_{\gamma}\omega,$
where $\gamma$ is any path from a fixed base point to $(x,y)$ , and using simple connectedness plus the mixed-partial condition to prove path-independence.

Thermodynamic significance: checking exactness (or exactness after an integrating factor; see integrating factor lemma ) is the mathematical core of turning differential heat/work statements into state functions and potentials consistent with the first law and second law .