Solution
= Solution
With $F=U-TS$, the <thermodynamics of a liquid film> gives
$$
\boxed{dF=-S\,dT+\gamma\,dA.}
$$
Equality of mixed derivatives of $F(T,A)$ gives the <Maxwell relation>
$$
\boxed{\left(\frac{\partial S}{\partial A}\right)_T
=-\left(\frac{\partial\gamma}{\partial T}\right)_A.}
$$
Use the <area> convention already specified by $\gamma\,dA$; do not insert an additional factor for two film faces.