In thermodynamics one often meets equations of the form \[ \sum df_i\wedge dF_i= 0\tag{\(*\)} \] relating the differentials of functions \(f_1,\dots,f_n\) and \(F_1,\dots,F_n\), from which certain reciprocity relations between the partial derivatives of the functions follow. Well-known examples are the Maxwell relations and the Clausius-Clapeyron equation. Usually these reciprocity relations are derived via so-called thermodynamic potentials like enthalpy, free enthalpy, free energy, or by considering certain cyclic processes.
G. Job [Zur Vereinfachung thermodynamischer Rechnungen. Das “Stürzen” einer partiellen Ableitung, Z. Naturforsch. 25a, 1502–1508 (1970)] introduced a method by means of which all reciprocity relations can be obtained directly from equation \((*)\) without considering thermodynamic potentials, which simplifies the theory substantially. The proof is, however, subdivided into many different cases. In this paper the author generalises Job’s method to functional determinants and gives a unified proof based on exterior forms.