Maxwell Relations in Three Dimensions

Reciprocity relations that guarantee exactness of three-dimensional Pfaffians are 

That is, if these three relationships are satisfied simultaneously for a given Pfaffian, the Pfaffian is exact, and some function F(.x. y. r) exists such that the total differential dF= bQ. 

The necessary (and sufficient) condition for a three-dimensional Pfaffian to be inexact but integrable is 

If neither of the two sets of relations holds, then the Pfaffian expression can be considered to be inexact and non-integrable. 

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