Exact differential

In multivariate calculus, a differential is said to be exact or perfect, as contrasted with an inexact differential, if it is of the form "dQ", for some differentiable function "Q".
Overview.
Definition.
We work in three dimensions, with similar definitions holding in any other number of dimensions. In three dimensions, a form of the type
is called a differential form. This form is called "exact" on a domain formula_2 in space if there exists some scalar function formula_3 defined on formula_4 such that
throughout D. This is equivalent to saying that the vector field formula_7 is a conservative vector field, with corresponding potential formula_8.
One dimension.
In one dimension, a differential form
is exact as long as formula_10 has an antiderivative (but not necessarily one in terms of elementary functions). If formula_10 has an antiderivative, let formula_8 be an antiderivative of formula_10 and this formula_8 satisfies the condition for exactness. If formula_10 does "not" have an antiderivative, we cannot write formula_16 and so the differential form is inexact.
Two and three dimensions.
By symmetry of second derivatives, for any "well-behaved" (non-pathological) function formula_8, we have
Hence, it follows that in a simply-connected region "R" of the "xy"-plane, a differential
is an exact differential if and only if the following holds:
For three dimensions, a differential
is an exact differential in a simply-connected region "R" of the "xyz"-coordinate system if between the functions "A", "B" and "C" there exist the relations:
These conditions are equivalent to the following one: If "G" is the graph of this vector valued function then for all tangent vectors "X",Y of the "surface" "G" then "s"("X", "Y") = 0 with "s" the symplectic form.
These conditions, which are easy to generalize, arise from the independence of the order of differentiations in the calculation of the second derivatives. So, in order for a differential "dQ", that is a function of four variables to be an exact differential, there are six conditions to satisfy.
In summary, when a differential "dQ" is exact:
In thermodynamics, when "dQ" is exact, the function "Q" is a state function of the system. The thermodynamic functions "U", "S", "H", "A" and "G" are state functions. Generally, neither work nor heat is a state function. An "exact differential" is sometimes also called a 'total differential', or a 'full differential', or, in the study of differential geometry, it is termed an exact form.
Partial differential relations.
If three variables, formula_26, formula_27 and formula_28 are bound by the condition formula_29 for some differentiable function formula_30, then the following total differentials exist
Substituting the first equation into the second and rearranging, we obtain
Since formula_27 and formula_28 are independent variables, formula_38 and formula_39 may be chosen without restriction. For this last equation to hold in general, the bracketed terms must be equal to zero.
Reciprocity relation.
Setting the first term in brackets equal to zero yields
A slight rearrangement gives a reciprocity relation,