The integral of an exact differential over any integral path is path-independent, and this fact is used to identify state functions in thermodynamics.
Even if we work in three dimensions here, the definitions of exact differentials for other dimensions are structurally similar to the three dimensional definition.
This form is called exact on an open domain
in space if there exists some differentiable scalar function
is the general differential displacement vector, if an orthogonal coordinate system is used.
is a conservative vector field for the corresponding potential
), then this integral path independence can also be proved by using the vector calculus identity
is simply connected open space (roughly speaking, a single piece open space without a hole within it), then any irrotational vector field (defined as a
) has the path independence by the Stokes' theorem, so the following statement is made; In a simply connected open region, any
The equality of the path independence and conservative vector fields is shown here.
is commonly used to represent heat in physics.
It should not be confused with the use earlier in this article as the parameter of an exact differential.)
In one dimension, a differential form is exact if and only if
has an antiderivative (but not necessarily one in terms of elementary functions).
By symmetry of second derivatives, for any "well-behaved" (non-pathological) function
is a differentiable (smoothly continuous) function along
are differentiable (again, smoothly continuous) functions along
For three dimensions, in a simply-connected region R of the xyz-coordinate system, by a similar reason, a differential is an exact differential if and only if between the functions A, B and C there exist the relations These conditions are equivalent to the following sentence: 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 (the combination
is one-to-one (injective) for each independent variable, e.g.,
Substituting the first equation into the second and rearranging, we obtain Since
For this last equation to generally hold, the bracketed terms must be equal to zero.
[2] The left bracket equal to zero leads to the reciprocity relation while the right bracket equal to zero goes to the cyclic relation as shown below.
Setting the first term in brackets equal to zero yields A slight rearrangement gives a reciprocity relation, There are two more permutations of the foregoing derivation that give a total of three reciprocity relations between
Setting the second term in brackets equal to zero yields Using a reciprocity relation for
on this equation and reordering gives a cyclic relation (the triple product rule), If, instead, reciprocity relations for
are used with subsequent rearrangement, a standard form for implicit differentiation is obtained: (See also Bridgman's thermodynamic equations for the use of exact differentials in the theory of thermodynamic equations) Suppose we have five state functions
Suppose that the state space is two-dimensional and any of the five quantities are differentiable.
gives the triple product rule: