In multivariate calculus, a differential is said to be exact or perfect, as contrasted with an inexact differential, if it is of the form
Contents
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
throughout D. This is equivalent to saying that the vector field
One dimension
In one dimension, a differential form
is exact as long as
Two and three dimensions
By symmetry of second derivatives, for any "nice" (non-pathological) function
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,
Substituting the first equation into the second and rearranging, we obtain
Since
Reciprocity relation
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
Cyclic relation
The cyclic relation is also known as the cyclic rule or the Triple product rule. Setting the second term in brackets equal to zero yields
Using a reciprocity relation for
If, instead, a reciprocity relation for
Some useful equations derived from exact differentials in two dimensions
(See also Bridgman's thermodynamic equations for the use of exact differentials in the theory of thermodynamic equations)
Suppose we have five state functions
but also by the chain rule:
and
so that:
which implies that:
Letting
Letting
Letting
using (