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In mathematics, a differential dQ is said to be exact, as contrasted with an inexact differential, if the differentiable function Q exists. However, if dQ is arbitrarily chosen, a corresponding Q might not exist.
In one dimension, a differential
is always exact. In two dimensions, in order that a differential
be an exact differential in a simply-connected region R of the xy-plane, it is necessary and sufficient that between A and B there exists the relation:
In 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, 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
For three variables, x, y and z bound by some differentiable function F(x,y,z), the following total differentials exist
Substituting the first equation into the second and rearranging, we obtain
Since y and z are independent variables, dy and dz may be chosen without restriction. For this last equation to hold in general, the bracketed terms must be equal to zero.
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 x, y and z. Reciprocity relations show that the inverse of a partial derivative is equal to its reciprocal.
Setting the second term in brackets equal to zero yields
If, instead, a reciprocity relation for is used with subsequent rearrangement, a standard form for implicit differentiation is obtained:
Some useful equations derived from exact differentials in two dimensions
Suppose we have five state functions z,x,y,u, and v. Suppose that the state space is two dimensional and any of the five quantites are exact differentials. Then by the chain rule
but also by the chain rule:
which implies that:
Letting v = y gives:
Letting u = y, v = z gives:
using ( gives the triple product rule:
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Exact_differential". A list of authors is available in Wikipedia.|