Download Exact differential A mathematical differential is said to be exact, as

Exact differential
A mathematical differential is said to be exact, as contrasted with an inexact differential, if it is
of the form dQ, for some differentiable function Q.
The form A(x, y, z) dx + B(x, y, z) dy + C(x, y, z) dz is called a differential form. A differential
form is exact on a domain D in space if A dx + B dy + C dz = df for some scalar function f
throughout D. This is equivalent to saying that the field is conservative.
Overview
For one dimension, a differential
is always exact.
For 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:
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:
;
;
Note: The subscripts outside the parenthesis indicate which variables are being held
constant during differentiation. Due to the definition of the partial derivative, these
subscripts are not required, but they are included as a reminder.
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:


the function Q exists;
independent of the path followed.
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.
[edit] Partial differential relations
For three variables, x, y and z bound by some differentiable function F(x,y,z), the following
total differentials exist[1]:667&669
Substituting the first equation into the second and rearranging, we obtain[1]:669
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.[1]:669
[edit] Reciprocity relation
Setting the first term in brackets equal to zero yields[1]:670
A slight rearrangement gives a reciprocity relation,[1]:670
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.
[edit] Cyclic relation
Setting the second term in brackets equal to zero yields[1]:670
Using a reciprocity relation for
triple product rule),[1]:670
on this equation and reordering gives a cyclic relation (the
If, instead, a reciprocity relation for is used with subsequent rearrangement, a standard form
for implicit differentiation is obtained:
[edit] 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 z,x,y,u, and v. Suppose that the state space is two
dimensional and any of the five quantities are exact differentials. Then by the chain rule
but also by the chain rule:
and
so that:
which implies that:
Letting v
= y gives:
Letting u
= y gives:
Letting u
= y, v = z gives:
using (

gives the triple product rule: