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.

Contents

OverviewEdit

DefinitionEdit

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   in space if there exists some scalar function   defined on   such that

     

throughout D. This is equivalent to saying that the vector field   is a conservative vector field, with corresponding potential  .

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.

One dimensionEdit

In one dimension, a differential form

 

is exact as long as   has an antiderivative (but not necessarily one in terms of elementary functions). If   has an antiderivative, let   be an antiderivative of   and this   satisfies the condition for exactness. If   does not have an antiderivative, we cannot write   and so the differential form is inexact.

Two and three dimensionsEdit

By symmetry of second derivatives, for any "nice" (non-pathological) function   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(XY) = 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.

Partial differential relationsEdit

If three variables,  ,   and   are bound by the condition   for some differentiable function  , then the following total differentials exist[1]:667&669

 
 

Substituting the first equation into the second and rearranging, we obtain[1]:669

 
 
 

Since   and   are independent variables,   and   may be chosen without restriction. For this last equation to hold in general, the bracketed terms must be equal to zero.[1]:669

Reciprocity relationEdit

Setting the first term in brackets equal to zero yields[1]

 

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  ,   and  . Reciprocity relations show that the inverse of a partial derivative is equal to its reciprocal.

Cyclic relationEdit

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[1]:670

 

Using a reciprocity relation for   on this equation and reordering gives a cyclic relation (the triple product rule),[1]:670

 

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 dimensionsEdit

(See also Bridgman's thermodynamic equations for the use of exact differentials in the theory of thermodynamic equations)

Suppose we have five state functions  , and  . 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   gives:

 

Letting   gives:

 

Letting  ,   gives:

 

using (  gives the triple product rule:

 

See alsoEdit

ReferencesEdit

  1. ^ a b c d e f g Çengel, Yunus A.; Boles, Michael A. (1998) [1989]. "Thermodynamics Property Relations". Thermodynamics - An Engineering Approach. McGraw-Hill Series in Mechanical Engineering (3rd ed.). Boston, MA.: McGraw-Hill. ISBN 0-07-011927-9.

External linksEdit