In multivariate calculus, a differential or differential form is said to be exact or perfect (exact differential), as contrasted with an inexact differential, if it is equal to the general differential for some differentiable function  in an orthogonal coordinate system (hence is a multivariable function whose variables are independent, as they are always expected to be when treated in multivariable calculus).

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.

The integral of an exact differential over any integral path is path-independent, and this fact is used to identify state functions in thermodynamics.

Overview

Definition

Even if we work in three dimensions here, the definitions of exact differentials for other dimensions are structurally similar to the three dimensional definition. In three dimensions, a form of the type

is called a differential form. This form is called exact on an open domain in space if there exists some differentiable scalar function defined on such that

 

throughout , where are orthogonal coordinates (e.g., Cartesian, cylindrical, or spherical coordinates). In other words, in some open domain of a space, a differential form is an exact differential if it is equal to the general differential of a differentiable function in an orthogonal coordinate system.

Note: In this mathematical expression, 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 explicitly shown here as reminders.

Integral path independence

The exact differential for a differentiable scalar function defined in an open domain is equal to , where is the gradient of , represents the scalar product, and is the general differential displacement vector, if an orthogonal coordinate system is used. If is of differentiability class (continuously differentiable), then is a conservative vector field for the corresponding potential by the definition. For three dimensional spaces, expressions such as and can be made.

The gradient theorem states

that does not depend on which integral path between the given path endpoints and is chosen. So it is concluded that the integral of an exact differential is independent of the choice of an integral path between given path endpoints (path independence).

For three dimensional spaces, if defined on an open domain is of differentiability class (equivalently is of ), then this integral path independence can also be proved by using the vector calculus identity and the Stokes' theorem.

for a simply closed loop with the smooth oriented surface in it. If the open domain 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 vector field which curl is zero, i.e., ) has the path independence by the Stokes' theorem, so the following statement is made; In a simply connected open region, any vector field that has the path-independence property (so it is a conservative vector field.) must also be irrotational and vise versa. The equality of the path independence and conservative vector fields is shown here.

Thermodynamic state function

In thermodynamics, when is exact, the function is a state function of the system: a mathematical function which depends solely on the current equilibrium state, not on the path taken to reach that state. Internal energy , Entropy , Enthalpy , Helmholtz free energy , and Gibbs free energy are state functions. Generally, neither work nor heat is a state function. (Note: 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.)

One dimension

In one dimension, a differential form

is exact if and only if has an antiderivative (but not necessarily one in terms of elementary functions). If has an antiderivative and let be an antiderivative of so , then obviously satisfies the condition for exactness. If does not have an antiderivative, then we cannot write with for a differentiable function so is inexact.

Two and three dimensions

By symmetry of second derivatives, for any "well-behaved" (non-pathological) function , we have

Hence, in a simply-connected region R of the xy-plane, where are independent,[1] a differential form

is an exact differential if and only if the equation

holds. If it is an exact differential so and , then is a differentiable (smoothly continuous) function along and , so . If holds, then and are differentiable (again, smoothly continuous) functions along and respectively, and is only the case.

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(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 (the combination ) to satisfy.

Partial differential relations

If a differentiable function is one-to-one (injective) for each independent variable, e.g., is one-to-one for at a fixed while it is not necessarily one-to-one for , then the following total differentials exist because each independent variable is a differentiable function for the other variables, e.g., .

Substituting the first equation into the second and rearranging, we obtain

Since and are independent variables, and may be chosen without restriction. 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.

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 , and .

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 on this equation and reordering gives a cyclic relation (the triple product rule),

If, instead, reciprocity relations for and are used with subsequent rearrangement, a standard form for implicit differentiation is obtained:

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 , and . Suppose that the state space is two-dimensional and any of the five quantities are differentiable. Then by the chain rule

 

 

 

 

(1)

but also by the chain rule:

 

 

 

 

(2)

and

 

 

 

 

(3)

so that (by substituting (2) and (3) into (1)):

 

 

 

 

(4)

which implies that (by comparing (4) with (1)):

 

 

 

 

(5)

Letting in (5) gives:

 

 

 

 

(6)

Letting in (5) gives:

 

 

 

 

(7)

Letting and in (7) gives:

 

 

 

 

(8)

using ( gives the triple product rule:

 

 

 

 

(9)

See also

References

  1. ^ If the pair of independent variables is a (locally reversible) function of dependent variables , all that is needed for the following theorem to hold, is to replace the partial derivatives with respect to or to , by the partial derivatives with respect to and to involving their Jacobian components. That is: is an exact differential, if and only if:
  2. ^ Çengel, Yunus A.; Boles, Michael A.; Kanoğlu, Mehmet (2019) [1989]. "Thermodynamics Property Relations". Thermodynamics - An Engineering Approach (9th ed.). New York: McGraw-Hill Education. pp. 647–648. ISBN 978-1-259-82267-4.