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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a differential dQ is said to be exact, as contrasted with an inexact differential
Inexact differential

In thermodynamics, an inexact differential or imperfect differential is any quantity, particularly heat Q and thermodynamic work W, that are not state functions, in that their values depend on how the thermodynamic process is carried out....
, if the differentiable function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 Q exists. However, if dQ is arbitrarily chosen, a corresponding Q might not exist.

lways 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 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.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, a differential dQ is said to be exact, as contrasted with an inexact differential
Inexact differential

In thermodynamics, an inexact differential or imperfect differential is any quantity, particularly heat Q and thermodynamic work W, that are not state functions, in that their values depend on how the thermodynamic process is carried out....
, if the differentiable function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
 Q exists. However, if dQ is arbitrarily chosen, a corresponding Q might not exist.

Overview


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 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 function
State function

In thermodynamics, a state function, state quantity, or a function of state, is a physical quantity of a system that depends only on the current Thermodynamic state, not on the way in which the system got to that state....
s. 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, , and bound by some differentiable function , the following total differentials exist

. Note: The subscripts outside the parenthesis indicate which variables are being held constant during differentiation. Due to the definition of the partial derivative
Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant ....
, these subscripts are not required, but they are included as a reminder.

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 hold in general, the bracketed terms must be equal to zero.

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 . Reciprocity relations
Inverse functions and differentiation

In mathematics, the inverse of a function is a function that, in some fashion, "undoes" the effect of . The inverse of is denoted . The statements y=f and x=f -1 are equivalent....
 show that the inverse of a partial derivative is equal to its reciprocal.

Cyclic Relation

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
Triple product rule

The triple product rule, known variously as the cyclic chain rule, cyclic relation, or Leonhard Euler chain rule, is a formula which relates partial derivative of three interdependent variables....
),

.

If, instead, a reciprocity relation for is used with subsequent rearrangement, a standard form for implicit differentiation
Implicit function

In mathematics, an implicit function is a function in which the dependent variable has not been given "explicitly" in terms of the independent variable....
 is obtained:

.

Some useful equations derived from exact differentials in two dimensions


(See also Bridgman's thermodynamic equations
Bridgman's thermodynamic equations

In thermodynamics, Bridgman's thermodynamic equations are a basic set of thermodynamic equations, derived using a method of generating a large number of thermodynamic identities involving a number of thermodynamic quantities....
 for the use of exact differentials in the theory of thermodynamic equations
Thermodynamic equations

In thermodynamics, there are a large number of equations relating the variousthermodynamic quantities. In chemical thermodynamics, which is a sub-branch of thermodynamics, for example, there are millions of useful equations....
)

Suppose we have five state functions , and . Suppose that the state space is two dimensional and any of the five quantites are exact differentials. Then by the chain rule
Chain rule

In calculus, the chain rule is a formula for the derivative of the functional composition of two function .In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of Mathematics#Change of y with respect to x can be computation as the rate of chan...


but also by the chain rule:

and

so that:


which implies that:

Letting gives:

Letting , gives:

using ( gives the triple product rule
Triple product rule

The triple product rule, known variously as the cyclic chain rule, cyclic relation, or Leonhard Euler chain rule, is a formula which relates partial derivative of three interdependent variables....
:

See also

  • Closed and exact differential forms
    Closed and exact differential forms

    In mathematics, especially vector calculus and differential topology, a closed form is a differential form a whose differential is zero , and an exact form is a differential form that is the differential of another differential form ....
     for a higher-level treatment
  • Differential
    Differential

    Differential may refer to:...
  • Inexact differential
    Inexact differential

    In thermodynamics, an inexact differential or imperfect differential is any quantity, particularly heat Q and thermodynamic work W, that are not state functions, in that their values depend on how the thermodynamic process is carried out....
  • Integrating factor
    Integrating factor

    In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given ordinary differential equation....
     for solving non-exact differential equations by making them exact


External links

  • – from Wolfram MathWorld
  • – University of Arizona
  • – University of Texas
  • – from Wolfram MathWorld