Exact differential

In multivariate calculus, a differential or differential form is said to be exact or perfect (so called an exact differential), as contrasted with an inexact differential, if it is equal to the general differential dQ for some differentiable function Q.

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 integral path-independent, and this fact is used to identity state functions in thermodynamics.

Overview

Definition

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

A(x,y,z)\,dx+B(x,y,z)\,dy+C(x,y,z)\,dz

is called a differential form. This form is called exact on a domain $D\subset \mathbb {R} ^{3}$ in space if there exists some differentiable scalar function $Q=Q(x,y,z)$ defined on $D$ such that

dQ\equiv \left({\frac {\partial Q}{\partial x}}\right)_{y,z}\,dx+\left({\frac {\partial Q}{\partial y}}\right)_{x,z}\,dy+\left({\frac {\partial Q}{\partial z}}\right)_{x,y}\,dz,

dQ=A\,dx+B\,dy+C\,dz

throughout $D$ . In other words, in some domain of three dimensional space, a differential form is an exact differential if it is equal to the general differential of a differentiable function.

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 included as a reminder.

Integral path independence

The exact differential for a differentiable function $Q$ is equal to $dQ=\nabla Q\cdot d\mathbf {r}$ , that is the scalar product between the conservative vector field $(A,B,C)=\nabla Q$ (where the right hand side is the gradient of $Q$ ) for the corresponding potential $Q$ and the general differential displacement vector $(dx,dy,dz)=d\mathbf {r}$ . The gradient theorem states $\int _{i}^{f}dQ=\int _{i}^{f}\nabla Q(\mathbf {r} )\cdot d\mathbf {r} =Q\left(f\right)-Q\left(i\right)$ that is integral path-independent.

In thermodynamics, when dQ is exact, the function Q is a state function (a mathematical function depending only on the current equilibrium state, not depending on the system path taken to reach the equilibrium state) of the system. The thermodynamic functions U (internal energy), S (entropy), H (enthalpy), A (Helmholtz free energy), and G (Gibbs free energy) are state functions. Generally, neither work nor heat is a state function.

One dimension

In one dimension, a differential form

A(x)\,dx

is exact if and only if $A$ has an antiderivative (but not necessarily one in terms of elementary functions). If $A$ has an antiderivative, let $Q$ be an antiderivative of $A$ and this $Q$ satisfies the condition for exactness. If $A$ does not have an antiderivative, we cannot write $dQ=A(x)\,dx$ and so the differential form is inexact.

Two and three dimensions

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

{\frac {\partial ^{2}Q}{\partial x\,\partial y}}={\frac {\partial ^{2}Q}{\partial y\,\partial x}}.

Hence, in a simply-connected region R of the xy-plane, a differential

A(x,y)\,dx+B(x,y)\,dy

is an exact differential if (not "if and only if") the equation

\left({\frac {\partial A}{\partial y}}\right)_{x}=\left({\frac {\partial B}{\partial x}}\right)_{y}

holds since this equation is satisfied if and only if $A={\frac {\partial Q}{\partial x}}$ and $B={\frac {\partial Q}{\partial y}}$ , or $\left({\frac {\partial A}{\partial y}}\right)_{x}={\frac {\partial ^{2}Q}{\partial y\partial x}}={\frac {\partial ^{2}Q}{\partial x\partial y}}=\left({\frac {\partial B}{\partial x}}\right)_{y}$ .

For three dimensions, in a simply-connected region R of the xyz-coordinate system, by a similar reason, a differential

dQ=A(x,y,z)\,dx+B(x,y,z)\,dy+C(x,y,z)\,dz

is an exact differential if between the functions A, B and C there exist the relations

\left({\frac {\partial A}{\partial y}}\right)_{x,z}\!\!\!=\left({\frac {\partial B}{\partial x}}\right)_{y,z}

;

\left({\frac {\partial A}{\partial z}}\right)_{x,y}\!\!\!=\left({\frac {\partial C}{\partial x}}\right)_{y,z}

;

\left({\frac {\partial B}{\partial z}}\right)_{x,y}\!\!\!=\left({\frac {\partial C}{\partial y}}\right)_{x,z}.

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(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 (the combination $C(4,2)=6$ ) to satisfy.

Partial differential relations

If three variables, $x$ , $y$ and $z$ are bound by the condition $F(x,y,z)={\text{constant}}$ for some differentiable function $F(x,y,z)$ , then the following total differentials exist^[1]^: 667&669

dx={\left({\frac {\partial x}{\partial y}}\right)}_{z}\,dy+{\left({\frac {\partial x}{\partial z}}\right)}_{y}\,dz

dz={\left({\frac {\partial z}{\partial x}}\right)}_{y}\,dx+{\left({\frac {\partial z}{\partial y}}\right)}_{x}\,dy.

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

dz={\left({\frac {\partial z}{\partial x}}\right)}_{y}\left[{\left({\frac {\partial x}{\partial y}}\right)}_{z}dy+{\left({\frac {\partial x}{\partial z}}\right)}_{y}dz\right]+{\left({\frac {\partial z}{\partial y}}\right)}_{x}dy,

dz=\left[{\left({\frac {\partial z}{\partial x}}\right)}_{y}{\left({\frac {\partial x}{\partial y}}\right)}_{z}+{\left({\frac {\partial z}{\partial y}}\right)}_{x}\right]dy+{\left({\frac {\partial z}{\partial x}}\right)}_{y}{\left({\frac {\partial x}{\partial z}}\right)}_{y}dz,

\left[1-{\left({\frac {\partial z}{\partial x}}\right)}_{y}{\left({\frac {\partial x}{\partial z}}\right)}_{y}\right]dz=\left[{\left({\frac {\partial z}{\partial x}}\right)}_{y}{\left({\frac {\partial x}{\partial y}}\right)}_{z}+{\left({\frac {\partial z}{\partial y}}\right)}_{x}\right]dy.

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

Reciprocity relation

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

{\left({\frac {\partial z}{\partial x}}\right)}_{y}{\left({\frac {\partial x}{\partial z}}\right)}_{y}=1.

A slight rearrangement gives a reciprocity relation,^[1]^: 670

{\left({\frac {\partial z}{\partial x}}\right)}_{y}={\frac {1}{{\left({\frac {\partial x}{\partial z}}\right)}_{y}}}.

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.

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^[1]^: 670

{\left({\frac {\partial z}{\partial x}}\right)}_{y}{\left({\frac {\partial x}{\partial y}}\right)}_{z}=-{\left({\frac {\partial z}{\partial y}}\right)}_{x}.

Using a reciprocity relation for ${\tfrac {\partial z}{\partial y}}$ on this equation and reordering gives a cyclic relation (the triple product rule),^[1]^: 670

{\left({\frac {\partial x}{\partial y}}\right)}_{z}{\left({\frac {\partial y}{\partial z}}\right)}_{x}{\left({\frac {\partial z}{\partial x}}\right)}_{y}=-1.

If, instead, a reciprocity relation for ${\tfrac {\partial x}{\partial y}}$ is used with subsequent rearrangement, a standard form for implicit differentiation is obtained:

{\left({\frac {\partial y}{\partial x}}\right)}_{z}=-{\frac {{\left({\frac {\partial z}{\partial x}}\right)}_{y}}{{\left({\frac {\partial z}{\partial y}}\right)}_{x}}}.

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

dz=\left({\frac {\partial z}{\partial x}}\right)_{y}dx+\left({\frac {\partial z}{\partial y}}\right)_{x}dy=\left({\frac {\partial z}{\partial u}}\right)_{v}du+\left({\frac {\partial z}{\partial v}}\right)_{u}dv

1

but also by the chain rule:

dx=\left({\frac {\partial x}{\partial u}}\right)_{v}du+\left({\frac {\partial x}{\partial v}}\right)_{u}dv

2

and

dy=\left({\frac {\partial y}{\partial u}}\right)_{v}du+\left({\frac {\partial y}{\partial v}}\right)_{u}dv

3

so that:

{\begin{aligned}dz=&\left[\left({\frac {\partial z}{\partial x}}\right)_{y}\left({\frac {\partial x}{\partial u}}\right)_{v}+\left({\frac {\partial z}{\partial y}}\right)_{x}\left({\frac {\partial y}{\partial u}}\right)_{v}\right]du\\+&\left[\left({\frac {\partial z}{\partial x}}\right)_{y}\left({\frac {\partial x}{\partial v}}\right)_{u}+\left({\frac {\partial z}{\partial y}}\right)_{x}\left({\frac {\partial y}{\partial v}}\right)_{u}\right]dv\end{aligned}}

4

which implies that:

\left({\frac {\partial z}{\partial u}}\right)_{v}=\left({\frac {\partial z}{\partial x}}\right)_{y}\left({\frac {\partial x}{\partial u}}\right)_{v}+\left({\frac {\partial z}{\partial y}}\right)_{x}\left({\frac {\partial y}{\partial u}}\right)_{v}

5

Letting $v=y$ gives:

\left({\frac {\partial z}{\partial u}}\right)_{y}=\left({\frac {\partial z}{\partial x}}\right)_{y}\left({\frac {\partial x}{\partial u}}\right)_{y}

6

Letting $u=y$ gives:

\left({\frac {\partial z}{\partial y}}\right)_{v}=\left({\frac {\partial z}{\partial y}}\right)_{x}+\left({\frac {\partial z}{\partial x}}\right)_{y}\left({\frac {\partial x}{\partial y}}\right)_{v}

7

Letting $u=y$ , $v=z$ gives:

\left({\frac {\partial z}{\partial y}}\right)_{x}=-\left({\frac {\partial z}{\partial x}}\right)_{y}\left({\frac {\partial x}{\partial y}}\right)_{z}

8

using ( $\partial a/\partial b)_{c}=1/(\partial b/\partial a)_{c}$ gives the triple product rule:

\left({\frac {\partial z}{\partial x}}\right)_{y}\left({\frac {\partial x}{\partial y}}\right)_{z}\left({\frac {\partial y}{\partial z}}\right)_{x}=-1

9

References

^ ^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.

Perrot, P. (1998). A to Z of Thermodynamics. New York: Oxford University Press.
Dennis G. Zill (14 May 2008). A First Course in Differential Equations. Cengage Learning. ISBN 978-0-495-10824-5.

External links

Inexact Differential – from Wolfram MathWorld
Exact and Inexact Differentials – University of Arizona
Exact and Inexact Differentials – University of Texas
Exact Differential – from Wolfram MathWorld

[Cengel1998-1] ^ ^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.

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