Exact differential

In mathematics, a differential dQ is said to be exact, as contrasted with an inexact differential, if the function Q exists. It is always possible to calculate the differential dQ of a given function Q(x, y, z). However, if dQ is arbitrarily given, the function Q generally does not exist.

Overview

In one dimension, a differential

$dQ=A(x)dx\,$

is always exact. In two dimensions, in order that a differential

$dQ=A(x,y)dx+B(x,y)dy\,$

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:

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

In three dimensions, a differential

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

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:

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

These conditions, 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;
• $\int _{i}^{f}dQ=Q(f)-Q(i)$, 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.

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 quantites are exact differentials. Then by the chain rule

$(1)~~~~~dz=\left({\frac {\partial z}{\partial x}}\right)_{y}dx+\left({\frac {\partial z}{\partial y}}\right)_{x}dy=\left({\frac {\partial z}{\partial u}}\right)_{v}du+\left({\frac {\partial z}{\partial v}}\right)_{u}dv$

but also by the chain rule:

$(2)~~~~~dx=\left({\frac {\partial x}{\partial u}}\right)_{v}du+\left({\frac {\partial x}{\partial v}}\right)_{u}dv$

and

$(3)~~~~~dy=\left({\frac {\partial y}{\partial u}}\right)_{v}du+\left({\frac {\partial y}{\partial v}}\right)_{u}dv$

so that:

$(4)~~~~~dz=\left[\left({\frac {\partial z}{\partial x}}\right)_{y}\left({\frac {\partial x}{\partial u}}\right)_{v}+\left({\frac {\partial z}{\partial y}}\right)_{x}\left({\frac {\partial y}{\partial u}}\right)_{v}\right]du$

$+\left[\left({\frac {\partial z}{\partial x}}\right)_{y}\left({\frac {\partial x}{\partial v}}\right)_{u}+\left({\frac {\partial z}{\partial y}}\right)_{x}\left({\frac {\partial y}{\partial v}}\right)_{u}\right]dv$

which implies that:

$(5)~~~~~\left({\frac {\partial z}{\partial u}}\right)_{v}=\left({\frac {\partial z}{\partial x}}\right)_{y}\left({\frac {\partial x}{\partial u}}\right)_{v}+\left({\frac {\partial z}{\partial y}}\right)_{x}\left({\frac {\partial y}{\partial u}}\right)_{v}$

Letting $v=y$ gives:

$(6)~~~~~\left({\frac {\partial z}{\partial u}}\right)_{y}=\left({\frac {\partial z}{\partial x}}\right)_{y}\left({\frac {\partial x}{\partial u}}\right)_{y}$

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

$(7)~~~~~\left({\frac {\partial z}{\partial y}}\right)_{x}=-\left({\frac {\partial z}{\partial x}}\right)_{y}\left({\frac {\partial x}{\partial y}}\right)_{z}$

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

$(8)~~~~~\left({\frac {\partial z}{\partial x}}\right)_{y}\left({\frac {\partial x}{\partial y}}\right)_{z}\left({\frac {\partial y}{\partial z}}\right)_{x}=-1$

See also

• Closed and exact differential forms for a higher-level treatment
• Differential
• Inexact differential
• Integrating factor for solving non-exact differential equations by making them exact

References

• Perrot, P. (1998). A to Z of Thermodynamics. New York: Oxford University Press.
• Zill, D. (1993). A First Course in Differential Equations, 5th Ed. Boston: PWS-Kent Publishing Company.

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