Exact differential

In mathematics, a differential "dQ" is said to be "exact", as contrasted with an inexact differential, if the differentiable function "Q" exists. However, if "dQ" is arbitrarily chosen, a corresponding "Q" might 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:

:$left(\; frac\{partial\; A\}\{partial\; y\}\; ight)\_\{x\}\; =\; left(\; frac\{partial\; B\}\{partial\; x\}\; ight)\_\{y\}$

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:

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

These conditions are equivalent to the following one: If G is the graph of this vector valued function then for all tangent vectors X,Yof 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;
*$int\_i^f\; dQ=Q(f)-Q(i)$, 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.

Partial Differential Relations

For three variables, $x$, $y$ and $z$ bound by some differentiable function $F(x,y,z)$, the following total differentials existcite book |last=Çengel |first=Yunus A. |authorlink= |coauthors=Boles, Michael A. |title=Thermodynamics - An Engineering Approach |origyear=1989 |edition=3rd |series=McGraw-Hill Series in Mechanical Engineering |year=1998 |publisher=McGraw-Hill |location=Boston, MA. |isbn=0-07-011927-9 |chapter=Thermodynamics Property Relations] rp|667&669

:$d\; x\; =\; \{left\; (\; frac\{partial\; x\}\{partial\; y\}\; ight\; )\}\_z\; d\; y\; +\; \{left\; (\; frac\{partial\; x\}\{partial\; z\}\; ight\; )\}\_y\; dz$:$d\; z\; =\; \{left\; (\; frac\{partial\; z\}\{partial\; x\}\; ight\; )\}\_y\; d\; x\; +\; \{left\; (\; frac\{partial\; z\}\{partial\; y\}\; ight\; )\}\_x\; dy$.::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.

Substituting the first equation into the second and rearranging, we obtainrp|669

:$d\; z\; =\; \{left\; (\; frac\{partial\; z\}\{partial\; x\}\; ight\; )\}\_y\; left\; [\; \{left\; (\; frac\{partial\; x\}\{partial\; y\}\; ight\; )\}\_z\; d\; y\; +\; \{left\; (\; frac\{partial\; x\}\{partial\; z\}\; ight\; )\}\_y\; dz\; ight\; ]\; +\; \{left\; (\; frac\{partial\; z\}\{partial\; y\}\; ight\; )\}\_x\; dy$,:$d\; z\; =\; left\; [\; \{left\; (\; frac\{partial\; z\}\{partial\; x\}\; ight\; )\}\_y\; \{left\; (\; frac\{partial\; x\}\{partial\; y\}\; ight\; )\}\_z\; +\; \{left\; (\; frac\{partial\; z\}\{partial\; y\}\; ight\; )\}\_x\; ight\; ]\; d\; y\; +\; \{left\; (\; frac\{partial\; z\}\{partial\; x\}\; ight\; )\}\_y\; \{left\; (\; frac\{partial\; x\}\{partial\; z\}\; ight\; )\}\_y\; dz$,:$left\; [\; 1\; -\; \{left\; (\; frac\{partial\; z\}\{partial\; x\}\; ight\; )\}\_y\; \{left\; (\; frac\{partial\; x\}\{partial\; z\}\; ight\; )\}\_y\; ight\; ]\; dz\; =\; left\; [\; \{left\; (\; frac\{partial\; z\}\{partial\; x\}\; ight\; )\}\_y\; \{left\; (\; frac\{partial\; x\}\{partial\; y\}\; ight\; )\}\_z\; +\; \{left\; (\; frac\{partial\; z\}\{partial\; y\}\; ight\; )\}\_x\; ight\; ]\; d\; y$.

Since $y$ and $z$ are independent variables, $d\; y$ and $d\; z$ may be chosen without restriction. For this last equation to hold in general, the bracketed terms must be equal to zero.rp|669

Reciprocity Relation

Setting the first term in brackets equal to zero yieldsrp|670

:$\{left\; (\; frac\{partial\; z\}\{partial\; x\}\; ight\; )\}\_y\; \{left\; (\; frac\{partial\; x\}\{partial\; z\}\; ight\; )\}\_y\; =\; 1$.

A slight rearrangement gives a reciprocity relation,rp|670

:$\{left\; (\; frac\{partial\; z\}\{partial\; x\}\; ight\; )\}\_y\; =\; frac\{1\}$left ( frac{partial x}{partial z} ight )}_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

Setting the second term in brackets equal to zero yieldsrp|670

:$\{left\; (\; frac\{partial\; z\}\{partial\; x\}\; ight\; )\}\_y\; \{left\; (\; frac\{partial\; x\}\{partial\; y\}\; ight\; )\}\_z\; =\; -\; \{left\; (\; frac\{partial\; z\}\{partial\; y\}\; ight\; )\}\_x$.

Using a reciprocity relation for $frac\{partial\; z\}\{partial\; y\}$ on this equation and reordering gives a cyclic relation (the triple product rule),rp|670

:$\{left\; (\; frac\{partial\; x\}\{partial\; y\}\; ight\; )\}\_z\; \{left\; (\; frac\{partial\; y\}\{partial\; z\}\; ight\; )\}\_x\; \{left\; (\; frac\{partial\; z\}\{partial\; x\}\; ight\; )\}\_y\; =\; -1$.

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

:$\{left\; (\; frac\{partial\; y\}\{partial\; x\}\; ight\; )\}\_z\; =\; -\; frac\; \{\; \{left\; (\; frac\{partial\; z\}\{partial\; x\}\; ight\; )\}\_y\; \}\{\; \{left\; (\; frac\{partial\; z\}\{partial\; y\}\; ight\; )\}\_x\; \}$.

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 quantites are exact differentials. Then by the chain rule

$(1)~~~~~\; dz\; =\; left(frac\{partial\; z\}\{partial\; x\}\; ight)\_y\; dx+\; left(frac\{partial\; z\}\{partial\; y\}\; ight)\_x\; dy\; =\; left(frac\{partial\; z\}\{partial\; u\}\; ight)\_v\; du\; +left(frac\{partial\; z\}\{partial\; v\}\; ight)\_u\; dv$

but also by the chain rule:

$(2)~~~~~\; dx\; =\; left(frac\{partial\; x\}\{partial\; u\}\; ight)\_v\; du\; +left(frac\{partial\; x\}\{partial\; v\}\; ight)\_u\; dv$

and

$(3)~~~~~\; dy=\; left(frac\{partial\; y\}\{partial\; u\}\; ight)\_v\; du\; +left(frac\{partial\; y\}\{partial\; v\}\; ight)\_u\; dv$

so that:

$(4)~~~~~\; dz\; =\; left\; [\; left(frac\{partial\; z\}\{partial\; x\}\; ight)\_y\; left(frac\{partial\; x\}\{partial\; u\}\; ight)\_v\; +\; left(frac\{partial\; z\}\{partial\; y\}\; ight)\_x\; left(frac\{partial\; y\}\{partial\; u\}\; ight)\_v\; ight]\; du$

:::$+\; left\; [\; left(frac\{partial\; z\}\{partial\; x\}\; ight)\_y\; left(frac\{partial\; x\}\{partial\; v\}\; ight)\_u\; +\; left(frac\{partial\; z\}\{partial\; y\}\; ight)\_x\; left(frac\{partial\; y\}\{partial\; v\}\; ight)\_u\; ight]\; dv$

which implies that:

$(5)~~~~~\; left(frac\{partial\; z\}\{partial\; u\}\; ight)\_v\; =\; left(frac\{partial\; z\}\{partial\; x\}\; ight)\_y\; left(frac\{partial\; x\}\{partial\; u\}\; ight)\_v\; +\; left(frac\{partial\; z\}\{partial\; y\}\; ight)\_x\; left(frac\{partial\; y\}\{partial\; u\}\; ight)\_v$

Letting $v=y$ gives:

$(6)~~~~~\; left(frac\{partial\; z\}\{partial\; u\}\; ight)\_y\; =\; left(frac\{partial\; z\}\{partial\; x\}\; ight)\_y\; left(frac\{partial\; x\}\{partial\; u\}\; ight)\_y$

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

$(7)~~~~~\; left(frac\{partial\; z\}\{partial\; y\}\; ight)\_x\; =\; -\; left(frac\{partial\; z\}\{partial\; x\}\; ight)\_y\; left(frac\{partial\; x\}\{partial\; y\}\; ight)\_z$

using ($partial\; a/partial\; b)\_c\; =\; 1/(partialb/partial\; a)\_c$ gives the triple product rule:

$(8)~~~~~\; left(frac\{partial\; z\}\{partial\; x\}\; ight)\_y\; left(frac\{partial\; x\}\{partial\; y\}\; ight)\_z\; left(frac\{partial\; y\}\{partial\; z\}\; ight)\_x\; =-1$

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

* [http://mathworld.wolfram.com/InexactDifferential.html Inexact Differential] – from Wolfram MathWorld
* [http://www.chem.arizona.edu/~salzmanr/480a/480ants/e&idiff/e&idiff.html Exact and Inexact Differentials] – University of Arizona
* [http://farside.ph.utexas.edu/teaching/sm1/lectures/node36.html Exact and Inexact Differentials] – University of Texas
* [http://mathworld.wolfram.com/ExactDifferential.html Exact Differential] – from Wolfram MathWorld