2.3: Mathematical Interlude - Exact and Inexact Differentials
( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand\Dalpha
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[1], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dbeta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[2], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dgamma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[3], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Ddelta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[4], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Depsilon
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[5], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dvarepsilon
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[6], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dzeta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[7], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Deta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[8], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dtheta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[9], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dvartheta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[10], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Diota
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[11], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dkappa
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[12], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dlambda
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[13], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Dvarpi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[14], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\DGamma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[15], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\DDelta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[16], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\DTheta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[17], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vmu
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[18], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vnu
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[19], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vxi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[20], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vom
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[21], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vpi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[22], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vvarpi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[23], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vrho
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[24], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vvarrho
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[25], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vsigma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[26], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vvarsigma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[27], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vtau
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[28], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vupsilon
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[29], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vphi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[30], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vvarphi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[31], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vchi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[32], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vpsi
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[33], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\Vomega
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[34], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\VGamma
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[35], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\VDelta
Callstack: at (Template:MathJaxArovas), /content/body/div/p[1]/span[36], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\newcommand\BI{\mib I}}
\)
\newcommand { M}
\newcommand { m}
}
\( \newcommand\tcb{\textcolor{blue}\)
\( \newcommand\tcr{\textcolor{red}\)
1$#1_$
\newcommand\SZ{\textsf Z}} \( \newcommand\kFd{k\ns_{\RF\dar}\)
\newcommand\mutB{\tilde\mu}\ns_\ssr{B}
\( \newcommand\xhihOZ
Callstack: at (Template:MathJaxArovas), /content/body/div/span[1], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
\( \newcommand\labar
Callstack: at (Template:MathJaxArovas), /content/body/div/span[2], line 1, column 1 at template() at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02:_Thermodynamics/2.03:_Mathematical_Interlude_-_Exact_and_Inexact_Differentials), /content/body/p[1]/span, line 1, column 23
The differential dF=\sum_{i=1}^k A\ns_i\,dx\ns_i \label{dFeqn} is called exact if there is a function F(x\ns_1,\ldots,x\ns_k) whose differential gives the right hand side of Equation \ref{dFeqn}. In this case, we have A\ns_i={\pz F\over\pz x\ns_i} \qquad\Longleftrightarrow\qquad {\pz A_i\over\pz x\ns_j} = {\pz A_j\over\pz x\ns_i} \quad \forall\ i,j\ . For exact differentials, the integral between fixed endpoints is path-independent: \int\limits_{\RA}^{\RB}\!\!dF = F(x^{\ssr{B}}_1,\ldots,x^{\ssr{B}}_k)-F(x^{\ssr{A}}_1,\ldots,x^{\ssr{A}}_k)\ , from which it follows that the integral of dF around any closed path must vanish: \oint\!dF=0\ .
When the cross derivatives are not identical, when \pz A\ns_i/\pz x\ns_j\ne \pz A\ns_j/\pz x\ns_i, the differential is inexact. In this case, the integral of dF is path dependent, and does not depend solely on the endpoints.

As an example, consider the differential dF=K\ns_1\,y\,dx + K\ns_2\,x\,dy\ . \label{dFe} Let’s evaluate the integral of dF, which is the work done, along each of the two paths in Fig. [work_path]: \begin{aligned} W^\ssr{(I)}&=K\ns_1\!\int\limits_{x\ns_\RA}^{x\nd_\RB}\!\!dx\>y\subA + K\ns_2\!\int\limits_{y\ns_\RA}^{y\nd_\RB}\!\!dy\>x\subB= K\ns_1\,y\subA \,(x\subB-x\subA) + K\ns_2\,x\subB\,(y\subB-y\subA )\\ W^\ssr{(II)}&=K\ns_1\!\int\limits_{x\ns_\RA}^{x\nd_\RB}\!\!dx\>y\subB + K\ns_2\!\int\limits_{y\ns_\RA}^{y\nd_\RB}\!\!dy\>x\subA = K\ns_1\,y\subB\,(x\subB-x\subA ) + K\ns_2\,x\subA \,(y\subB-y\subA )\ .\end{aligned} Note that in general W^\ssr{(I)}\ne W^\ssr{(II)}. Thus, if we start at point A, the kinetic energy at point B will depend on the path taken, since the work done is path-dependent.
The difference between the work done along the two paths is W^\ssr{(I)}-W^\ssr{(II)}=\oint\!dF=(K\ns_2-K\ns_1)\,(x\subB-x\subA)\,(y\subB-y\subA)\ . \label{Wdiff} Thus, we see that if K\ns_1=K\ns_2, the work is the same for the two paths. In fact, if K\ns_1=K\ns_2, the work would be path-independent, and would depend only on the endpoints. This is true for any path, and not just piecewise linear paths of the type depicted in Fig. [work_path]. Thus, if K\ns_1=K\ns_2, we are justified in using the notation dF for the differential in Equation [dFe]; explicitly, we then have F=K\ns_1\,xy. However, if K\ns_1\ne K\ns_2, the differential is inexact, and we will henceforth write \dbar F in such cases.