2.14: Appendix I- Integrating Factors

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\( \newcommand\rok{\Br\ns_1\,,\,\ldots\,,\,\Br\ns_k}\)
\( \newcommand\xhiOZ{\xhi^\ssr{OZ}}\)
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Suppose we have an inexact differential \[\dbar W=A\ns_i\,dx\ns_i\ .\] Here I am adopting the ‘Einstein convention’ where we sum over repeated indices unless otherwise explicitly stated; \(A\ns_i\,dx\ns_i=\sum_i A\ns_i\,dx\ns_i\). An integrating factor  \(e^{L(\Vx)}\) is a function which, when divided into \(\dbar F\), yields an exact differential: \[dU=e^{-L}\,\dbar W = {\pz U\over \pz x\ns_i}\,dx\ns_i\ .\] Clearly we must have \[{\pz^2 U\over\pz x\ns_i\,\pz x\ns_j} ={\pz\over\pz x\ns_i}\,\big(e^{-L}\,A\ns_j\big) = {\pz\over\pz x\ns_j}\,\big(e^{-L}\,A\ns_i\big)\ .\] Applying the Leibniz rule and then multiplying by \(e^L\) yields \[{\pz A\ns_j\over\pz x\ns_i} - A\ns_j\,{\pz L\over\pz x\ns_i} = {\pz A\ns_i\over\pz x\ns_j} - A\ns_i\,{\pz L\over\pz x\ns_j} \ .\] If there are \(K\) independent variables \(\{x\ns_1,\ldots,x\ns_K\}\), then there are \(\half K(K-1)\) independent equations of the above form – one for each distinct \((i,j)\) pair. These equations can be written compactly as \[\Omega\ns_{ijk}\,{\pz L\over\pz x\ns_k} = F\ns_{ij}\ ,\] where \[\begin{aligned} \Omega\ns_{ijk} &= A\ns_j\,\delta\ns_{ik} - A\ns_i\,\delta\ns_{jk} \vph\\ F\ns_{ij}&={\pz A\ns_j\over\pz x\ns_i} - {\pz A\ns_i\over \pz x\ns_j}\ .\end{aligned}\] Note that \(F\ns_{ij}\) is antisymmetric, and resembles a field strength tensor, and that \(\Omega\ns_{ijk}=-\Omega\ns_{jik}\) is antisymmetric in the first two indices (but is not totally antisymmetric in all three).

Can we solve these \(\half K(K-1)\) coupled equations to find an integrating factor \(L\)? In general the answer is no. However, when \(K=2\) we can always find an integrating factor. To see why, let’s call \(x\equiv x\ns_1\) and \(y\equiv x\ns_2\). Consider now the ODE \[{dy\over dx} = -{A\ns_x(x,y)\over A\ns_y(x,y)}\ .\] This equation can be integrated to yield a one-parameter set of integral curves, indexed by an initial condition. The equation for these curves may be written as \(U\ns_c(x,y)=0\), where \(c\) labels the curves. Then along each curve we have \[\begin{split} 0={dU\ns_c\over dx}&={\pz U\ns_x\over\pz x} + {\pz U\ns_c\over\pz y}\,{dy\over dx}\vph\\ &={\pz U\ns_c\over \pz x} - {A\ns_x\over A\ns_y}\,{\pz U\ns_c\over\pz y}\ . \end{split}\] Thus, \[{\pz U\ns_c\over\pz x}\,A\ns_y = {\pz U\ns_c\over\pz y}\,A\ns_x \equiv e^{-L} A\ns_x\,A\ns_y\ .\] This equation defines the integrating factor \(L\,\): \[L=-\ln\!\bigg({1\over A\ns_x}\,{\pz U\ns_c\over\pz x}\bigg) = -\ln\!\bigg({1\over A\ns_y}\,{\pz U\ns_c\over\pz y}\bigg) \ .\] We now have that \[A\ns_x=e^{L}\,{\pz U\ns_c\over \pz x} \qquad,\qquad A\ns_y=e^{L}\,{\pz U\ns_c\over \pz y} \ ,\] and hence \[e^{-L}\,\dbar W = {\pz U\ns_c\over \pz x} \,dx + {\pz U\ns_c\over \pz y} \,dy=dU\ns_c\ .\]