2.14: Appendix I- Integrating Factors

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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 $$e^{-L}\,\dbar W = {\pz U\ns_c\over \pz x} \,dx + {\pz U\ns_c\over \pz y} \,dy=dU\ns_c\ .$$