Integrating factor lemma

Sufficient conditions for a non-exact 1-form M dx + N dy to become exact after multiplication by a scalar integrating factor; used for entropy/temperature as an integrating factor for heat.

Integrating factor lemma

Statement

Let $D\subset\mathbb R^2$ be an open domain and let $M,N\in C^1(D)$ . Consider the $1$ -form

\omega = M(x,y)\,dx + N(x,y)\,dy.

A nonvanishing function $\mu$ on $D$ is an integrating factor for $\omega$ if the rescaled form $\mu\,\omega$ is exact (equivalently, $\mu\,\omega=dF$ for some potential $F$ ).

A common sufficient criterion (two-variable, one-function dependence)

Assume $N\neq 0$ on $D$ and define

\Phi(x,y)=\frac{1}{N(x,y)}\left(\frac{\partial M}{\partial y}-\frac{\partial N}{\partial x}\right).

If $\Phi(x,y)$ depends only on $x$ , i.e. $\Phi(x,y)=f(x)$ , then

\mu(x)=\exp\!\left(\int^x f(s)\,ds\right)

is an integrating factor on $D$ .

Similarly, if $M\neq 0$ on $D$ and

\Psi(x,y)=\frac{1}{M(x,y)}\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)

depends only on $y$ , i.e. $\Psi(x,y)=g(y)$ , then

\mu(y)=\exp\!\left(\int^y g(t)\,dt\right)

is an integrating factor.

Key hypotheses

$M,N$ are $C^1$ on $D$ .
$\mu$ is nonzero on $D$ (so multiplication does not change the set of admissible directions).
For the explicit formulas above, the indicated dependence restriction holds (e.g. $\Phi(x,y)=f(x)$ ).

Key conclusions

Under the criterion, $\mu\,\omega$ satisfies the exact differential criterion , hence $\mu\,\omega=dF$ for some state function $F$ .
The integrating factor is determined (up to an overall multiplicative constant) by a one-dimensional integral.