How can the addition of two(or more) inexact differential give an exact differential? Moreover, if an exact differential represents a linear map, what does an inexact differential represent(analogously)?
Lets take the 2-D Case.
We know Green's Theorem: $$ \oint\limits_C P dx+Q dy= \iint\limits_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dx \ dy $$
In a particular resource(Richard Fitzpatrick, Thermodynamic notes) it was mentioned that if $\frac{\partial Q}{\partial x} \ne \frac{\partial P}{\partial y}$, then the differential is not exact.
This is a sufficient condition. However, is it a necessary condition? What is the necessary and sufficient condition?