I'm curious, coming from a math scholar, what would be an inexact differential? I am not nearly good at math, but a mathematical function, applied to another function, how can that be inexact? For example 4+5, or 9x3, their results are exact, right? I suppose, 9 divided by 5, yeah that's not exact. But the differentials, I may be missing what would be inexact, so I'd like to see a theoretical inexact differential of a function.
I suppose, given that calculus was invented/discovered to describe real physical phenomena, that the derivatives have to likes "make sense." But I can't say for sure that requires the differential to be exact.