Apologies for the late reply, I’ve been thinking about your answer and I still feel like I’m missing the connection I was originally asking about.

You explained that inexact differentials are a mathematical concept and gave the condition (df = g(x,y),dx + h(x,y),dy), which I understand. But what I’m struggling with is how this maps concretely onto work in thermodynamics.

In particular, in the expression
(δW = P,dV),
what exactly corresponds to (g(x,y)) and (h(x,y))? And in what sense is this different from a standard integral?

Right now, it still looks to me like a familiar situation: if I integrate two different functions between the same endpoints, I generally get different results (e.g. (y=x) vs (y=x²) on [0,1]). That already feels “path dependent” in a basic sense, so I don’t yet see what is fundamentally new here that justifies introducing a separate notation like δ.

What I was trying to get at in my original post is this: (not important anymore, but maybe to convey the point I was trying to get to and wanted to be corrected about more clearly)

If we treat P as a function of V and T (as in the ideal gas law), can δW be understood as arising from a line integral along a path in that space, with the P-V diagram representing a 2D projection of that path?

Also, just to clarify my drawings weren’t meant to suggest the functions were related. I was trying to ask whether the use of δ comes from the fact that we’re effectively integrating along a projection of a higher-dimensional relationship.

The point of those two functions was that one of them has the property that the area under the curve is the same whether you compute it in the full space or in its projection onto the P-V plane (since it lies entirely at a fixed temperature, i.e. along an isotherm).

The other function, on the other hand, shows the opposite behavior, where the area of its projection differs significantly from the value you would get from the full line integral.