A couple of recent questions about antiderivatives reminded me of the following, which I can't recall seeing tackled explicitly anywhere to my satisfaction and that I sketched to an ambitious calculus student long ago. I hope it is not completely trivial...

Consider $f: \mathbb{R} \rightarrow \mathbb{R}$ nice. An antiderivative $F$ of $f$ is defined by $f \ dx = dF$. Let $\Omega = [a,b]$. The Stokes theorem is basically that $d$ and $\partial$ are adjoint w/r/t the standard pairing (a/k/a the integral), viz. $\langle dF,\Omega \rangle = \langle F, \partial \Omega \rangle$.

This is obviously just an overly fancy formulation of the fundamental theorem of calculus (*à la* Spivak, for example). But its generality is a virtue in motivating the following question (which also betrays my long-decayed knowledge of the most trivial bits of algebraic topology): **is there a general formulation of the notion of antiderivative that incorporates basic information about de Rham (co)homology?** As a followup, is it necessary to bring bundles into the picture?