#math.fa

This paper lays the groundwork for an operator calculus based on a spectral pair ('B, O) where 'B is a complete locally convex topological vector space of "differential chains" and O is a rich algebra of continuous operators acting on 'B. The covariant, constructive theory of chains 'B dualizes to the contravariant, analytic theory of differential forms B which is the Fr\'echet space of smooth differential forms, each with a uniform bound on each of its directional derivatives. Discrete infinitesimal chains are dense in the predual space 'B, making the resulting calculus ideal for multiscale analysis. In the spirit of Whitney, chains come first.

We announce new primitive theorems of calculus underlying the classical integral theorems of calculus, as well as two new fundamental theorems of calculus for chains in a flow.