  In this paper we present a new theory of calculus over $k$-dimensional
domains in a smooth $n$-manifold, unifying the discrete, exterior, and
continuum theories. The calculus begins at a single point and is extended to
chains of finitely many points by linearity, or superposition. It converges to
the smooth continuum with respect to a norm on the space of ``pointed chains,''
culminating in the chainlet complex. Through this complex, we discover a broad
theory of coordinate free, multivector analysis in smooth manifolds for which
both the classical Newtonian calculus and the Cartan exterior calculus become
special cases. The chainlet operators, products and integrals apply to both
symmetric and antisymmetric tensor cochains. As corollaries, we obtain the full
calculus on Euclidean space, cell complexes, bilayer structures (e.g., soap
films) and nonsmooth domains, with equal ease. The power comes from the
recently discovered prederivative and preintegral that are antecedent to the
Newtonian theory. These lead to new models for the continuum of space and time,
and permit analysis of domains that may not be locally Euclidean, or locally
connected, or with locally finite mass.
