Domain Theory and Multi-Variable Calculus

                     Abbas Edalat (Imperial College)
           (joint work with Andre Lieutier and Dirk Pattinson)

A domain is a partially ordered set equipped with a so-called Scott
topology and notions of completeness and approximation. An example is
the collection of the non-empty compact intervals of the real line
ordered by reverse inclusion. We introduce a domain-theoretic model
for multi-variable differential calculus, extending the existing
framework for functions of one variable. The domain-theoretic
derivative is shown to give the best hyper-rectangular approximation
to the generalized (Clarke's) gradient of a locally Lipschitz
function. We construct a domain for locally Lipschitz functions of
several variables, which also serves as a domain for piecewise
differentiable functions of several variables. This domain is
constructed by pairing consistent information about the locally
Lipschitz function and its differential properties. We then introduce
a domain-theoretic notion of path integral, which we use to establish
a main result of our work: a necessary and sufficient condition for an
interval-valued Scott continuous vector function to be integrable,
extending the classical Green's Theorem for a vector field to be a
gradient. The main result is to show that consistency is decidable on
the rational basis elements of the domain for locally Lipschitz
functions, giving an effective framework for multi-variable
differential calculus.