A computational model for multi-variable differential calculus

Abstract

We develop a domain-theoretic computational model for multi-variable differential calculus, which for the first time gives rise to data types for piecewise differentiable or more generally Lipschitz functions, by constructing an effectively given continuous Scott domain for real-valued Lipschitz functions on finite dimensional Euclidean spaces. The model for real-valued Lipschitz functions of n variables is built as a sub-domain of the product of two domains by tupling together consistent information about locally Lipschitz functions and their differential properties as given by their L-derivative or equivalently Clarke gradient, which has values given by non-empty, convex and compact subsets of Rn. To obtain a computationally practical framework, the derivative information is approximated by the best fit compact hyper-rectangles in Rn. In this case, we show that consistency of the function and derivative information can be decided by reducing it to a linear programming problem. This provides an algorithm to check consistency on the rational basis elements of the domain, implying that the domain can be equipped with an effective structure and giving a computable framework for multi-variable differential calculus. We also develop a domain-theoretic, interval-valued, notion of line integral and show that if a Scott continuous function, representing a non-empty, convex and compact valued vector field, is integrable, then its interval-valued integral over any closed piecewise C1 path contains zero. In the case that the derivative information is given in terms of compact hyper-rectangles, we use techniques from the theory of minimal surfaces to deduce the converse result: a hyper-rectangular valued vector field is integrable if its interval-valued line integral over any piecewise C1 path contains zero. This gives a domain-theoretic extension of the fundamental theorem of path integration. Finally, we construct the least and the greatest piecewise linear functions obtained from a pair of function and hyper-rectangular derivative information. When the pair is consistent, this provides the least and greatest maps to witness consistency.