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@article{Edalat:2012:10.1016/j.ic.2012.11.006,
author = {Edalat, A and Lieutier, A and Pattinson, D},
doi = {10.1016/j.ic.2012.11.006},
journal = {Informarion and Computation},
pages = {22--45},
title = {A Computational Model for Multi-Variable Differential Calculus},
url = {http://dx.doi.org/10.1016/j.ic.2012.11.006},
volume = {224},
year = {2012}
}

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TY  - JOUR
AB  - We develop a domain-theoretic computational model for multi-variabledifferential calculus, which for the first time gives rise to datatypes 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 asub-domain of the product of two domains by tupling together consistent information aboutlocally Lipschitz functions and theirdifferential properties as given by their L-derivative or equivalently Clarke gradient, which has values given by non-empty, convex and compact subsets of $\R^n$. To obtain a computationally practical framework, the derivative information is approximated by the best fit compact hyper-rectangles in $\R^n$. 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 rationalbasis elements of the domain, implying that the domain can be equipped with aneffective structure and giving a computable framework formulti-variable differential calculus.We also develop a domain-theoretic, interval-valued, notion of line integraland 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 $C^1$ path contains zero. In the case that the derivative information is given interms 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 $C^1$ path contains zero.  This gives a domain-theoretic extension of the fundamental theorem of path integration. Finally, weconstruct the least and the greatest piecewise linear functionst
AU  - Edalat,A
AU  - Lieutier,A
AU  - Pattinson,D
DO  - 10.1016/j.ic.2012.11.006
EP  - 45
PY  - 2012///
SN  - 0890-5401
SP  - 22
TI  - A Computational Model for Multi-Variable Differential Calculus
T2  - Informarion and Computation
UR  - http://dx.doi.org/10.1016/j.ic.2012.11.006
UR  - http://hdl.handle.net/10044/1/14153
VL  - 224
ER  -

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