Domain Theory and Differential Calculus
We introduce domain theory in differential calculus. Based on a new
structure in domain theory, we define the derivative of a Scott
continuous function on the domain of intervals, which is itself a
Scott continuous function. This leads to a domain-theoretic
generalization of the fundamental theorem of calculus. We then
construct a domain for differentiable real valued functions of a real
variable. The classical C^1 functions, equipped with their C^1 norm,
is embedded into the set of maximal elements of this domain, which is
a countably based bounded complete continuous domain. The construction
can be generalized to C^k and C^infty functions and to real-valued
functions of several variables. It can also be extended to analytic
functions. As an immediate application, we present a domain-theoretic
generalization of Picard's theorem, which provides a data type for
solving differential equations.
Prof Edalat will also be speaking at the Pure and Applied Logic
Colloquium on Thursday, 28 September, at 5409 Wean Hall, 4:30 PM
(refreshments at 3:30) on the topic "A Data Type for Solid Modeling
and Computational Geometry". [Please note: owing to a misunderstanding
the POP and PAL talks were originallly scheduled in the other order.
The talks are related, but the first on Thursday will be more
introductory.] For appointments with Edalat on Thursday and Friday,
please contact Phyllis Pomerantz , x8-7897.
September 29, 2000