The preprint: "Differential Restriction Categories" is now available from http://pages.cpsc.ucalgary.ca/~gscruttw/publications.html Authors: Robin Cockett, Geoff Cruttwell, and Jonathan Gallagher The paper takes the setting for differential calculus described by Blute, Cockett, and Seely (TAC, Vol22, 209) and shows how to extend it to partially defined maps. A basic model is the category of smooth maps defined on open subsets of R^n, but the paper also describes another class of important models. To achieve this extension, the paper combines Cartesian differential categories with restriction categories. This involves describing, initially, what "left additive" structure is for restriction categories and what it means for a partial map to be "additive". Once this is established can one add, and axiomatize, a differential operator much as for Cartesian differential categories. The paper presents -- in detail -- the differential restriction category of rational functions over an arbitrary commutative ring. The example is, from the point of view of algebraic geometry, relatively standard. However, the development allows the structure to be viewed from a more unified categorical perspective than was hitherto possible. Differentiation in this example is the usual formal differentiation of rational functions with partiality arising from "poles". Much of the technical difficulty arises in giving an explicit description of the underlying restriction structure. Somewhat surprisingly this works best at the level of commutative rigs: negatives are only needed for differentiation of fractions. The paper ends by showing that differential restriction structure is closed to two important constructions: the join completion and the classical completion. The former is important as it is the jumping off point for examining differential structure in manifold categories, and so for describing general "tangent bundles". The latter is somewhat surprising: it shows that one can always reason classically about smooth maps; however, it also means one must allow functions defined on domains consisting of a single "point" to be differentiable! - Geoff Cruttwell [For admin and other information see: http://www.mta.ca/~cat-dist/ ]