  This is a Research and Instructional Development Project from the U. S. Naval
Academy. In this monograph, the basic methods of nonstandard analysis for
n-dimensional Euclidean spaces are presented. Specific rules are deveoped and
these methods and rules are applied to rigorous integral and differential
modeling. The topics include Robinson infinitesimals, limited and infinite
numbers; convergence theory, continuity, *-transfer, internal definition,
hyprefinite summation, Riemann-Stieltjes integration over Jordan-measurable
regions; modeling with integrals via Infinite Sum Theorems; the nonstandard
integral modeling rules - the method of constants, the methods of max. and
min., and the hyperfinite sum; pure infinitesimal integral modeling via
geometric elements; microgeometry, line and surface integrals, gauge integrals,
fluxions and their relations to dynamic geometry. These rigorous results are
also applied to lengths of curves, areas and volumes for Jordan-measurable
regions, incompressible fluids, moments of inertia, electric field vectors,
force fields and very elementary physics. Also presented are rigorous
differential derivations for the partial differential equations for internal
heat transfer and the vibrating membrane. All results are rigorously
established and, mostly, all proofs appear in the appendix.
