We approach the study of differentiable manifolds modeled on Banach spaces by means of Nonstandard Analysis. We stay inside the category of classical manifolds and using nonstandard analysis techniques, we present some new nonstandard characterizations for the tangent bundle, differentiable function, differential of a function, directional derivatives, etc. We establish some relations between o...

We present Nonstandard Analysis by three axioms: the Extension, Transfer and Saturation Principles in the framework of the superstructure of a given innnite set. We also present several applications of this axiomatic approach to point-set topology. Some of the topological topics such as the Hewitt realcompactiication and the nonstandard characterization of the sober spaces seem to be new in the...

The theorem prover Isabelle is used to formalise and reproduce some of the styles of reasoning used by Newton in his Principia. The Principia's reasoning is resolutely geometric in nature but contains \in nitesimal" elements and the presence of motion that take it beyond the traditional boundaries of Euclidean Geometry. These present di culties that prevent Newton's proofs from being mechanised...

This application of nonstandard analysis utilizes the notion of the highly-saturated enlargement. These nonstandard methods are applied to the theory of generalized functions (distributions) and demonstrates how such analysis clarifies many aspects of this theory. 1. Additional Modeling Concepts. In what follows, the basic notation and definitions are as they appear in [3]. However, except as m...