Differentiation in Bundles with a Hyperspace Base

TL;DR

This paper develops a method to regularize the differentiation operation within the framework of extrafunctions, using fiber bundles over hyperspaces of differential vector spaces and algebras, to extend calculus operations.

Contribution

It introduces a novel regularization technique for differentiation in extrafunctions using fiber bundle constructions over hyperspaces.

Findings

Regularization method for differentiation in extrafunctions.

Extension of differentiation operation to extrafunctions.

Application of fiber bundles over hyperspaces in calculus.

Abstract

It is possible to perform some operations with extrafunctions applying these operations separately to each coordinate. Operations performed in this manner are called regular. It is proved that it is possible to extend several operations with functions to regular operations with extrafunctions. Examples of such operations are addition of real functions and multiplication of real functions by numbers. However, there are operations with functions the extension of which by coordinates does not work because their application is not invariant with respect to representations of extrafunctions. One of such operations is differentiation, which is important for calculus, differential equations and many applications. In this work, a method of regularization of irregular operations is developed and applied to differentiation. The main constructions are put together in the context of fiber bundles over hyperspaces of differential vector spaces and differential algebras.