**Authors:** Yakov A. Iosilevskii

A concise rigorous axiomatic algebraico-functional theory of a real affine Euclidean space of any given dimension n>=1 (nDRAfES), which is an underlying discipline of differential and integral calculus and particularly of my recent theory of nD wave fields, presented in arXiv:1510.00328, is developed from an algebraic system, called an affine additive group (AAG). The latter consists of a certain underlying set of points, called an affine additive group manifold (AAGM), and of a certain commutative [abstract] additive group (CAG), called the adjoint group of the AAG, whose elements, called vectors, are related to pairs of points of AAGM by a binary surjection, satisfying the appropriate version of the Chasle, or triangle, law. An AAG is illustrated by an nD primitive (Bravais) affine lattice. When the CAG is successively supplemented by the appropriate additional attributes to become ultimately an nD real abstract vector Euclidean space (nDRAbVES), the AAG is automatically self-adjusted to all current metamorphoses of its adjoint CAG to become ultimately an nDRAfES, of which the above nDRAbVES is adjoint. Relative to its any orthonormal basis, the nDRAbVES, adjoint of the nDRAfES, is isomorphic to the nD real arithmetical vector Euclidean space (nDRArVES), whose underlying set consists of ordered n-tuples of real numbers, being coordinates of the respective abstract vectors of the underlying vector set of the nDRAbVES. A time continuum (TC) is a special interpretation of 1DRAfES. A real-valued functional form (FF) that is initially defined on a certain region of the direct product (DP) of a 1DRAfES and a nDRAfES can rigorously be mapped onto a certain real-valued FF defined on a certain region of the DP of the TC and nDRArVES and vice versa.

**Comments:** 68 Pages.

**Download:** **PDF**

[v1] 2016-04-06 07:25:43

[v2] 2016-05-02 05:47:55

**Unique-IP document downloads:** 56 times

Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful. *