**Author:**

Kalanov, Temur Z.

**Category:**

Research Papers

**Sub-Category:**

Mathematics and Applied Mathematics

**Date Published:**

December 4, 2019

**Keywords:**

general mathematics, pure mathematics, philosophy of mathematics, methodology of mathematics, general applied mathematics, dimensional analysis, Euclidean and projective geometries, vector, vector calculus, vector fields, history of mathematics

**Abstract:**

The critical analysis of the foundations of vector calculus and classical electrodynamics is proposed. Methodological basis of the analysis is the unity of formal logic and rational dialectics. The main results arethefollowingstatements: (1) a vector is a property of the motion and of the interaction of material objects, i.e. the concept of a vector is the concept of a physical property. Therefore, the concept of a vector is a general and abstract concept;(2) a vector is depicted in the form of an arrow (i.e., “straight-line segment with arrowhead”) in a real (material) coordinate system. A vector drawn (depicted) in a coordinate system does not have the measure “meter”. Therefore, a vector is a pseudo-geometric figure in a coordinate system. A vector is an imaginary (fictitious) geometric figure; (3) geometrical constructions containingvectors (as pseudo-geometric figures) and vector operations in a coordinate system are fictitious actions;(4) the scalar and vector products of vectors represent absurd because vectors (as abstract concepts, as fictional geometric figures that have different measures) cannot intersect at the material point of the coordinate system;(5) the concepts of gradient, divergence, and rotor as the basic concepts of vector analysis are a consequence of the main mathematical error in the foundations of differential and integral calculus. .../...

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