Mathematical Physics

2608 Submissions

[2] viXra:2608.0093 [pdf] submitted on 2026-08-27 21:24:20

Divergence Verification and Doubts About the Theory of Limits in Calculus

Authors: Hongyuan Ye
Comments: 8 Pages.

In the two definitions of divergence，the flux-density divergence provides the physical definition of divergence, whereas the nabla-operator divergence serves as a mathematical tool for calculation and derivation. Using examples derived from Coulomb’s law of electrostatics, and based on the theory of limits, this paper reveals that the flux-density divergence and the nabla-operator divergence are non-equivalent. The differential form of Gauss’s flux theorem in Maxwell’s equations does not hold. Furthermore, keeping or discarding relatively higher-order infinitesimals yields different results of the flux-density divergence, which contradicts the theory of limits in calculus. This paper proposes the effective order infinitesimal theorem to enhance the theory of limits. In the expression of the flux-density divergence, the effective order of its infinitesimal variables is 3. However, during the mathematical derivation of the nabla-operator divergence from the flux-density divergence, only 2nd-order infinitesimal variables are retained. Gauss’s flux theorem in differential form does not hold, and the theory of limits in calculus exhibits deficiencies. Physics and mathematics are undergoing a revolutionary scientific transformation.
Category: Mathematical Physics

[1] viXra:2608.0044 [pdf] submitted on 2026-08-11 02:04:43

Gauss’s Flux Theorem in Differential Form Has no Definite Physical Significance

Authors: Hongyuan Ye
Comments: 13 Pages.

In classical electromagnetism, Gauss’s flux theorem in integral form states that the electric flux through a closed surface is proportional to the charge enclosed by that surface. Recent research points out that Gauss’s flux theorem in integral form applies to static electric fields but not to time-varying electric fields. This paper reveals that when Gauss’s flux, characterized by three-dimensional space, is compressed to a single point, the divergence of a field no longer retains its original meaning of the flux. Based on the divergence of the electric field at a point in space, Gauss's flux theorem in differential form contradicts Coulomb's law and violates the superposition principle of the electric fields. Therefore, the differential and integral forms of Gauss’s flux theorem are not equivalent. In conclusion，Gauss’s flux theorem in differential form does not hold in both static and time-varying electric fields, and has no definite mathematical and physical significance. Theoretical physics and mathematics will undergo a revolutionary scientific transformation.
Category: Mathematical Physics