Integral Property of Laplace Operator

TL;DR

This paper establishes integral relations involving the Laplace operator, linking scalar and vector functions to electric fields, and demonstrates their consistency with classical electrostatics through generalized function theory.

Contribution

It introduces a novel integral property of the Laplace operator and applies it to derive electric field expressions within the framework of generalized functions.

Findings

Derived integral relations connecting functions and Laplace operator

Expressed electric fields in terms of charge gradients

Validated the approach with explicit electric field calculations

Abstract

Relations have been derived which establish connection between a scalar or a vector functions and the integral of Laplace operator of these functions (the integral property of Laplace operator). The integral property of Laplace operator was employed to obtain relations expressing the electric field in terms of the charge gradient. It is demonstrated that these relations define the same field which is described by the well known classical expressions for the electric field. It is shown that these proposed expressions represent actually a particular case of the general theory of generalized functions. This approach is illustrated by calculation of an electric field performed with the proposed formalism.