On Transforming Functions of a Certain Dot-Product Gradient Operator

TL;DR

This paper presents a method to transform functions involving a specific dot-product gradient operator into a double integral form, facilitating the analysis of complex differential expressions and their Fourier transforms.

Contribution

It introduces a novel transformation of functions of a dot-product gradient operator into a double integral, simplifying the handling of complex differential expressions.

Findings

Transformation to double integral form is possible for functions of the dot-product gradient.

Inner integral corresponds to a Fourier transform of the operator function.

Method aids in simplifying one-dimensional differential problems.

Abstract

In this paper it is shown that a function of the constant dot product of the gradient operator acting on an arbitrary function can be transformed to a double three-dimensional integral. The inner one of them is a Fourier transform of the operator function. The result converted to one-dimensional problems is also useful in transforming complex differential expressions.