Inverse Vector Operators

TL;DR

This paper introduces inverse vector operators in orthogonal curvilinear coordinates to facilitate finding parent vector or scalar functions from known curl, divergence, or gradient, addressing a common challenge in physics.

Contribution

It defines inverse curl, divergence, and gradient operators in orthogonal curvilinear coordinates, providing a new mathematical tool for physics and engineering applications.

Findings

Defined inverse curl, divergence, and gradient operators.

Operators are applicable in orthogonal curvilinear coordinate systems.

Facilitates solving inverse vector calculus problems.

Abstract

In different branches of physics, we frequently deal with vector del operator ($\vec{\nabla}$). This del operator is generally used to find curl or divergence of a vector function or gradient of a scalar function. In many important cases, we need to know the parent vector whose curl or divergence is known or require to find the parent scalar function whose gradient is known. But the task is not very easy, especially in case of finding vector potential whose curl is known. Here,'inverse curl', 'inverse divergence' and 'inverse gradient' operators are defined to solve those problems easily. All operators are defined in the orthogonal curvilinear co-ordinate system.