Representation of vector fields

TL;DR

This paper provides an explicit formula to reconstruct a smooth vector field in three-dimensional space from its divergence and curl, including extensions to bounded domains, enhancing understanding of vector field representations.

Contribution

It introduces a simple proof and explicit formula for recovering vector fields from divergence and curl, with extensions to bounded domains, advancing theoretical tools in vector calculus.

Findings

Explicit formula for vector field reconstruction from divergence and curl

Representation of vector fields as sum of gradient and divergence-free components

Extensions to bounded smooth domains

Abstract

A simple proof is given for the explicit formula which allows one to recover a $C^2-$smooth vector field $A=A(x)$ in $\mathbb{R}^3$, decaying at infinity, from the knowledge of its $\nabla \times A$ and $\nabla \cdot A$. The representation of $A$ as a sum of the gradient field and a divergence-free vector fields is derived from this formula. Similar results are obtained for a vector field in a bounded $C^2-$smooth domain.