Abstract
<p>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.</p>
Highlights
In fluid mechanics and electrodynamics one is often interested in the following questions: Q1
A simple proof is given for the explicit formula which allows one to recover a C2−smooth vector field A = A(x) in R3, decaying at infinity, from the knowledge of its ∇ × A and ∇ · A
The representation of A as a sum of the gradient field and a divergence-free vector fields is derived from this formula
Summary
In fluid mechanics and electrodynamics one is often interested in the following questions: Q1. Let A(x), x ∈ R3, be a twice differentiable in R3 vector field vanishing at infinity together with its two derivatives. Given ∇ × A and ∇ · A, can one recover A(x) uniquely? Can one give an explicit formula for A(x)?. Can one find a scalar field u = u(x) and a divergence-free vector field B(x), ∇ · B = 0, such that. These questions were widely discussed in the literature, for example, in [1] - [3]. Our aim is to give a simple answer to these questions.
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