(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

In a volume V, enclosed by a surface S, the vector fieldsXandYsatisfy the coupled equations

∇×∇×X=X+Y

∇×∇×Y=Y−X

If the values of ∇×Xand ∇×Yare given on S, show thatXandYare unique in V.

2. Relevant equations

∇.(A×B)=B.(∇×A)−A.(∇×B)

∇×(A×B)=∇(∇.A)−∇^{2}A

3. The attempt at a solution

I am assuming that I need to show that ∇^{2}Xand ∇^{2}Yare equal to zero to satisfy the uniqueness theorem for Poisson's equation. But am unsure of a good way to get there, so before I write my scribbles if someone could point me in the right direction it would be great.

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# Homework Help: Prove the functions are unique in a volume, vector calculus problem

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