1. The problem statement, all variables and given/known data In a volume V, enclosed by a surface S, the vector fields X and Y satisfy the coupled equations ∇×∇×X=X+Y ∇×∇×Y=Y−X If the values of ∇×X and ∇×Y are given on S, show that X and Y are unique in V. 2. Relevant equations ∇.(A×B)=B.(∇×A)−A.(∇×B) ∇×(A×B)=∇(∇.A)−∇2A 3. The attempt at a solution I am assuming that I need to show that ∇2X and ∇2Y are 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.