I am working through the following problem, but finding it hard to know where to go.
Using the Divergence theorem and the following identities
$\nabla .(A \times B) = B.(\nabla \times A) - A.(\nabla \times B)$
$\nabla \times (\nabla \times A) = \nabla (\nabla . A) - \nabla ^2 A$
In a volume V, enclosed by a surface S, the vector fields X and Y satisfy the coupled equations
$\nabla \times \nabla \times X = X+Y$
$\nabla \times \nabla \times Y = Y-X$
If the values of $\nabla \times X$ and $\nabla \times Y$ are given on S, show that X and Y are unique in V.
I am assuming that I need to show that $\nabla ^2 X$ and $\nabla ^2 Y$ are equal to zero and that X and Y are zero on S 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 write direction it would be great.
Any help, pointing in the right direction would be very helpful.
EDIT: Fixed the second expression, original didn't make sense.