Like as is written in the title, I'm trying to solve,
$\mathbf J \times \mathbf B = \mathbf C$
$\mathbf J = \nabla \times \mathbf B$
for $\mathbf J$, in terms of $\mathbf B$ and $\mathbf C$ if possible. The answer can be a differential equation, I just want $\mathbf J$ on one side. I've done some stuff with $\mathbf B$:
$$ (\nabla \times \mathbf B) \times \mathbf B = (\mathbf B \cdot \nabla) \mathbf B -\nabla (B^2/2) = \mathbf C $$
If I take the curl of this I can get to:
$$ \nabla \times (\mathbf B \cdot \nabla ) \mathbf B = \nabla \times \mathbf C$$
But I don't know how to deal with the left term... Any help?