Prove the following uniqueness theorem: If the current density J is specified throughout a volume ν, and either the potential A or the magnetic field B is specified on the surface S bounding ν , then the magnetic field itself is uniquely determined throughout ν . [Hint: First use the divergence theorem to show that
\int\{(\nabla \times U ) \cdot(\nabla \times V )- U \cdot[\nabla \times(\nabla \times V )]\} d \tau=\oint[ U \times(\nabla \times V )] \cdot d a ,
for arbitrary vector functions U and V.]