How do I reexpress the equation $\nabla \times (\nabla \times gs)) \times (\nabla \times \nabla(f\nabla \cdot t))$?

Question

How do I go about reexpressing

$\nabla \times (\nabla \times bs)) \times (\nabla \times \nabla(c\nabla \cdot t))$

where s and t are vector properties and b and c are scalar.

I don't know where to even start on this one, I'm just asking if anyone can give me some hints/point me in the right direction. I've looked at a lot of grad/div/curl combination rules but it's quite overwhelming for a beginner

Thank you

Answer 1

You just need to know that for any scalar function $f$: $$ \nabla \times \nabla f = {\rm curl}\, {\rm grad}\, f = 0$$ That means that $$ \nabla \times \nabla(c \nabla\cdot t) = 0$$ $$ \big((\nabla \times (\nabla \times bs)\big) \times \big(\nabla \times \nabla(c \nabla\cdot t)\big) = \big((\nabla \times (\nabla \times bs)\big) \times 0 = 0$$