Einstein Summation Notation Interpretation A vector field is called irrotational if its curl is zero. A vector field is called solenoidal if its divergence is zero. If A and B are irrotational, prove that A $ \times $ B is solenoidal.
I'm having a hard time the proof equation that is required, and the steps that would go with it. I am defining V as a vector.
$ \nabla \times V = 0 $ = irrotational
$ \nabla \cdot V = 0 $ = solenoidal
$ \nabla \times A = 0 $
$ \nabla \times B = 0 $ 
so therefore, ($ \nabla \times A  $)+ ($ \nabla \times B) = \nabla \cdot (A \times B) $
would this be a correct setup? I'm having a hard time expanding this to E.S. form.
 A: This doesn't work; $(\nabla\times A)+(\nabla\times B)$ doesn't correspond to anything, and $\nabla\cdot(A\times B)$ doesn't expand to it (as noted by Henning Makholm in a comment, one of these is effectively a scalar and one effectively a vector).  Instead, you want a form of the triple product identity $A\cdot(B\times C) = B\cdot(C\times A) = C\cdot (A\times B)$ - but this is where you need to be at least a little careful, because $\nabla$ isn't 'really' a vector.
As for Einstein summation form, the most important piece to keep in mind is the form for the cross-product: if $A\times B = C$, then $C^i = \epsilon^i\ _{jk}A^jB^k$ where $\epsilon$ is the so-called Levi-Civita symbol which essentially represents the sign of the permutation of its coordinates (i.e., $\epsilon_{ijk}=0$ if any two of $i,j,k$ are pairwise equal, $\epsilon_{012}=\epsilon_{120}=\epsilon_{201}=1$, and $\epsilon_{021}=\epsilon_{210}=\epsilon_{102}=-1$).  Writing out the triple-product identity in terms of this notation should make it clear how it works, and then substituting in your hypotheses should show you how to draw your conclusion.
A: Note that your left side is a vector (which happens to be zero) and the right side is a scalar (which you are hoping to prove is zero) so they can't be equal .  The right side is $\varepsilon_{ijk}\frac d{dx_ i}A_jB_k$ while $\nabla \times A = 0$ is $\varepsilon_{ijk}\frac d{dx_i}A_j=0$ but those look a lot the same.  If you expand out the product you should be able to find a way to make use of the irrotational nature of $A$ and $B$.
