I am to show
int{A . grad(phi)}dV=0 where A is a solinoidal field and normal component of A at the boundary vanishes.
I expressed A as curl F and suspect that the (curl F).(grad phi)=0
So,I am done.Is it correct?
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You aren't supposed to show the integrand vanishes. In general, it doesn't. You need to find something whose divergence is A.grad(phi) and then argue that integrating that something over the boundary vanishes.
I hyope i got it.I took the divergence of (phi A),and what comes out is that the given integral transforms to Int{phi A}.dS where I can apply the additional condition