I have been trying to prove this equivalence but I could not manage to do it. I have used all kinds of identities with no success.
I want to prove that, being $\mathbf n$ a unit surface normal vector, the expression
$$ \nabla \times \mathbf A = \mathbf k \times \mathbf n $$
is equivalent to
$$ \frac{\partial \mathbf A }{\partial n} =-\mathbf k $$
Note that I am not sure if the equivalence actually holds.
Thanks in advance.
EDIT: As @Rahul has proved with a counterexample the equivalence does not hold. Rafa has found that his proof is flawed because the incorrent elimination of the Levi-Civita symbol. I will try to understand things better and keep posting.