Suppose $\vec{F}(x,y,z)=\vec{f}(x,y)\exp(ikz)$ and $\vec{F}$ satisfies the equations $\nabla \cdot \Re{\vec{F}}=0$ where $\Re{\vec{F}}$ is the real part of $\vec{F}$. It also satisfies $\nabla \times \Re\vec{F}=\Re\vec{G}$ for $\Re\vec{G}=\vec{g}(x,y)\exp(ikz)$
If I want to express these equations in terms of $\vec{f},\vec{g},k$, Is it possible to simplify do better than $\Re\left[\left({\partial \vec{f}\over \partial x}+{\partial \vec{f}\over \partial y}+ik\vec{f}\right)\exp(ikz)\right]=0$ for the first one? As for the second one I really have no idea how to express it in the desired form except trivially substituting the entire expression of $\vec{F}$ into the equation.
Could anyone help?
Added: $F,f$ are both 3-D vectors the $x,y,z$ are just their arguments.
Thanks.