Answer
$3y\textbf{k}$
Work Step by Step
curl $\textbf{F}=\nabla\times\textbf{F}=\nabla\times\langle x^{2}-y^{2},xy,z\rangle$
$=(\frac{\partial (z)}{\partial y}-\frac{\partial (xy)}{\partial z})\textbf{i}+(\frac{\partial(x^{2}-y^{2})}{\partial z}-\frac{\partial(z)}{\partial x})\textbf{j}+(\frac{\partial(xy)}{\partial x}-\frac{\partial(x^{2}-y^{2})}{\partial y})\textbf{k}$
$=(0-0)\textbf{i}+(0-0)\textbf{j}+(y-(0-2y))\textbf{k}$
$=3y\textbf{k}$
