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