Answer
As an example: ${\bf{F}} = \left( {yz,xz,xy} \right)$
We have $div\left( {\bf{F}} \right) = 0$ and $curl\left( {\bf{F}} \right) = {\bf{0}}$
Work Step by Step
As an example: ${\bf{F}} = \left( {{F_1},{F_2},{F_3}} \right) = \left( {yz,xz,xy} \right)$.
1. $div\left( {\bf{F}} \right) = \frac{{\partial {F_1}}}{{\partial x}} + \frac{{\partial {F_2}}}{{\partial y}} + \frac{{\partial {F_3}}}{{\partial z}}$
$div\left( {\bf{F}} \right) = 0$
2. $curl\left( {\bf{F}} \right) = \left| {\begin{array}{*{20}{c}}
{\bf{i}}&{\bf{j}}&{\bf{k}}\\
{\frac{\partial }{{\partial x}}}&{\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial z}}}\\
{yz}&{xz}&{xy}
\end{array}} \right| = 0{\bf{i}} - 0{\bf{j}} + 0{\bf{k}}$
$curl\left( {\bf{F}} \right) = {\bf{0}}$
