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