Answer
$div\left( {\bf{F}} \right) = - 6\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right) + \left( {4{x^2} + 4{y^2} + 4{z^2}} \right)\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right)$
$curl\left( {\bf{F}} \right) = {\bf{0}}$
Work Step by Step
We have ${\bf{F}} = \left( {{F_1},{F_2},{F_3}} \right) = \nabla \left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right)$.
So,
${\bf{F}} = \left( {{F_1},{F_2},{F_3}} \right) = {\bf{i}}\frac{\partial }{{\partial x}}\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right) + {\bf{j}}\frac{\partial }{{\partial y}}\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right) + {\bf{k}}\frac{\partial }{{\partial z}}\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right)$
${\bf{F}} = \left( {{F_1},{F_2},{F_3}} \right)$
$ = - 2x\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right){\bf{i}}$
$ - 2y\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right){\bf{j}}$
$ - 2z\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right){\bf{k}}$
1. $div\left( {\bf{F}} \right) = \frac{{\partial {F_1}}}{{\partial x}} + \frac{{\partial {F_2}}}{{\partial y}} + \frac{{\partial {F_3}}}{{\partial z}}$
$div\left( {\bf{F}} \right) = - 2\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right) + 4{x^2}\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right)$
$ - 2\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right) + 4{y^2}\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right)$
$ - 2\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right) + 4{z^2}\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right)$
$div\left( {\bf{F}} \right) = - 6\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right) + \left( {4{x^2} + 4{y^2} + 4{z^2}} \right)\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right)$
2. $curl\left( {\bf{F}} \right) = \left| {\begin{array}{*{20}{c}}
{\bf{i}}&{\bf{j}}&{\bf{k}}\\
{\frac{\partial }{{\partial x}}}&{\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial z}}}\\
{ - 2x\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right)}&{ - 2y\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right)}&{ - 2z\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right)}
\end{array}} \right|$
$curl\left( {\bf{F}} \right) = \left( {4yz\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right) - 4yz\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right)} \right){\bf{i}}$
$ - \left( {4xz\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right) - 4xz\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right)} \right){\bf{j}}$
$ + \left( {4xy\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right) - 4xy\left( {{{\rm{e}}^{ - {x^2} - {y^2} - {z^2}}}} \right)} \right){\bf{k}}$
$curl\left( {\bf{F}} \right) = {\bf{0}}$
