Answer
$F=\lt -x e^{-x^2-y^2-z^2},-y e^{-x^2-y^2-z^2}, -z e^{-x^2-y^2-z^2} \gt$
Work Step by Step
The vector field $F$ can be computed as: $F=\lt \dfrac{\partial \phi}{\partial x}, \dfrac{\partial \phi}{\partial y}, \dfrac{\partial \phi}{\partial z}\gt$
Here, we have $\phi (x, y,z)=\dfrac{1}{2} e^{-x^2-y^2-z^2}$
Thus,our required vector field $F$ is:
$F=\lt \dfrac{\partial }{\partial x} (\dfrac{1}{2} e^{-x^2-y^2-z^2}), \dfrac{\partial}{\partial y} (\dfrac{1}{2} e^{-x^2-y^2-z^2}), \dfrac{\partial}{\partial z} (\dfrac{1}{2} e^{-x^2-y^2-z^2})\gt$
or, $F=\lt -x e^{-x^2-y^2-z^2},-y e^{-x^2-y^2-z^2}, -z e^{-x^2-y^2-z^2} \gt$
