Answer
$F(x,y)=\dfrac{-xi-yj}{(x^2+y^2)^{3/2}}$
Work Step by Step
The gradient field of $f(x,y,z)$ can be expressed as:
$\nabla f(x,y,z) =\dfrac{\partial (f(x,y,z))}{\partial x} i+\dfrac{\partial (f(x,y,z))}{\partial y} j+\dfrac{\partial (f(x,y,z))}{\partial z}k $
The vector field is: $F(x,y)=-xi-yj$
and $|F(x,y)|=\sqrt {x^2+y^2}$
The vector field $F(x,y)=-xi-yj$ must have a magnitude of $\dfrac{1}{x^2+y^2}$ .
So, we need to divide the vector field
$F(x,y)=-xi-yj$ by $(x^2+y^2)^{3/2}$
So, $F(x,y)=\dfrac{-xi-yj}{(x^2+y^2)^{3/2}}$
