Answer
$\phi(x, y)=\lt \dfrac{1}{2} \sqrt {\dfrac{ y}{ x}}, \dfrac{1}{2} \sqrt {\dfrac{x}{ y}} \gt$
Work Step by Step
Our aim is to compute the gradient vector field.
We are given that $\phi(x, y)=\lt \sqrt {xy} \gt$
or, $=\lt \dfrac{\partial (\sqrt {xy} ) }{\partial x}, \dfrac{\partial ( \sqrt {xy} ) }{\partial y}\gt$
or, $=\lt y^{1/2} \times \dfrac{\partial (\sqrt x) }{\partial x}, x^{1/2} \times \dfrac{\partial (\sqrt y) }{\partial y} \gt$
Thus, our required result is: $\phi(x, y)=\lt \dfrac{1}{2} \sqrt {\dfrac{ y}{ x}}, \dfrac{1}{2} \sqrt {\dfrac{x}{ y}} \gt$