Answer
Please see the figure attached.
Work Step by Step
We use a computer algebra system to produce a contour plot of $f\left( {x,y} \right) = {x^2} - 3xy + y - {y^2}$ together with its gradient vector field on the domain $\left[ { - 4 \times 4} \right] \times \left[ { - 4 \times 4} \right]$. The result is shown in the figure attached.
