The vector field F=[({{x}, {-y}})] is not the gradient field of any function f(x,y).  We can plot representative vectors here as well as representative flow lines.  Here, the vector field has divergence 0 everywhere.  What does this mean?

VectorPlot[{x,-y},{x,-3,3},{y,-3,3}]
StreamPlot[{x,-y},{x,-3,3},{y,-3,3}]

f(x,y)=(x^2)+(y^2) has the gradient field ∇f=[({{2x}, {2y}})] .  We can plot representative vectors here as well as representative flow lines.  Note that this vector field is expanding since its divergence is 4 everywhere.  Can you see this?

VectorPlot[{2x,2y},{x,-3,3},{y,-3,3}]
StreamPlot[{2x,2y},{x,-3,3},{y,-3,3}]

The vector field G=[({{(x^3)}, {x sin(x y) }})] is shown below.  Is it expanding, contracting, or neither, on the plane?

VectorPlot[{x^3,x*Sin[x*y]},{x,-2,2},{y,-2,2}]
StreamPlot[{x^3,x*Sin[x*y]},{x,-2,2},{y,-2,2}]