Two vector Fields

F = [x³ - x, y³ - y ]

This is a gradient field. You can see that from the formula or that Q_x - P_y = 0. A potential is f(x,y) = x⁴/4 - x²/2 + y⁴/4 - x^y/2.

F = [y³ - y, x³ - x ]

This is not a gradient field. You see that the curl Q_x - P_y = 3x^2-3y^2 is nonzero. You see that if x is bigger than y, then the curl is positive and you see the vector field turning counter clockwise there.. If y is bigger than x, you see that the curl is negative. The vector field turns clockwise.