Problem 1403 (difficulty: 5/10)

Which of the following vector fields are gradient fields? For those that are not give a closed curve on which the line integral of the field does not vanish.

\(\displaystyle (x,y) \qquad (y,x) \qquad \left(\frac{-y}{x^2+y^2}, \frac{x}{x^2+y^2}\right) \qquad \left(\frac{x}{\sqrt{x^2+y^2}}, \frac{y}{\sqrt{x^2+y^2}}\right) \)

Give me another random problem!