Show that a particular force field with $\operatorname{curl} = 0$ is not conservative

Question

$\newcommand{\dd}{\partial}\newcommand{\Vec}[1]{\mathbf{#1}}$Show that the force field $$ -Q\, \Vec{i} + P\, \Vec{j} = -\frac{y}{x^{2} + y^{2}}\, \Vec{i} + \frac{x}{x^{2} + y^{2}}\, \Vec{j} $$ is not conservative, even though it satisfies the condition $$ \frac{\dd P}{\dd x} = \frac{\dd Q}{\dd y} = \frac{(-x^{2} + y^{2})}{(x^{2} + y^{2})^{2}}. $$

Answer 1

If $\vec F$ is your field and $\gamma$ is the unit circle centered at zero, then $$ \int_\gamma \vec F \cdot d\vec r=2\pi\neq 0 $$ so that $\vec F$ is not conservative.