Solution
= Solution
The origin is excluded by the denominator. On the punctured plane, direct differentiation gives
$$
d\left(-\frac{x}{x^2+y^2}\right)=\frac{(x^2-y^2)dx+2xy\,dy}{(x^2+y^2)^2}.
$$
Hence the form is \b[exact on its whole stated domain], with $\boxed{f=-x/(x^2+y^2)+C}$. Although the punctured plane is not simply connected, the explicitly constructed single-valued potential establishes global exactness directly; a topological sufficiency criterion is unnecessary here.