Consider the vector field $F(x,y)=\frac{-yi+xj}{x^2+y^2}=($:$P(x,y),Q(x,y)$:$).$

$a\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{y}}=\frac{\text{d}\text{e}\text{l}\text{Q}}{\text{d}\text{e}\text{l}\text{x}}.$

$b{\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dris$ not independent $of$ the path $CF=+Qjbe$ a vector field defined $onan$ open, simply$-$connected region $D.$

Suppose $P$ and $Q$ have continuous first$-$order partial derivatives throughout $D,$ and

$\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{y}}=\frac{\text{d}\text{e}\text{l}\text{Q}}{\text{d}\text{e}\text{l}\text{x}}$

holds everywhere $inD.$ Then $Fis$ conservative.