$\mathbf F(x,y)=\frac{-y}{x^2+y^2}\mathbf i+\frac{x}{x^2+y^2}\mathbf j$ when the region D is $D=\mathbf R^2 \vert (0,0) $
How do I proof that the vector field F has no potential in region D.
I know that:
$F_1=\frac{-y}{x^2+y^2}$ and $F_2=\frac{x}{x^2+y^2}$
and after that $\frac{dF_1}{dy}=\frac{dF_2}{dx} = \frac{y^2-x^2}{(y^2+x^2)^2}$
Line integral over unit circle:
$\oint_C \mathbf F \cdot \mathbf {dr}$
what is F and dr in this case. Limit values are $[0,2\pi]$
Hint: Consider the line integral along the unit circle!