Let a vector field $F$ what it is defined by $F(x,y)=(\frac{-y}{(x-1)^2+y^2},\frac{x^2+y^2-x}{(x-1)^2+y^2})\ \forall \ (x,y)\epsilon\mathbb{R}^2$\ {$(1,0)$} then... is the vector field $F$ a conservative field in $\mathbb{R}^2$\ { $(1,0)$}? Why? I dont know how I can prove that, I know that if I calculate its rotational I cant say nothing, so what can I do?
2026-04-03 23:16:27.1775258187
Conservative field?
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If you shift $x$ by $1$ to get $G(x,y)=F(x+1,y)=(\frac{-y}{x^2+y^2},1+\frac{x}{x^2+y^2})$ then it is easy to see that $G$ is not conservative, e.g. calculate the curve integral along the unit circle to see it is not zero. Neglecting the constant component $(0,1)$ it is the electromagnetic field around $z$-axis.