$F := (\frac{x}{x^2+y^2+1} ,\frac{y}{x^2+y^2+1})$ and $Γ := (x, y) : x^2 + y^2 − 2x = 1$ , $Γ$ traversed counterclockwise
$F := (2xy^2z, 2x^2yz, x^2y^2 - 2z)$ and $Γ :=$ {$(cos(t),\frac{\sqrt{3}}{2} sin (t),\frac{1}{2} sin (t)$}, $0 ≤ t ≤ 2π$
The question is to compute $\int_ΓF$
In both cases I got that $\int_ΓF = 0$ where in 1. I used Green's theorem, that is $\int_ΓF = \int_Γ F_{2x} - F_{1y}$, where $F_{2x} = F_{1x}$ are the partial derivatives of $F$ which is conservative.
And in 2. I just applied the line integral formula $\int_ΓF = \int_a^b F(r(t))\frac{dr}{dt} dt$
Is it okay that I got 0 as an answer?