I have the vector field $X(x,y,z) = (\frac {-y}{x^2+y^2+z^2}, \frac{x}{x^2+y^2+z^2},0)$ on $\mathbb{R} ^3$ \ $\left\{ (0,0,0) \right\}$. I need to compute $\int_C (X|dx)$ over a closed circle of radius $1$ being parallel to the $xy$-plane “at height” $z_0$.
I guess because $z=0$ one can represent that as:
$$\int_C (X|dx)=\int_C\big(\frac{-y}{x^2+y^2+z_0^2}\big)dx+\big(\frac{x}{x^2+y^2+z_0^2}\big)dy$$
where $C:=\{(x,y)|x+y^2=1\}$
Is it correct?