Parametrization the curve of intersection of sphere $x^2+y^2+z^2=5$ and cylinder $x^2+\left(y-\frac{1}{2}\right)^2=\left(\frac{1}{2}\right)^2$. I've seen similar posts here but none of the answers helped me. I am trying to parametrize a curve of intersection of a (top half $z>0$) sphere $x^2+y^2+z^2=5$ and cylinder $x^2+\left(y-\frac{1}{2}\right)^2=\left(\frac{1}{2}\right)^2$.
I tried $$x=\frac{1}{2}\cos t $$ $$y=\frac{1}{2}+\frac{1}{2}\sin t$$ $$z=\sqrt{5-\left(\frac{1}{2}\cos t\right)^2-\left(\frac{1}{2}+\frac{1}{2}\sin t\right)^2}=\sqrt{\frac{9}{2}-\frac{1}{2}\sin t}$$ for $t \in (0,2\pi)$,  but I don't think it is correct. Even if it was, is there a better (more simple) approach?
Note: I need to find the circulation of a field $F=(y+z,x-z,0)$ over this curve so I need a good parametrizatian so that I was able to integrate it.

The cylinder is
$$x^2+y^2=y,$$ implying (by the equation of the sphere)
$$y+z^2=5.$$
In terms of $y$,
$$\begin{cases}x^2=y-y^2,\\z^2=5-y.\end{cases}$$
Anyway, I don't know what you expect from a "good parameterization". (Will be hard to find a continuous parameterization that describes the two disconnected curves.)
