$\renewcommand{\dd}[1]{\,\mathrm{d}#1}$Use Stokes theorem to evaluate $$ \iint_\Sigma y \dd{y} \dd{z} + z \dd{z} \dd{x} + x \dd{x} \dd{y} $$ where $\Sigma$ is upper half-sphere oriented outwardly. Hint: $$ (y,z,x) = \text{Curl}(-xy, -yz, -zx) $$
Now, I've used Stokes Theorem and the hint:
$$
\iint_\Sigma y \dd{y} \dd{z} + z \dd{z} \dd{x} + x \dd{x} \dd{y} = \oint_\Gamma -xy \dd{x} - yz \dd{y} - zx \dd{z}
$$
but I instantly got stuck. What should I do next? I know how to calculate line integrals, but only when parametrization is given. $\Gamma$ seems to be circle with arbitrary radius R, but I can't think of any parametrization which would take z into account.
Any help appreciated.
The circle is used because that is the boundary of the upper-half sphere. Because that circle is in the $xy-$ plane, $z=0$. A parameterization of the circle could then be $$x=\cos t$$ $$y=\sin t$$ $$z=0$$