Parameterize [tex]S[/tex] by
[tex]\vec r(u,v)=u\,\vec\imath+v\,\vec\jmath+(2-u^2-v^2)\,\vec k[/tex]
with [tex]0\le u\le1[/tex] and [tex]0\le v\le1[/tex]. Take the normal vector to [tex]S[/tex] to be
[tex]\vec r_u\times\vec r_v=2u\,\vec\imath+2v\,\vec\jmath+\vec k[/tex]
Then the flux of [tex]\vec F[/tex] across [tex]S[/tex] is
[tex]\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\int_0^1\int_0^1(uv\,\vec\imath+v(2-u^2-v^2)\,\vec\jmath+u(2-u^2-v^2)\,\vec k)\cdot(\vec r_u\times\vec r_v)\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle\int_0^1\int_0^1(2u^2v+(u+2v^2)(2-u^2-v^2))\,\mathrm du\,\mathrm dv=\boxed{\frac{293}{180}}[/tex]
