Let $S$ be the upwards$-$oriented surface of the square pyramid with vertices $(0,0,1),(1,1,0),(1,-1,0),(-1,1,0),$ and $(-1,-1,0)$ which is above the $xy-$plane, not including the base. Given that $(y^2+z^2,x^2+z^2,x^2+y^2$:$)=$ curl vec$(G)$ for some vector field vec$(G),$ calculate ${}_S($:$y^2+z^2,x^2+z^2,x^2+y^2$:$)*dvec(S).$