{-y,x,z}

{x,y,(x^2)+(y^2)}

?Region*

RegionPlot3D[And[z-((x^2)+(y^2))>0,z<1],{x,-1,1},{y,-1,1},{z,0,1.1},PlotPoints->{100,100,50}]

r{Cos[θ] Sin[ϕ],Sin[θ] Sin[ϕ],Cos[ϕ]}+{x0,y0,z0}

D[r{Cos[θ] Sin[ϕ],Sin[θ] Sin[ϕ],Cos[ϕ]}+{x0,y0,z0},θ]

D[r{Cos[θ] Sin[ϕ],Sin[θ] Sin[ϕ],Cos[ϕ]}+{x0,y0,z0},ϕ]

FullSimplify[-Cross[D[r{Cos[θ] Sin[ϕ],Sin[θ] Sin[ϕ],Cos[ϕ]}+{x0,y0,z0},θ],D[r{Cos[θ] Sin[ϕ],Sin[θ] Sin[ϕ],Cos[ϕ]}+{x0,y0,z0},ϕ]]]

The vector field on the surface

{-y,x,z}/.{x->x0+r Cos[θ] Sin[ϕ],y->y0+r Sin[θ] Sin[ϕ],z->z0+r Cos[ϕ]}

Integrate[Integrate[{-y0-r Sin[θ] Sin[ϕ],x0+r Cos[θ] Sin[ϕ],z0+r Cos[ϕ]}.{(r^2) Cos[θ] ((Sin[ϕ])^2),(r^2) Sin[θ] ((Sin[ϕ])^2),(r^2) Cos[ϕ] Sin[ϕ]},{θ,0,2Pi}],{ϕ,0,Pi}]

(1/((4 π (r^3))/3))Integrate[Integrate[({(x^2),(y^3),z}/.{x->x0+r Cos[θ] Sin[ϕ],y->y0+r Sin[θ] Sin[ϕ],z->z0+r Cos[ϕ]}).{(r^2) Cos[θ] ((Sin[ϕ])^2),(r^2) Sin[θ] ((Sin[ϕ])^2),(r^2) Cos[ϕ] Sin[ϕ]},{θ,0,2Pi}],{ϕ,0,Pi}]

Limit[(1/5) (5+3 (r^2)+10 x0+15 (y0^2)),r->0]

(1/((4 π (r^3))/3))Integrate[Integrate[({(x^2),(y^3),z}/.{x->x0+r Cos[θ] Sin[ϕ],y->y0+r Sin[θ] Sin[ϕ],z->z0+r Cos[ϕ]}).{(r^2) Cos[θ] ((Sin[ϕ])^2),(r^2) Sin[θ] ((Sin[ϕ])^2),(r^2) Cos[ϕ] Sin[ϕ]},{θ,0,2Pi}],{ϕ,0,Pi}]