(1/((((z^2)+1))^2)){-y,x,0}+((-z)/((((z^2)+1))^2)){x,y,0}-(1/((z^2)+1)){0,0,1}

{-(y/(((1+(z^2)))^2))-((x z)/(((1+(z^2)))^2)),(x/(((1+(z^2)))^2))-((y z)/(((1+(z^2)))^2)),-(1/(1+(z^2)))}

{Cos[θ]Sin[ϕ],Sin[θ]Sin[ϕ],Cos[ϕ]}

-FullSimplify[Cross[D[{Cos[θ]Sin[ϕ],Sin[θ]Sin[ϕ],Cos[ϕ]},θ],D[{Cos[θ]Sin[ϕ],Sin[θ]Sin[ϕ],Cos[ϕ]},ϕ]]]

FullSimplify[{-(y/(((1+(z^2)))^2))-((x z)/(((1+(z^2)))^2)),(x/(((1+(z^2)))^2))-((y z)/(((1+(z^2)))^2)),-(1/(1+(z^2)))}/.{x-> Cos[θ]Sin[ϕ],y-> Sin[θ]Sin[ϕ],z-> Cos[ϕ]}]

FullSimplify[{-(((Cos[θ] Cos[ϕ]+Sin[θ]) Sin[ϕ])/(((1+((Cos[ϕ])^2)))^2)),(((Cos[θ]-Cos[ϕ] Sin[θ]) Sin[ϕ])/(((1+((Cos[ϕ])^2)))^2)),-(2/(3+Cos[2 ϕ]))}.{Cos[θ] ((Sin[ϕ])^2),Sin[θ] ((Sin[ϕ])^2),Cos[ϕ] Sin[ϕ]}]

Integrate[-((4 Sin[2 ϕ])/(((3+Cos[2 ϕ]))^2)),{θ,0,2Pi}]

Integrate[-((4 Sin[2 ϕ])/(((3+Cos[2 ϕ]))^2)),{ϕ,Pi/2,Pi}]