Vector Calculus Stokes Theorem?

x² + y² + z² = 1

r² = 1

F = r x {i,j,k} = {x,y,z} x {i,j,k} = {y-z , z-x , x-y}

stokes theorem,

∮ F • dr = ∫ ∫ (∇ x F) • dS

where ∇ x F = {∂/∂x , ∂/∂y , ∂/∂z} x {y-z , z-x , x-y}

∇ x F = 2{-1,-1,-1}

∮ F • dr = ∫ ∫ (∇ x F) • dS = 2 ∫ ∫ {-1,-1,-1} • dS

∮ F • dr = -2 (Area of sphere) = -2(4πr²) = -2(4π)(1)

∮ F • dr = -8π

this is incorrect.