Math needed help.

Question

If F (x,y,z)=tan z i +(x+z)j+xsiny k and σ is the portion of the cylinder z=9-x²-y² above the xy-plane oriented so that the unit normal vector n points to the outside of the cylinder, use Stoke's theorem to evaluate ∫∫_σ(∇ x F) • n dS.

Answer 1

∫∫_σ(∇ x F) • n dS = ∫_CF•dr = ∫_C(tanzdx + (x + z)dy + xsinydz), where C is circle x² + y² = 9, z = 0, x = 3cosθ, y = 3sinθ, z = 0, 0 ≤ θ ≤ 2π.

∫∫_σ(∇ x F) • n dS = ∫₀^2π(3cosθ·3cosθ)dθ = 9(1/2θ + 1/4sin2θ)₀^2π = 9π