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Multivariable & Vectorhard
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Let SSS be the portion of the paraboloid z=9−x2−y2z = 9 - x^2 - y^2z=9−x2−y2 that lies above the plane z=5z = 5z=5, oriented upwards. Let F⃗=⟨yez,x2z,xy⟩\vec{F} = \langle y e^z, x^2 z, x y \rangleF=⟨yez,x2z,xy⟩. Evaluate the flux integral ∬S(∇×F⃗)⋅dS⃗\iint_S (\nabla \times \vec{F}) \cdot d\vec{S}∬S​(∇×F)⋅dS.