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Multivariable & Vectorhard
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A vector field F⃗\vec{F}F is given by F⃗=⟨z2,x2,y2⟩\vec{F} = \langle z^2, x^2, y^2 \rangleF=⟨z2,x2,y2⟩. Using Stokes' Theorem, compute ∬S(∇×F⃗)⋅dS⃗\iint_S (\nabla \times \vec{F}) \cdot d\vec{S}∬S​(∇×F)⋅dS where SSS is the portion of the paraboloid z=1−x2−y2z = 1 - x^2 - y^2z=1−x2−y2 that lies above the xyxyxy-plane, oriented upwards.