Use the surface integral in Stokes’ Theorem to calculate the flux of the curl of the field F across the surface S in the direction of the outward unit normal n.

F = 2zi + 3xj + 5yk

S: r(r, θ) = (r cos θ)i + (r sin θ)j + (4 - r²)k, 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π

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THEOREM 6-Stokes' Theorem Let S be a piecewise smooth oriented surface having a piecewise smooth boundary curve C. Let F = Mi + Nj + Pk be a vector field whose components have continuous first partial derivatives on an open region containing S. Then the circulation of F around C in the direction counterclockwise with respect to the surface's unit normal vector n equals the integral of the curl vector field VX F over S: fF.dr = [[ S Counterclockwise circulation VXF ndo Curl integral (4)