Let S be the cylinder x² + y² = a², 0 ≤ z ≤ h, together with its top, x² + y² ≤ a², z = h. Let F = -yi + xj + x²k. Use Stokes’ Theorem to find the flux of ∇ * F outward through S.

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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: & F ·dr = [[ F.dr S Counterclockwise circulation VXF n do Curl integral (4)