quiz12' - Quiz 12 Solutions 1. Let F(x, y, z ) = 4yz, −xz...

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Unformatted text preview: Quiz 12 Solutions 1. Let F(x, y, z ) = 4yz, −xz 2 , sin(xyz ) . Compute the flux of curl F across the the part of the paraboloid z = x2 + y 2 lying below the plane z = 4, oriented upward. By Stokes’ Theorem, we can compute the line integral of F along the boundary of the surface instead. This boundary can be parametrized by r(t) = 2 cos t, 2 sin t, 4 , where t ∈ [0, 2π ]. Then r (t) = −2 sin t, 2 cos t, 0 and F(r(t)) = 32 sin t, −32 cos t, sin(16 sin t cos t) . Thus, the line integral is 2π F · dr = C 0 F(r(t)) · r (t) dt 2π = −64 0 cos2 t + sin2 t dt = −128π. 2. Find the flux of the vector field F(x, y, z ) = ecos x , sin(xy ), z (sin x)ecos x − xz cos(xy ) through the surface x2 + y 2 + z 2 = 9, oriented outward. By the Divergence Theorem, we can instead compute the integral of div F on the interior of the surface. Since div F = −(sin x)ecos x + x cos(xy ) + (sin x)ecos x − x cos(xy ) = 0, the flux integral is 0. 1 ...
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