Given F ((x, y, z) = y2i + xj + z2 k; S is part of the paraboloid z = x2 + y2 th

Question

Given F ((x, y, z) = y2i + xj + z2 k; S is part of the paraboloid z = x2 + y2 that lies below the plane z = 1 and is oriented upward. Verify that Stoke's Theorem is true for this vector field F and surface S. Evaluate the surface integral for this given vector field F anti the oriented surface S. Use the Divergence Theorem to calculate the flux of F across S.

Explanation / Answer

Surface Integral: (i) curl F = (ii) Parameterize S by R(t, z) = for u in [0, 2p], v in [0, 3]. ==> R_t x R_z = So, ??s curl F * dS = ??s curl F * (R_t x R_z) dA = ?(v in [0, 3]) ?(u in [0, 2p]) * du dv = ?(v in [0, 3]) ?(u in [0, 2p]) (4v^2 cos u - 4v^2 sin u - 2v) du dv = ?(v in [0, 3]) (0 - 4p v) dv = -18p. ------------------------------------- Line integral: The boundary curve C is x^2 + y^2 = 9, which may be parameterized c(t) = for t in [0, 2p]. (note that this is clockwise orientation, as this is under the surface!) So, ?c F * dr = ?(t in [0, 2p]) * dt = ?(t in [0, 2p]) -9 dt = -18p. --------------

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