(Solution) - For each of the following evaluate S curl F -(2025 Original AI-Free Solution)

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For each of the following, evaluate ??S curl F ? nd?.
a) S is the "bottomless" surface in the upper half-space z > 0 bounded by y = x2, z = 1 - y, n is the outward-pointing normal, and F(x, y, z) = (x sin z3, y cos z3, x3 + y3 + z3).
b) S is the truncated paraboloid z = 3 - x2 - y2, z > 0, n is the outward-pointing normal, and F(x, y, z) = (y, xyz, y).
c) S is the hemisphere z = ?10 - x2 - y2, n is the inward-pointing normal, and F(x, y, z) = (x, x, x2y3 log(z + 1)).
d) S is the "bottomless" tetrahedron in the upper half-space z > 0 bounded by x = 0, y = 0, x + 2y + 3z = 1, z > 0, n is the outward-pointing normal, and F(x, y, z) = (xy, yz, xz).