Evaluate the surface integral S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = x i + y j + 7 k S is the boundary of the region enclosed by the cylinder x2 + z2 = 1 and the planes y = 0 and x + y = 6

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Mathematics

6 September, 14:02

Evaluate the surface integral S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F (x, y, z) = x i + y j + 7 k S is the boundary of the region enclosed by the cylinder x2 + z2 = 1 and the planes y = 0 and x + y = 6

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Humberto Hansen · Answer 1

V is the 3D region enclosed by S.

div (F) = ∂ (x) / ∂x + ∂ (y) / ∂y + ∂ (10) / ∂z = 2

∫ = ∫∫∫ [V] div (F) dV = 2 ∫∫∫ [V] dV

Using cylindrical coords, (rcos (θ), y, rsin (θ)), dV=rdrdθdy

∫ = 2 ∫∫∫ rdrdθdy, [r=0,1], [θ=0,2π], [y=0,7-rcos (θ) ] = 14π (easily integrated)