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.Example. The outward flux of vector field F(x, y, z) = xi + yj + zk. across the sphere x 2 + y 2 + z 2 − 2x − 2y − 2z = 3 is .Solution. Let D be the solid bounded by the sphere, then fromdivergence theorem we have the outward flux of F is∫∫∫∫∫∫∫∫F · n dS = ∇ · F dV = ( ∂xSDD ∂x + ∂x∂x + ∂x∂x∫∫∫D) dV = 3 dV= 3 volume of D = 3 × 4( √ 6) 3 π/3 = 24π √ 6.. . . . . .
.Example. Let F(x, y, z) = (x, y, z), and T be the surface defined by{ (x, y, z) ∈ R 3 | |x| + |y| + |z| = 1 }, with outward ∫∫ pointing unitnormal n on T. Evaluate the surface integral F · ndS..TSolution. Let D be the solid bounded by the surface T, i.e.D = { (x, y, z) ∈ R 3 | |x| + |y| + |z| ≤ 1 }. One can check that thecondition of the divergence theorem holds, and hence∫∫∫∫∫∫∫∫F · ndS = divFdV = ( ∂xTDD ∂x + ∂x∂x + ∂x∂x ) dV∫∫∫= 3 × dV = 3Vol(D) = 3 × 8 × 1 3! × 1 × 1 × 1 = 4.D. . . . . .
Page 1 and 2: .Chapter 15 Vector Calculus.15.1 Ve
Page 3: .Definition. Let S be a parameteriz
Page 7 and 8: .Example. Let S be a parametric sur
Page 9: .Surface Integral of Vector Field.L
Page 13 and 14: .Example. Find the flux of F(x, y,
Page 15 and 16: .Example. Suppose that a uniform so
Page 17 and 18: .Example. Let S be the surface (wit
Page 19 and 20: .Example. Let S be the sphere x 2 +
Page 21: .Example. Show that the divergence
Page 25 and 26: .Stokes’ Theorem.Definition. A pi
Page 27 and 28: .Example. ∮ Let F(x, y, z) = 3zi
Page 29 and 30: .∫∫Example. Evaluate the surfac
Page 31 and 32: .Proposition. Let F be a continuous
Page 33 and 34: .Summary of Line Integral of Vector
Page 35 and 36: .Example. Let T be the solid bounde
Page 37: .Example. Let F(x, y, z) = f (x, y,

lecture notes 13

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