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.Example. Let F(x, y, z) = (2y, −2x, sin z). Let S be the upper half ofthe sphere x 2 + y 2 + z 2 = 9. Evaluate the outward (i.e., upward) fluxof . curl F through S.Solution. The surface S is given by{ (x, y, z) | x 2 + y 2 + z 2 = 9, z ≥ 0 }, then its boundary C is the circlex 2 + y 2 = 9 in the xy-plane oriented counterclockwise. So C isparameterized ∫∫by ∮r(t) = (3 cos t, 3 sin t, 0) for t ∈ [0, 2π]. The flux iscurlF · n dS = F · T ds==S∫ 2π∫02π0C(2 · 3 sin t, −2 · 3 cos t, 0) · (−3 sin t, 3 cos t, 0) dt−18(cos 2 t + sin 2 t) dt = −36π.Remark. As n is already fixed in the question, you can use the sameidea of a man walking along the boundary C so that the surfaceappears in his left hand side to determine the positive orientation of Cwith respect to n of S.. . . . . .
.Example. ∮ Let F(x, y, z) = 3zi + 5xj − 2yk. Evaluate the line integralF · T ds where C is the ellipse in which the plane z = y + 3Cintersects the cylinder x 2 + y 2 = 1. Orient C counterclockwise as viewfrom . above.Solution. Let S be the region on the plane z = y + 3 bounded by thecylinder x 2 + y 2 = 1. Then S = { (x, y, z) | z − y = 3, x 2 + y 2 ≤ 1 } ispart of a level surface z − y = 3, with a unit normal vector field∇(z − y) (0, −1, 1)n(x, y, z) = = √ on S. Next the∥∇(z − y)∥ 2curl F(x, y, z) = i j k∂ ∂ ∂∂x ∂y ∂z = −2i + 3j + 5k, hence3z 5x −2ycurl F · n = (−2, 3, 5) · (0, −1, 1)/ √ 2 = (−3 + 5)/ √ 2 = √ ∫∫∫∫∫2.√ √ F · T ds = (curl F) · n dS = 2 dS = 2Area of (S) =√C√ SS2 × 1 × 2π = 2π.Remark. S can be parameterized by r(x, y) = (x, y, y + 3) defined onD = { (x, y) | x 2 + y 2 ≤ 1 }.. . . . . .
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 and 22: .Example. Show that the divergence
Page 23 and 24: .Example. Let F(x, y, z) = (x, y, z
Page 25: .Stokes’ Theorem.Definition. A pi
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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