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.Example. Let T be the solid cylinder defined by the inequalitiesx 2 + y 2 ≤ 1 and 0 ≤ z ≤ 4. Let S be the entire boundary of T. Let. F(x, y, z) = (5z + 3)k. Find the outward flux of F through S.Solution I. Let S 1 = { (x, y, 4) | x 2 + y 2 ≤ 4 } be the top of T, andS 2 = { (x, y, 0) | x 2 + y 2 ≤ 4 } be the bottom of T. The outer unitnormal vector field on S 1 and S 2 are n 1 = k, and n 2 = −krespectively. The outward flux of F throughS 1 = ∫∫ SF1· n dS = ∫∫ S(51× 4 + 3)k · k dS = 23×Area of S 1 = 23π.The outward flux of F throughS 2 = ∫∫ SF2· n dS = ∫∫ S(52× 0 + 3)k · (−k) dS = 3×Area of S 2 = 3π.The flux through the curved part of the boundary is 0 since n ishorizontal there, so F · n = 0, So the total flux of F across S is23π − 3π = 20π.Solution II. We have div F ∫∫ = 0 + 0 + 5 = ∫∫∫ 5, so by the divergencetheorem, the flux equals F · n dS = ∇ · F dV = 5× Volume ofT= 5 × π × 1 × 4 = 20π.ST. . . . . .
.Stokes’ Theorem.Definition. A piecewise smooth surface S in space is called anoriented surface, if one chooses a continuous unit normal vector fieldon each smooth piece S.A positive orientation of the boundary C of an oriented surface is aunit . tangent vector T such that n × T always points into S.Remark. Think of a man walks along C as T, and heads up in n, thenthe region is always on the left hand side of the man..Stokes’ Theorem. Let S be an oriented, bounded, and piecewisesmooth surface in space with positive oriented boundary C withrespect to the unit normal vector field n on S. Suppose that T is apositively oriented unit vector field tangent to C in the right orientationof C. If F is a continuously∫differentiable∫∫vector field defined in aregion containing S, then F · T ds = (curl F) · n dS..CSRemark. The condition that the positive orientation of C agrees withthe choice of unit normal vector field n is essential.. . . . . .
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: .Example. Let F(x, y, z) = (x, y, z
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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