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.Example. Let F(x, y, z) = (x + y)i + (y − x)j + zk be vector field. LetS be the hemisphere x 2 + y 2 + z 2 = 4, z ≤ 0, ∫∫and n be the outwardpointing unit normal vector field of S. Then curlF · ndσ is ..SSolution. The boundary of S is the circle given byC : r(t) = ( x(t), y(t), z(t) ) = (2 cos t, 2 sin t, 0), 0 ≤ t ≤ 2π. From theStokes’ theorem, we have∫∫∫curlF · ndσ = F · Tds===S∫ 2πt=0∫ 2π0∫ 2π0C[ (x(t) + y(t))x ′ (t) + (y(t) − x(t))y ′ (t) + z(t)z ′ (t) ]dt−4(cos t + sin t) sin t + 4(sin t − cos t) cos tdt(−4 sin 2 t − 4 cos 2 t)dt = −8π.. . . . . .
.Example. Let F(x, y, z) = f (x, y, z)k, where f (x, ∫∫y, z) is a differentiablefunction defined in R 3 . Then the outward flux curlF · n dS of curlFacross the upper hemisphereS = { (x, y, z) | x 2 + y 2 + z 2 = 1, z ≥ 0 } is. A. 0 B. f (0, 0, 0) C. grad f (0, 0, 0) D. None of the above.Solution. Let C be the unit circle x 2 + y 2 = 1 in the usualcounterclockwise ∫∫ direction, one∫can apply Stokes’ ∫ theorem to obtainthe following curlF · n dS = F · T ds = 0 ds = 0.SCCS. . . . . .
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 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: .Example. Let T be the solid bounde

lecture notes 13

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