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.Conservative and Irrotational Fields.Definition. A differentiable vector field F is called irrotational if. ∇ × F = 0..−yi + xjExample. The vector field F(x, y) =x 2 + y 2 is irrotational onD . = R 2 \ {(0, 0)}. , but we had proved that F is not conservative on D..Definition. Let D be a region in space, D is called simply connected, ifevery simple closed curve in D can be continuously shrunk to a pointwhile . staying inside D..Examples. (a) The entire plane, entire space are simply connected;(b) Rectangle, the sphere and ball are simply connected;(c) The interior of a torus, and the punctured plane are not a simply. connected.. . . . . .
.Proposition. Let F be a continuous vector field defined on a region D,prove ∫ that the line integral of F is independent of path if and only ifF · T ds = 0 for any piecewise smooth closed curve C in D.. CProof. Suppose that line integral of F is independent of path, then letC be any closed curve with the same starting and terminal point A,then the constant path C ′ with A for all t is also a curve with the samestarting and terminal point ∫ A. It follows from the path ∫ independence ofthe line integral of F that F · T ds =C∫C F · T ds = F · 0 ds = 0.′ C ′Conversely, suppose C 1 and C 2 are two paths, both of them startsfrom the same point A, and terminates at point B. Let C = C 1 ∪ (−C 2 )be a closed path from A to B via C 1 , and back from B to A via −C 2 (inreverse direction of C 2 ). Then C is a piecewise smooth closed curvein D, ∫hence one has ∫∫∫∫0 = F · T ds = F · T ds + F · T ds = F · T ds − F · T ds.C ∫C 1 ∫−C 2 C 1 C 2Hence F · T ds = F · T ds. So the line integral of F isC 1 C 2independence of path.. . . . . .
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: .∫∫Example. Evaluate the surfac
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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