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.Theorem. Let F be a continuously differentiable vector field in asimply connected region D in space. Then the vector field F isirrotational if and only if it is conservative; that is ∇ × F = 0 in D if and. only if F = ∇f for some scalar function f defined on D.Solution. If F = ∇f , then ∇ × F = ∇ × ∇f = 0 on D. Conversely,suppose that F is irrotational on D, one wants to define a functionf (x, y, z) such that ∇f = F on D. Following the same idea in theplanar case, Let A(a, b, c) and B(x, y, z) be two points in D, and C be apath from A to B. If one can show that the line integral of F is pathindependent, then define f (x, y, z) =∫ (x,y,z)(a,b,c)∇F · T ds, which dependsonly on the end points of C. Then it follows from continuity of thevector field F that ∇f = F on D. It remains to show that the lineintegral ∮ of F is independent of path, which is equivalent to the fact thatF · T ds = 0 for any closed path C in D. As D is simply connected, itCfollows one can shrink the curve C continuously to a path in such waythat ∮ C bounds ∫∫a surface S via the∫∫shrinking, thenF · T ds = (∇ × F) · n dS = 0 · n dS = 0.CSS. . . . . .
.Summary of Line Integral of Vector FieldLet F be a vector field defined on a region D. Then we can summaryour important result as follows:..F is conservative on D ⇐⇒ F = ∇f for some function f∮⇕⇓ (−) xy = (−) yxF · T ds = 0∇ × F = 0 on DCfor any closed path in DSimply connected⇐=Stokes ThmRemark. Conservative vector field is irrotational; but the converse isnot true. In fact, it depends on the domain of the vector field ((−y, x)compare the lower horizontal arrow). An example is F(x, y) =x 2 +y 2on R 2 \ {(0, 0)}.. . . . . .
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: .Proposition. Let F be a continuous
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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