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vcprac04s - The University of Sydney School of Mathematics...

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Unformatted text preview: The University of Sydney School of Mathematics and Statistics Solutions to Practice Session 4 MATH2061: Vector Calculus Summer School 2012 1. Find ∇ · F in each of the following: (a) F = i + j + k (b) F = − y radicalbig x 2 + y 2 i + x radicalbig x 2 + y 2 j (c) F = x i + y j + z k Solution: (a) ∇ · F = 0 (b) ∇ · F = 0 (c) ∇ · F = 3 2. Find the flux of the vector field F = 2 x i − 3 y j outward across the ellipse x = cos t, y = 4 sin t, ≤ t ≤ 2 π. Solution: Let C be the ellipse x = cos t, y = 4 sin t, ≤ t ≤ 2 π , let R be the interior of C and let n be the outward unit normal to C . x y 1 4 R n The flux of F across C is contintegraldisplay C F · n ds and by Green’s Theorem (divergence form) this is integraldisplayintegraldisplay R ∇ · F dx dy. Now ∇ · F = ∂ (2 x ) ∂x + ∂ ( − 3 y ) ∂y = − 1 , and so Flux = contintegraldisplay C F · n ds = integraldisplayintegraldisplay R ∇ · F dx dy = − 1 × (Area of R ) = − 4 π. (Observe that the negative value means that the net flux is inward.) 3. Let F = x i + y j . Show that the flux of F across any simple closed curve C in R 2 is equal to twice the area of the region enclosed by C ....
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