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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 . Solution: Let C be the ellipse x = cos t, y = 4 sin t, t 2 , 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 Greens 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 . (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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