vcprac05 - z 1, the base z = 0 , x 2 + y 2 1 and the top z...

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The University of Sydney School of Mathematics and Statistics Practice Session 5 MATH2061: Vector Calculus Summer School 2012 1. Evaluate the following surface integral: ii S ( z + 2) dS, S : x 2 + y 2 + z 2 = a 2 . 2. Find the ±ux of F = x i + y j + z k across the surface S , where S is the triangular region with vertices (1 , 0 , 0) , (0 , 1 , 0) , (0 , 0 , 1) . 3. Let F = 2 i j + k . Find the ±ux of F (a) upwards through the hemisphere z = r 16 x 2 y 2 ; (b) outwards through the closed cylinder consisting of the curved surface x 2 + y 2 = 1 , 0
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Unformatted text preview: z 1, the base z = 0 , x 2 + y 2 1 and the top z = 1 , x 2 + y 2 1. 4. Evaluate the triple integrals: (a) i 3 i-2 i 1 ( x + 2 y ) 2 dx dy dz (b) i 3 i-x i x + y 2 y dz dy dx 5. Use a triple integral to nd the volume of the solid region bounded by the paraboloid z = x 2 + y 2 and the plane z = a 2 . Copyright c c 2012 The University of Sydney...
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