vcprac04 - R be the projection of S onto the xy-plane....

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The University of Sydney School of Mathematics and Statistics 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 r x 2 + y 2 i + x r x 2 + y 2 j (c) F = x i + y j + z k 2. Find the ±ux of the vector ²eld F = 2 x i 3 y j outward across the ellipse x = cos t, y = 4 sin t, 0 t 2 π. 3. Let F = x i + y j . Show that the ±ux of F across any simple closed curve C in R 2 is equal to twice the area of the region enclosed by C . 4. Find the surface area of the part of the sphere x 2 + y 2 + z 2 = 25 that lies between the planes z = 3 and z = 4. 5. Let S be the triangular portion of the plane 3 x + 3 y + 5 z = 30 in the ²rst octant – that is, the portion of the plane cut o³ by the planes x = 0, y = 0 and z = 0, for x 0, y 0 and z 0. (a) Let
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Unformatted text preview: R be the projection of S onto the xy-plane. Describe R , in terms of x and y . (b) Suppose that a thin plate in the shape of S has density ( x + y + z ) at each point ( x, y, z ). Find the mass of the plate. (Mass = ii S ( x + y + z ) dS .) 6. (a) Describe the surface S dened by: x = 3 cos , y = 3 sin , z = t , for : / 2 / 2 and t : 0 1. (b) Evaluate ii S xyz dS where S is the surface in part (a). 7. Prove that ( F ) = F + F where = ( x, y, z ) and F is a vector eld in R 3 . Copyright c c 2012 The University of Sydney...
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