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Unformatted text preview: Solution to Set 12 1. (a) Σ is composed of two pieces of smooth surfaces, a hemisphere and a ﬂat disk. (i) The hemisphere is the graph of the function f ( x, y ) = p 1 − x 2 − y 2 . ∂ x f = − x p 1 − x 2 − y 2 ∂ y f = − y p 1 − x 2 − y 2 The projection of the sphere on the xy-plane is the unit disk D centered at the origin. The surface integral over the hemisphere is given by Z Z F · ndS = Z Z D − x∂ x f − y∂ y f + p 1 − x 2 − y 2 dA = Z Z D 1 p 1 − x 2 − y 2 dA = Z 2 π Z 1 rdrdθ √ 1 − r 2 = 2 π (ii) On the disk, the outward unit normal is − k and F · n = − z = 0. Therefore, the ﬂux integral over this piece is 0. In total, Z Z Σ F · ndS = 2 π . (b) Σ is composed of three smooth pieces, a cylindrical surface and two ﬂat disks. (i) The cylindrical surface To evaluate the ﬂux integral over the cylindrical surface, one may cut the cylinder into two pieces and project them onto the xz-plane (or the yz-plane)....
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- Fall '09
- Multivariable Calculus