Tutorial_10 - NATIONAL UNIVERSITY OF SINGAPORE Department...

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NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA 1505 Mathematics I Tutorial 10 1. Evaluate RR S f ( x,y,z ) dS and RR S F dS , where f ( x,y,z ) = x + y + z and F = x 2 i + y 2 j + z 2 k and S is the surface defined parametrically by r ( u,v ) = (2 u + v ) i + ( u - 2 v ) j + ( u + 3 v ) k , (0 u 1 , 0 v 2) . The orientation of S is given by the normal vector r u × r v . Ans : 40 3; - 430 3 2. Evaluate RR S z dS , where S is the portion of the paraboloid z = 4 - x 2 - y 2 lying on and above the xy plane. Ans : 289 60 π 17 - 41 60 π 3. Evaluate RR S F dS , where F = y i + x 2 j + z 2 k and S is the portion of the plane 3 x +2 y + z = 6 in the first octant. The orientation of S is given by the upward normal vector. Ans : 31 4. Use Stoke’s Theorem to evaluate H C ( 1 2 y 2 dx + z dy + x dz ) , where C is the curve of inter- section of the plane x + z = 0 and the ellipsoid x 2 + 2 y 2 + z 2 = 1, oriented counterclockwise
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