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Unformatted text preview: 4 ,4 ± xy ± 4 } (see the picture below). Use the change of variables x = u + v , y = vu to compute the integral: I = ±± R (1 + x + y ) dxdy . (Hint: One of the main things you need to Fgure out is the limits of integration in the ( u, v ) coordinates. It helps to notice that the domain in the ( u, v ) coordinates will be a boxa ± u ± a anda ± v ± a . You just need to Fnd the correct value of a .) 5 Problem #5 ( pts. total) Compute the outward fux ± S ± F · ˆ n dS oF the vector±eld ± F = ( x 3 , y 3 , z ) over the entire surFace oF the cylinder S = { x 2 + y 2 ± 1 , z = 0 } ∪{ x 2 + y 2 = 1 , ± z ± 1 } ∪{ x 2 + y 2 ± 1 , z = 1 } . 6 Problem #6 ( pts. total) Please compute the outward fux oF the curl ± S ( ∇ × ± F ) · ˆ n dS over the upper spherical cap S = { x 2 + y 2 + z 2 = 1 , ± z } oF the vector±eld ± F = ( xy, x 2 , xe z ) ....
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This note was uploaded on 03/16/2010 for the course MATH 223B taught by Professor Staff during the Spring '09 term at Arizona.
 Spring '09
 STAFF
 Math

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