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a6f08 - ASSIGNMENT 6 1 Suppose that f R 3 R is of class C2...

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ASSIGNMENT # 6 1. Suppose that is of class C R R f 3 : 2 and that it satisfies the equation π 5 2 2 2 2 2 2 = + + z f y f x f . Find the flux of f = F across the entire boundary of the compact set V in 3 R bounded by the parabolic cylinder and 2 x z = the planes 0 , 0 2 , 2 = = = + y z y z x with n pointing outward. 2. Let F ( . Find the flux of F across the k j i z x z y z z y x + + + = ) 1 ln( ) ( tan ) , , 2 3 2 1 part of the paraboloid that lies above the plane z = 1 and is 2 2 2 = + + z y x oriented upward. Note that direct calculation might be a failure (this is also true for the next problem). 3. Evaluate , dz z x dy y z dx x y C ) arctan ( ) arctan ( ) arctan ( 3 2 + + + + + where C is the curve parametrized by ) 2 sin , cos , sin ( ) ( t t t t = g , π 2 0 t . 4. Show that the vector field F is conservative and find the ) 1 , 2 , 3 ( ) , , ( 2 y yz z y x + = work done by F on the particle that follows the path parametrized by r ( t ) = ( t t t sin , cos 1 2 , sin 1 2 + + ) , 2 0 π t .
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