237t3 - 1. Let S be the portion of the cone z 2 = x 2 + y 2...

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1. Let S be the portion of the cone 2 2 2 y x z + = with 2 0 z and 0 x . (a) [2 marks] Parametrize the surface S using cylindrical coordinates. (b) [6 marks] Compute dS x y x S ) ( 2 2 + + ∫ ∫ 2
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2. [8 marks] Evaluate dS curl S n F ∫ ∫ ) ( , where k i F xy yz + - = and S is the part of the paraboloid 2 2 z y x + = between the planes 1 = x and 4 = x , n pointing away from the x -axis . 3
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3. [9 marks] Find the flux of ) 1 , , ( ) , , ( 2 2 2 + - = z ye xe z y x z z F across the part of the sphere 2 2 2 2 = + + z y x with 0 , 0 z y , n pointing out of the origin. HINT: Direct calculation might be a failure, close the surface. 4
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4. (a) [2 marks] Determine the largest set(s) 2 R D on which the vector field j i F 3 2 2 2 1 y x y xy x + - + = is conservative. (b) [4 marks] Determine a potential function for F . (c) [4 marks] Find the work done by F in moving the particle along the curve C parametrized by g ( t ) = ) cos 2 3 , cos 1 ( t t + - , π t 0 . Argue that your answer is correct. 5
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5. (a) [2 marks] Formulate all the assumptions of Green’s Theorem.
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This note was uploaded on 10/21/2010 for the course MATHEMATIC MAT237Y1 taught by Professor Romauldstanczak during the Fall '09 term at University of Toronto- Toronto.

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237t3 - 1. Let S be the portion of the cone z 2 = x 2 + y 2...

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