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Unformatted text preview: t 1. 5. a) Explain why Q x = P y for a conservative vector eld ~ F ( x, y ) = h P ( x, y ) , Q ( x, y ) i . b) If ~ F = h P, Q i is a vector eld dened on all of R 2 , and Q x = P y , explain why H C ~ F d~ r = 0 for any closed curve C . 6. Compute R C cos y dx + x 2 sin y dy , where C is the rectangle through the vertices (0 , 0) , (0 , 2) , (5 , 2) , (5 , 0). 7. a) What does the curl of a vector eld have to do with whether or not it is conservative? b) For each of the following vector elds, compute the curl, divergence, and determine whether or not the eld is conservative. If it is, nd a function f such that ~ F = f . (i) ~ F ( x, y, z ) = h xy 2 z 2 , x 2 yz 2 , x 2 y 2 z i (ii) ~ F ( x, y, z ) = h e x , e y , e z i (iii) ~ F ( x, y, z ) = h cos( x + yz ) , z cos( x + yz ) + 2 y, y cos( x + yz ) + e z i (iv) ~ F ( x, y, z ) = h z, x, y i 1...
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 Winter '10
 Corbin

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