Unformatted text preview: ≤ t ≤ 1. 5. a) Explain why ∂Q ∂x = ∂P ∂y for a conservative vector ﬁeld ~ F ( x, y ) = h P ( x, y ) , Q ( x, y ) i . b) If ~ F = h P, Q i is a vector ﬁeld deﬁned 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 ﬁeld have to do with whether or not it is conservative? b) For each of the following vector ﬁelds, compute the curl, divergence, and determine whether or not the ﬁeld is conservative. If it is, ﬁnd 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
 Vector Calculus, Force, Englishlanguage films

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