MT2Rev - t 1. 5. a) Explain why Q x = P y for a...

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Math 32B - Calculus of Several Variables Midterm #2 - Practice Problems Disclaimer :- The choice of problems was entirely my own. I have no knowledge of what will appear on the midterm, nor how the exam will be structured. These questions are designed to cover what I think are some of the important parts of the course so far, but I do not claim that they cover everything you may be asked in the midterm. Finally, any mistakes are my own entirely. 1. Calculate the following integrals: a) RR D cosh ± y - x x ² dA , where D is bounded by the curves y = x - x , y = x + x , x = 0, and x = 1. b) RR D 4 y 2 x 2 + y 2 dA , where D is the region defined by 2 x y 3 x , 2 xy 3 2. A metal block has density given by ρ ( x, y, z ) = 3 z xy + y . A wire is cut out along the curve γ ( t ) = ( t 2 , 2 t, 2) for 1 t 2. What is the mass of the wire? 3. A force acting at the point ( x, y ) is given by ~ F ( x, y ) = h- y, x i . a) Sketch the force field b) A particle moves along the line y = x from (2 , 2) to (10 , 10). What is the work the force does on the particle? 4. Evaluate R γ ~ F · d~ r , where ~ F ( x, y ) = h ye xy , xe xy i and γ ( t ) = ( e t , e t 2 ) for 0
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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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