midterm2-practice

midterm2-practice - (1 − 2 and(4 6 3 4 32B MIDTERM 2...

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MATH 32B SECOND MIDTERM EXAMINATION Nov. 20th, 2006 Please show your work. You will receive little or no credit for a correct answer to a problem which is not accompanied by suFcient explanations. If you have a question about any particular problem, please raise your hand and one of the proctors will come and talk to you. At the completion of the exam, please hand the exam booklet to your TA. If you have any questions about the grading of the exam, please see the instructor within 15 calendar days of the examination. Name: Section: #1 #2 #3 #4 #5 Total 1
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2 32B MIDTERM 2 RADKO Problem 1. Sketch the vector feld F ( x, y ) = ( x y ) i + xj on the plane. (Here i is the unit vector along the x -axis and j is the unit vector along the y -axis).
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32B MIDTERM 2 Radko 3 Problem 2. Let F ( x, y, z ) = yz i + xz j + ( xy + 2 z ) k . (a) Find f ( x, y, z ) so that F ( x, y, z ) = f ( x, y, z ) . (b) Use f ( x, y, z ) to compute the line integral i C F · dr , where C is the straight line segment connecting the points
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Unformatted text preview: (1 , , − 2) and (4 , 6 , 3) . 4 32B MIDTERM 2 RADKO Problem 3. Rewrite the integral i i i E f ( x, y, z ) dV , where E = { ( x, y, z ) | x 2 + y 2 + z 2 ≤ a 2 ; x 2 + y 2 ≤ z 2 } in spherical coordinates. (Here f = f ( x, y, z ) is a given function and a > is a given number). 32B MIDTERM 2 Radko 5 Problem 4. Use the transformation x = √ 2 u − r 2 / 3 v , y = √ 2 u + r 2 / 3 v to evaluate the integral i i R ( x 2 − xy + y 2 ) dA , where R is the region bounded by the ellipse x 2 − xy + y 2 = 2 . 6 32B MIDTERM 2 RADKO Problem 5. Let C be a positively-oriented smooth simple closed curve on the xy-plane such that the area enclosed by this curve is equal to A . Use the Green’s formula to show that the area enclosed by the curve can be computed as A = 1 2 i C xdy − ydx ....
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midterm2-practice - (1 − 2 and(4 6 3 4 32B MIDTERM 2...

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