midterm3s

midterm3s - SOLUTIONS TO 18.02 MIDTERM #3 BJORN POONEN...

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Unformatted text preview: SOLUTIONS TO 18.02 MIDTERM #3 BJORN POONEN November 12, 2010, 2:05pm to 2:55pm (50 minutes). 1) For each of (a) and (b) below: If the statement is true, write TRUE. If the statement is false, write FALSE. (Please do not use the abbreviations T and F.) No explanations are required in this problem. (a) (5 pts.) Let R be the region obtained by removing the first quadrant from the plane, so R is the set ( x, y ) in R 2 such that x < 0 or y < 0 (or both); then R is simply connected. Solution: TRUE, because any curve in R can be continuously shrunk to a point by sliding it southwest into the third quadrant before shrinking it. (b) (5 pts.) For every region R in R 2 and for every continuously differentiable vector field F defined on R , if F is the gradient of some differentiable function on R , then curl F = 0 at every point of R . Solution: TRUE. This is one of the implications between the six conditions for conserva- tiveness. Here is the reason: If F = f = h f x , f y i , then curl F = ( f y ) x- ( f x ) y = 0. 2) Find one potential function (not all of them) for the conservative vector field F = (6 x 2- 10 xy ) i + (- 5 x 2 + 6 y ) j on R 2 by using a systematic method (not just guessing). Show what you are doing....
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midterm3s - SOLUTIONS TO 18.02 MIDTERM #3 BJORN POONEN...

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