practice3bs

practice3bs - SOLUTIONS TO 18.02 PRACTICE MIDTERM #3B BJORN...

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SOLUTIONS TO 18.02 PRACTICE MIDTERM #3B BJORN POONEN 1) For each of (a)-(c) 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) Let F be a continuously differentiable vector field on R 2 , and let C be a simple closed oriented curve. If div F = 0 at every point inside C , then H C F · d r = 0. Solution. FALSE. For example, if F is the rotational flow given by F ( x, y ) = h- y, x i , and C is the counterclockwise unit circle centered at the origin, then div F = 0 + 0 = 0 at every point, but H C F · d r is positive since F is parallel to the unit tangent vector at each point of C . (If we replace div F by curl F , then the statement is true by Green’s theorem.) ± (b) Any region obtained by removing a ray (half-line) from R 2 is simply connected. Solution. TRUE, because if a simple closed curve is contained in the region (i.e., does not intersect that ray), then its interior is contained in the region too: it’s impossible for a simple
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practice3bs - SOLUTIONS TO 18.02 PRACTICE MIDTERM #3B BJORN...

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