practice midterm 3

practice midterm 3 - SOLUTIONS TO 18.02 PRACTICE MIDTERM#3...

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Unformatted text preview: SOLUTIONS TO 18.02 PRACTICE MIDTERM #3 BJORN POONEN November 6, 2009, 1:05pm to 1:55pm (50 minutes) 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....
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This note was uploaded on 03/08/2010 for the course MATH 18.02 taught by Professor Auroux during the Fall '08 term at MIT.

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practice midterm 3 - SOLUTIONS TO 18.02 PRACTICE MIDTERM#3...

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