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Unformatted text preview: SOLUTIONS TO 18.02 MIDTERM #3 BJORN POONEN November 12, 2009, 1:05–1:55pm (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) The region in R 2 defined by  y  < x 2 + 1 is simply connected. Solution. TRUE. The region is obtained by removing from R 2 everything on or above the parabola y = x 2 + 1, and everything on or below its reflection in the xaxis. If a simple closed curve is contained in this region, so is its interior. This means that the region is simply connected. (b) If C is a positively oriented simple closed curve enclosing a region R , and F is a continuously differentiable vector field defined on all of R 2 , and I C F · n ds < 0, then div F < at some point of R . Solution. TRUE. If div F were nonnegative at every point of R , then RR R div F would be nonnegative, but actually Green’s theorem for flux shows that ZZ R div F = I C F · n ds < ....
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 Fall '08
 Auroux
 Multivariable Calculus, Vector Calculus, Manifold, 10 pts, −4y dy dx

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