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practice3as

# practice3as - SOLUTIONS TO 18.02 PRACTICE MIDTERM#3A BJORN...

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SOLUTIONS TO 18.02 PRACTICE MIDTERM #3A 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) 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 x -axis. 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 C F · n ds < 0, then div F < 0 at some point of R . Solution. TRUE. If div F were nonnegative at every point of R , then R div F would be nonnegative, but actually Green’s theorem for flux shows that R div F = C F · n ds < 0 . (c) If F ( x, y ) := ( x sin y ) i + (5 x + e 2 y ) j , then div F = (sin y ) i + (2 e 2 y ) j .

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practice3as - SOLUTIONS TO 18.02 PRACTICE MIDTERM#3A BJORN...

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