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midterm4s - SOLUTIONS TO 18.02 MIDTERM#4 BJORN POONEN...

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SOLUTIONS TO 18.02 MIDTERM #4 BJORN POONEN December 3, 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.) If you are looking at a flat disk S in R 3 , and its chosen unit normal vector is pointing straight at you, then the compatible orientation of its boundary curve would appear from your vantage point to be counterclockwise. Solution: TRUE. If the thumb of your right hand is pointing towards you, then the fingers appear to be pointing counterclockwise. (b) (5 pts.) If F = P, Q, R is a continuously differentiable 3D vector field on R 3 such that Q x = P y , then F = f for some function f ( x, y, z ) defined on R 3 . Solution: FALSE. Although Q x = P y would be enough for a 2D vector field on a simply connected 2D region, in 3D one needs all three components of curl F to be 0; that is, R y = Q z , P z = R x , and Q x = P y . The vector field F = 0 , 0 , y is an explicit example of one that satisfies
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