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Unformatted text preview: SOLUTIONS TO 18.02 PRACTICE MIDTERM #4 BJORN POONEN 1) For each of (a)-(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) (7 pts.) The 3-dimensional region T consisting of all ( x, y, z ) ∈ R 3 satisfying x 2 + y 2 + z 2 > 9 is simply connected. Solution. TRUE. Any simple closed curve in T can be shrunk to a point by sliding it around the missing ball. (b) (7 pts.) If the surface S is the Southern Hemisphere of the earth, oriented by choosing at each point the unit normal pointing into the sky (instead of into the earth), then the compatible orientation of the boundary of S is the westward direction along the Equator. Solution. TRUE. This can be seen from the right hand rule (if the right thumb is pointing out of the South Pole, then the fingers are pointing westward along the Equator). Or, observe that if you are walking westward along the Equator so that the surface S is to your left, then your head is pointing up. 2) (a) (15 pts.) Using a systematic method (not just guessing), find a function f = f ( x, y, z ) such that df = (6 x 2 + 7 yz ) dx + (7 xz + 1) dy + (7 xy + 2 z ) dz. Solution. We need f to satisfy f x = 6 x 2 + 7 yz f y = 7 xz + 1 f z = 7 xy + 2 z....
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- Fall '08
- Multivariable Calculus