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practice4a - 18.02 PRACTICE MIDTERM#4A BJORN POONEN Please...

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18.02 PRACTICE MIDTERM #4A BJORN POONEN Please turn cell phones off completely and put them away. No books, notes, or electronic devices are permitted during this exam. Generally, you must show your work to receive credit. Name: Student ID number: Recitation instructor’s last name: Recitation time (e.g., 10am): (Do not write below this line.) 1 out of 10 2 out of 25 3 out of 20 4 out of 30 5 out of 15 Total out of 100
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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) If C is a piecewise smooth curve in R 3 from ( a 1 , a 2 , a 3 ) to ( b 1 , b 2 , b 3 ), then C y dx = b 1 b 2 - a 1 a 2 . (b) If f ( x, y, z ) is a function with continuous second partial derivatives on R 3 , then div( f ) = 0 at every point.
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2) Let F = (cos x + 2 y 2 + 5 yz ) i + (4 xy + 5 xz ) j + (5 xy + 3 z 2 ) k on R 3 . Show that F is conservative, and find a potential function for F . For full credit, use a systematic method (not just guessing), and show your work.
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