practice4a

Practice4a - 18.02 PRACTICE MIDTERM#4A BJORN POONEN Please turn cell phones off completely and put them away No books notes or electronic devices

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Unformatted text preview: 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 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 R3 from (a1 , a2 , a3 ) to (b1 , b2 , b3 ), then b1 b2 − a1 a2 . C y dx = (b) If f (x, y, z ) is a function with continuous second partial derivatives on R3 , then div( f ) = 0 at every point. 2) Let F = (cos x + 2y 2 + 5yz )i + (4xy + 5xz )j + (5xy + 3z 2 )k on R3 . 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. 3) Assuming that the Earth is a solid sphere of constant density, how does the gravitational field at a point halfway to the center compare with the gravitational field at a point on the surface? 4) Consider the cone whose base is the disk x2 + y 2 ≤ 1 in the xy -plane and whose vertex is at (0, 0, 1). Let S be the lateral surface of the cone (i.e., not including the base). Let F(x, y, z ) = y i − xj. Compute both sides of Stokes’ theorem for F on S . 5) Find two different 3D vector fields F such that for any piecewise smooth positivelyoriented closed surface S in R3 the flux of F across S equals the volume enclosed by S . This is the end! If you finish during the last 5 minutes of the exam, please remain in your seat until the end, and then pass your exams towards the aisles, and continue to wait silently in your seat until the proctors have collected all exams. ...
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This note was uploaded on 09/14/2011 for the course MATH 18.02 taught by Professor Auroux during the Fall '08 term at MIT.

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Practice4a - 18.02 PRACTICE MIDTERM#4A BJORN POONEN Please turn cell phones off completely and put them away No books notes or electronic devices

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