This preview shows pages 1–6. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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. ...
View
Full
Document
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.
 Fall '08
 Auroux
 Multivariable Calculus

Click to edit the document details