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Unformatted text preview: UHIZHIZUUb 15:13 bllﬁb4224lb LILJH Mn'ﬂ'ulN LlHHIﬂuH‘r’ Hills”: Ulr'lﬁ! Department of Mathematics, University of California, Berkeley
Math 53
Alan Weinstein, Fall 2005 Final Examination, li‘riday1 December 16th 2005 Instructions. BE SURE TO WRITE YOUR NAME AND YOUR GSI’S NAME ON YOUR
BLUE BOOK. Read the problems very carefully to be sure that you understand the statements.
All work should be shown in the blue book; writing should be legible and clear. and there should
be enough work shown to justify your answers. Indicate the ﬁnal answers to problems by
circling them. [Point values of problems are in square brackets. There are eight problems, with
a total. point value is 120,. for 40% of your course grade] PLEASE HAND IN YOUR PREPARED NOTES ALONG WITH YOUR BLUE BOOK. YOU
SHOULD N 0T HAND IN THIS EXAM SHEET. 1. [13 points] The vector functions rﬂt) = {cos t, sin t, t2} and r2 = (cos t1 — sint, 0) describe two
curv which lie on the cylinder 3:2 + y2 = l.
(a) The two curves go through the same point P at t = 0. Find that point, and ﬁnd the tangent
vectors of the two curves at P.
(b) For t aé 0, there is a unique plane St through the points r1(t), raw), and P. Among all the
normal vectors to the plane 5,5, there is one which has the form
u(t) = (l,a{t),b(t)), where a(t) and Mt) are functions of it. Find u(t).
(c) Find the limiting position of the normal u(t) as t approaches zero. [lIint: use l‘Hdpital’s rule]
(d) Find a normal vector for the tangent plane at P to the cylinder 3:2 + y2 = l. '
(e) Comment on the relation between the answers to parts (c) and (d). Do you ﬁnd it surprising? 2. [12 points] The level curves of a certain function f(r,y) are the lines parallel to the line
y r— ﬁr, and the value of the function increases as one moves from lower left to upper right.
[9,) Where on the ellipse 1:2 + 23:2 = 9 does f attain its maximum value?
(13) Where does it attain the minimum value? 3. [15 points] Find the area. of the part of the paraboloid a a 9 — r2 — yz that lies above the
plane a z 5. 4. [15 points] (a) Describe the (solid) region of integration E for the integral :3 [1/25—32 “25—er —y”
ﬂ 0 es—eﬁ—yi V3.12 + y? + 22 dz d3; (b) Evaluate the integral by using spherical coordinates. 5. [15 points] In each of parts and (lb), evaluate as many as possible of the following ﬁve
expressions. If the expression is not deﬁned, say so. Be sure to distinguish between “zero” and
“not deﬁned”. (1) The divergence of the curl; (2) the curl of the gradient; (3) the gradient of the
curl; (a) the divergence of the gradient; (5) the gradient of the divergence. There should be ten answers in all. Please present them. in the order: la, 2a, 3a. 4a, 5a, lb, 2b.
as, 4b, 5b. (a) re. y. z) I4 i 4 y4k.
 0:») mm se— PLEASE TURN OVER THE PAGE FOR THE REMAINING PROBLEMS UHIZHIZUUIJ 15:13 bllﬁb4224lb LILJH Mn'ﬂ'ulN LlHHIﬂuH‘r’ Hills”: UZIUE E. [12 points] Let E be the solid .region between the sphere x2 + y2 + z2 = 1 and the larger sphere
(as  1)2 + (y — 2)2 + (a — 3)2 = 100. An expanding gas is ﬂowing into B through the small sphere,
and the ﬂux of the velocity ﬁeld F through that sphere (with respect to the normal pointing into
E from the hole in the middle} is equal to 20. Furthermore, the velocity ﬁeld F has divergence
everywhere equal to 3 on E. Find the ﬂux of 1“ through the larger sphere with respect to the
outward normal. 7. [15 points}
(a) Find a. vector ﬁeld F whose curl is (1 + 1:) k. [Hint guess a solution in the simplest possible
form, then check, and correct if necessary]
(1)) Let S be the surface which bounds the solid region which is inside the cylinder 3:2 + .712 = 1,
above the cry—plane, and below the plane 2: = 10 + 3:. We may think of S as s. “can with a slanted
top”. Use Stokes’ theorem (and NOT the divergence theorem) to show that the ﬂux of (1 + :r) 1:
through the elliptical top of the can is equal to the ﬂux of the same vector ﬁeld through the circular
bottom [with top and bottom both oriented by the “upward” normal). [Hintz don’t forget to take
the cylindrical “side” of the can into account]
(c) Find the value of this ﬂux. 3. [18 points] Let D be the Lshaped region in the plane whose boundary (7 consists of the
straight line Segments connecting the following points in the given order: (—11_1)1 (21—1): (211): (151)! (113): (—113)! (—1:_1)* Find the line integral around C of each of the following vector ﬁelds. (a)
V(:E2 + 1 — sin(mey))
(b)
yi — 3:1:j
(C) , _
y 1  no
me + ye [Hint replace C by a simpler curve] ...
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This note was uploaded on 10/31/2009 for the course STAT 131A taught by Professor Isber during the Spring '08 term at University of California, Berkeley.
 Spring '08
 ISBER

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