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Unformatted text preview: MATH 251, Fall 2006
RAICH EXAM 3  VERSION A Name (printed): ~ Section #:V “On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic
work.” 1 Signature: UIN: DIRECTIONS: I. The use of a calculator, laptop or computer is prohibited. 2. Please present your solutions in the space provided. Show all your work neatly and concisely
and clearly indicate your ﬁnal answer. You will be graded on the ﬁnal answer and the quality
and correctness of the work leading up to it. 3. There are questions on BOTH SIDES of the paper. Please be aware of this. DO NOT WRITE BELOW! Question Points Awarded 1Disputes about grades on this exam must be handled the day the exam is handed back and must be discussed before
you leave the room. If the exam leaves the room, it will not be re—evaluated (except for possible adding mistakes). 1. (a) (5 pts) Without ﬁnding a potential function, show that the vector ﬁeld 13(23, y, z) = (z, 2yz, :1;+ y2) is conservative. r y “\2 a
\ a/DX 2m a/bz Z XXVI _A C). ~
\ : <11v~1\{\~(\\\f0>
+1 (b) (10 pts) Find a function f so that F" = V f , i.e., ﬁnd a potential function. g (c) (5 pts) Using your answer from (b), ﬁnd the work done by the fOrce ﬁeld moving a particle from (2,1,2) to (0,1,5).
+3 Fiﬁ SUWM 0% = HOMSX SAZMZL ‘ w —
— = s—Wwa +1 2. (a) (10 pts) Find the Jacobian of the transformation a: i set, y = 36—15. ékx‘m V 6+ 56* ‘ _
+ (o (b) (10 pts) Find the image of the disk S given by U2 + v2 S 1' under the tranformation a: 2 au,
3/ 2 bv.  3. (20 pts) Evaluate the line integral [0 F  dr Where F(x,y, z) : azzi + myj + 22k and C’ is param—
eterized by r(t) : sin ti + cos tj + t2k, 0 g t S SﬁA? 3 FEWHVWHCXV i + ~10
_ g <SW‘V)SW€YC&T.ﬂ>a<60ﬁr5mrgév)d5§ 0 M7. ‘— 8 $9M cw ~ sw’ﬂrcmt + 1%)“ _ \ tam“
— §T\ +3
6 _ .‘n‘ 4. (20 pts) Find the equation of the tangent plane t6 the surface S parameterized by r(u,v) :
(u2,u — 112,112 +0) at the point (1, —3,6). u 1 20M \\ W1» U—V’ﬁ VEHA 1 ($1.36)
R = <0 )_1V»2\IH§ .—.\ T 3‘ Q:
. 9M \ o
0 —‘).v 2v H : <Qv,r\‘_§w\1m\, ‘L\U»V> 5. (20 pts) Evaluate // 3:2 dS Where S is the part of the surface 2 = 3:2 + y2 + 5 that lies inside
S the c linder 3:2 + 2 : 9. Hint: if you inte ate usin ﬂ olar coordinates, inte rate in 0 ﬁrst.
3’ y 81‘ g P g > , 6. (20 pts) Evaluate f (a: —— y2) dx + (x3 + 2y) dy Where C is the curve bounded by y = O, a: 2 2,
C and y : x3. Hint: this is a good candidate to use Green’s Theorem. bx
D 3"
1 >3 440 \> z
2 3 gym ow Ax = S ﬁve/My
o 0 {VB Y 0 W"
x5 + ‘ 3 1
._ 82 BXZV +V'L AX S Xi'va \w5 b ' 6 08 L\\
3.. I = 8*L\\{+\HQ\{ KAY
. \ 3X5+XL Ax = éxg+%x7\ \o (6 1 7‘3
0 y c. \ (i ‘— <gV% +%\lt
_ 72 +352 L‘ 331+ 6:229 0
=31+E§ :(o ‘3 d 7
7
v an
7, E?
,3 m
22% \Z 5’ ...
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 Spring '08
 Skrypka
 Force, Work, Trigraph, pts, The Work, Aggie

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