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Unformatted text preview: MATH 251, Fall 2006
RAICH EXAM 3  VERSION B Name (printed): \< eY *_ ‘3 \U6 Section #: “On my honor, as an Aggie, I have neither given nor received unauthorized aid on this academic
work.” 1 l ' Signature: UIN: DIRECTIONS: 1'. 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! 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). A l. (a) (5 pts) Show that the vector ﬁeld F(:v, y, z):
ﬁnding a potential function. .J'ry? VXP: a/Dx aim, 9192' +3 1y? 21 X73541} <2E *2}; ‘ (Rx21W O> .__\ :0 +1 (2mz,z2,a:2 + 2yz) is conservative Without (c) (5 pts) Using your answer from (b), ﬁnd the work done by the force ﬁeld moving a particle
from (2,1,0) to (1,2,3). 2. (a) (10 pts) Find the Jacobian of the transformation a: = SZCt, y = 36.”. ' C>lX '10 153 $265 $th +t i € 235*
: Lls 87° 3283? _. e p. = 362g“ ’H‘l 212222
(b) (10 pts) Find the 1mage of the solid ellipse 5' given byu— 7a + 2)? S 1 under the tranformation mzu/a, yZ’U/b. =(M a}: + mil“
\1 it +5 al ‘6’"
XIA‘V'i é" 3. (20 pts) Evaluate the line integral f0 F  dr Where F(m, y, z) = myi + y2j + 22k and C is param
eterized by r(t) = sin ti + cos tj + t3k, 0 S t S 335. \3
SC PM)“; ‘ g ﬂamewr ’
Z ' + L0 < armcpsﬁ @ﬁ, J? > ° QM. ~5Tvth 3*”) 0W / O “3 +7
ﬂ gm mgegmmﬁ + 31% aw r‘, w\
: Jgfq\> : JB—(EY 4. (20 pts) Find the equation of the tangent plane to the surface S parameterized by r(u,v) = (u2 — u)i + (u + v)j + 303k at the point (a 1,—3). 5. (20 pts) Evaluate // yz dS Where S is the part of the paraboioid z = m2 + 3/2 +3 that lies inside
 s the cylinder 2:2 + 3/2 : 4. Hint: if you use integrate using polar coordinates, integrate in 0 ﬁrst. 6. (20 pts) Evaluate f (m2 + 2y) dac + (2302 — y3) dy where C’ is the curve parameterized by y = 0,
C V :v 2 3, and y = 9:2. Hint: this is a good candidate for Green’s Theorem. PM“ = Xaly NW'W
46‘»: l (u ...
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 Spring '08
 Skrypka

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