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Test3KeyGreen - MATH 251, Fall 2006 RAICH EXAM 3 - VERSION...

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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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Test3KeyGreen - MATH 251, Fall 2006 RAICH EXAM 3 - VERSION...

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