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Test3KeyBlue - MATH 251 Fall 2006 RAICH EXAM 3 VERSION B...

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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}; ‘ (Rx-21W 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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