SetVectorCalculus2

# SetVectorCalculus2 - JANE PROFESSOR Sample WeBWorK...

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Unformatted text preview: JANE PROFESSOR Sample WeBWorK problems. WeBWorK assignment VectorCalculus2 due 5/4/08 at 2:00 AM. D. E. x y x2 xy 1 WW Prob Lib1 Summer 2000 6.(1 pt) Let C be the positively oriented circle x2 y2 1. Use Green’s Theorem to evaluate the line integral C 18y dx 3x dy. 7.(1 pt) Let C be the positively oriented square with vertices 0 0 , 1 0 , 1 1 , 0 1 . Use Green’s Theorem to evaluate the line integral C 3y2 x dx 5x2 y dy. 8.(1 pt) Find a parametrization of the curve x2 and use it to compute the area of the interior. 3 y2 3 2π. Note: Your answer should be an expression of x and y; e.g. ”3xy - y” B. Compute ∂P ∂y Note: Your answer should be an expression of x and y; e.g. ”3xy - y” C. Compute C F d r Note: Your answer should be a number D. Is F conservative? Type Y if yes, type N if no. 5.(1 pt) Determine whether the given set is open, connected, and simply connected. For example, if it is open, connected, but not simply connected, type ”YYN” standing for ”Yes, Yes, No.” A. x y x 1 y 2 B. C. x y 2x2 x y x2 12.(1 pt) Let F 1yi 1xj. Use the tangential vector form of Green’s Theorem to compute the circulation integral C F d r where C is the positively oriented circle x2 y2 4. 13.(1 pt) Let F 5xi 3yj and let n be the outward unit normal vector to the positively oriented circle x2 y2 16. Compute the ﬂux integral C F n ds. 14.(1 pt) A rock with a mass of 15 kilograms is put aboard an airplane in New York City and ﬂown to Boston. How much work does the gravitational ﬁeld of the earth do on the rock? Newton-meters 15.(1 pt) Suppose F F x y z is a gradient ﬁeld with F ∇ f , S is a level surface of f, and C is a curve on S. What is the value of the line integral C F d r? 1 y2 1 ©¨ £¢¢¡ y2 1 ©¨ ¤ ¤ £¢ ¡ ¤ £ ¢ ¡£ ¤ ¤ £ ¢£ ¡ ¡¡ 4.(1 pt) Let F x y cos t i sin t j, 0 t A. Compute ∂Q ∂x yi xj x2 y2 and let C be the circle r t ¤ £ ¡ ¤ ¤ ¤ " ©¨ ¤ 3.(1 pt) Suppose C is any curve from 0 0 0 to 1 1 1 and F xyz 3z 4y i 3z 4x j 3y 3x k. Compute the line integral C F d r. ¤ £ ¡¤£ ¡¤ ¤£ ¡ 2.(1 pt) If C is the curve given by r t 1 3 sin t i π 1 2 sin2 t j 1 5 sin3 t k, 0 t and F is the radial 2 xi yj zk, compute the work done by vector ﬁeld F x y z F on a particle moving along C. 9.(1 pt) Let F 6xi 3yj 3zk. Compute the divergence and the curl. A. div F B. curl F i j k 10.(1 pt) Let F 9yz i 9xz j 2xy k. Compute the following: A. div F B. curl F i j k C. div curl F Note: Your answers should be expressions of x, y and/or z; e.g. ”3xy” or ”z” or ”5” 11.(1 pt) Let F be any vector ﬁeld of the form F f x i g y j h z k and let G be any vector ﬁeld of the form F f y z i g x z j h x y k. Indicate whether the following statements are true or false by placing ”T” or ”F” to the left of the statement. 1. G is irrotational 2. F is irrotational 3. G is incompressible 4. F is incompressible ¤ # ¤# ¤ ¨ ¤¤ ¤ ¨ £¢¡£¢¡£¢¡£¢¡ 1.(1 pt) For each of the following vector ﬁelds F , decide whether it is conservative or not by computing curl F . Type in a potential function f (that is, ∇ f F). If it is not conservative, type N. A. F x y 12x 3y i 3x 8y j f xy B. F x y 6yi 7xj f xy C. F x y z 6xi 7yj k f xyz D. F x y 6 sin y i 6y 6x cos y j f xy E. F x y z 6x2 i 3y2j 4z2k f xyz Note: Your answers should be either expressions of x, y and z (e.g. “3xy + 2yz”), or the letter “N” !  ¤  £ ¢ !  " £ ¢  y2 1 x2 y2 4 ¡ ¡ §§ £ £¡     £ ¢ ¡ © £ ¤ ¤£ ¤ ¡¤£ ¤ £¢¢¡ £¢¢¡ ¡ ¤£ ¤ ¡ £¡ £¢ ¤ ¤ £ ¢ ¢ ¡ £ £ ¤ ¡ ¤ £ ¡ £ ¢ ¡ £¢ ¤ ¤ £ ¢ ¢ ¡ £ ¤ £ ¢ ¡ £ £ ¤ ¡ ¤ £ ¤ ¡ £ ¢ ¡ §§ ¤ ¤ £¢¢ ¤ ¥ ¤¡ ¦ ¦ !  " £ !  ¤ £   ¢  £  ©¨ ¨ ¢ ¡ ¢ ¡ ¢ ¡ ¡ £¢¢¡ ¡¤£ ¡ ¢¡ ¢¡ ¢¡ ¢¡ ¢¡ 1 ¤¥ Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c UR 2 \$ 16.(1 pt) A vector ﬁeld gives a geographical description of the ﬂow of money in a society. In the neighborhood of a political convention, the divergence of this vector ﬁeld is: \$ \$ A. positive B. negative C. zero % ...
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## This note was uploaded on 03/13/2010 for the course MATH 0314 taught by Professor Ivoklemes during the Spring '10 term at McGill.

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