MATH 52 FINAL EXAM DECEMBER 7, 2009 THIS IS A CLOSED BOOK, CLOSED NOTES EXAM. NO CALCULATORS OR OTHER ELECTRONIC DEVICES ARE PERMITTED. IF YOU NEED EXTRA SPACE, PLEASE USE THE BACK OF THE PREVIOUS PROB- LEM PAGE. SO EXTRA WORK FOR PROBLEM 1 WOULD GO ON THE BACK OF THIS COVER SHEET, ETC. MOST PROBLEMS ARE WORTH 10 POINTS. PROBLEM TEN IS 12 POINTS AND PROBLEM ELEVEN IS 8 POINTS. THE TOTAL IS 120 POINTS. THE TERMS C 1 OR C 2 FOR FUNCTIONS AND VECTOR FIELDS MEANS THAT THEY HAVE CONTINUOUS FIRST PARTIAL DERIVATIVES OR CONTINUOUS FIRST AND SECOND PARTIAL DERIVATIVES, RESPECTIVELY, ON THEIR DO- MAINS. Please sign the following, and make sure your name is legible: On my honor, I have neither given nor received any aid on this examination. I have furthermore abided by all other aspects of the honor code with respect to this examination. SIGNATURE: NAME: Prob/[pts] Score Prob/[pts] Score Prob/[pts] Score 1 [10] 5 [10] 9 [10] 2 [10] 6 [10] 10 [12] 3 [10] 7 [10] 11 [8] 4 [10] 8 [10] 12 [10] 1
1(a)[4]. Sketch the region of integration in the plane for 1 1+ x 0 1+x 3 f(x, y) dydx. (b)[6]. Reverse the order of integration for the integral in part (a). 2
2(a)[6]. Let W be the solid object in space bounded by the xy-plane, the cylinder x 2 +y 2 = 4, and the paraboloid z = x 2 + y 2. Set up AND EVALUATE a triple integral in cylindrical coordinates in the order dz dr dθ that gives the volume of the object. (b)[4]. Set up but DO NOT EVALUATE an integral in cylindrical coordinates in the order dr dz dθ that gives the volume of the same object from part (a). 3
3(a)[4]. A wire is in the shape of a coil described by (cos(2πt), sin(2πt), t), 0 t 4. Suppose the electric charge density along the wire is given by e(x, y, z) = z. Set up an explicit integral that gives the total electric charge on the wire. DO NOT EVALUATE. (b)[6]. Suppose surface S is the portion of the paraboloid x 2 + y 2 = 4z with 1 z 4. Parametrize S by T (s, t) = (2t cos(s), 2t sin(s), t 2 ) with 0 s 2π, 1 t 2. If the mass density at points of S is given by δ = z, Use the parametrization to express the moment of S with respect to the plane z = 0 as an explicit integral over a domain in the st-plane. DO NOT EVALUATE. 4
4. Let D be the triangle in the first quadrant of the xy-plane bounded by the x-axis, the y-axis, and the line x + y = 1. The transformation T (u, v) = (x, y) = (1 u, uv) maps the square 0 u 1, 0 v 1 in the uv-plane onto the triangle D in the xy-plane. (a)[3]. Find the Jacobian determinant (x, y) (u, v). (b)[3]. Fill in the limits of integration asterisks for (sin((1 x) 2 ) + y 1 x ) da = D (sin((1 x) 2 ) + y ) dx dy 1 x (c)[4]. Use the change of variables theorem to express D (sin((1 x)2 ) + y 1 x ) dxdy as an explicit integral D g(u, v) dudv over a region D in the uv-plane. DO NOT EVALUATE. Which looks easier to evaluate, the integral in (b) or (c)? 5
5(a)[3]. Find a potential function for the vector field F (x, y, z) = 1 + x 2 + y 2 + z 2 (x, y, z) on R 3 or give a reason no such potential function exists. (b)[3]. Is G = 2xy i + (x 2 y) j a conservative vector field on R 2? Why or why not? (c)[4]. Find the work done by the vector field G of part (b) on a particle that moves from ( 1, 1) to (1, 1) clockwise around the ellipse 2x 2 + 3y 2 = 5. 6
6. Let S be the portion of the sphere x 2 +y 2 +z 2 = 25 that lies on or below the plane z = 4. (So, S is most of the sphere.) Let F = (x + 3y 2 + z 2 ) i + (x 2 + y z 2 ) j + (xy 2z) k. (a)[2]. Calculate div( F ). (b)[8]. Compute the flux S F n ds, where n is the unit normal given by n = 1 5 (x, y, z) at points of S. Explain your work clearly. For your own protection, NO CREDIT will be given for attempts that try to parametrize S. [Hint: S is not a closed surface. But S together with some other surface does form a closed surface. Use part (a). You can also use symmetry and knowledge of certain areas to evaluate certain integrals without actually integrating.] 7
7. In this problem you will use the graph parametrization of a hemisphere given by T (x, y) = (x, y, a 2 x 2 y 2 ), for (x, y) in the disk D described by x 2 + y 2 a 2 in the xy-plane. (a)[3]. Give the standard normal vector N = T x T y for this parametrization and find the length N. [You can just give the answers if you remember, otherwise work it out.] (b)[3]. The surface area of S is given by the integral D N da. Is this integral an improper integral or a proper integral? Give a reason for your answer. (c)[4]. Showing your work, EVALUATE the integral D [You know the answer, but you must show work here.] N da. 8
8. In this problem, we consider the plane vector field y F = (P (x, y), Q(x, y)) = ( x 2 + y 2, x x 2 + y 2 + 2x) which satisfies Q/ x P/ y = 2 at all points where F is defined. (a)[5]. If C 1 is the circle x 2 + y 2 = 1 oriented counterclockwise, calculate C 1 F ds. (b)[5]. If C 2 is the ellipse x2 25 + y2 9 = 1 oriented counterclockwise, calculate C 2 F ds. [You may use the formula for the area inside an ellipse, but NO CREDIT for attempts that parametrize the ellipse. Remember Q/ x P/ y = 2 at all points where F is defined.] 9
9[10]. Write T (True) or F(False) in the margin to the left of each statement. There are ten statements, continuing on the next page. One point for each correct answer. (i). If f(x) and g(y) are two continuous functions of one variable then the double integral of the product f(x)g(y) over an x-simple region a x b, p(x) y q(x) can always be computed as the product ( b a f(x)dx)( q(x) p(x) g(y)dy). (ii). The region in the plane bounded by the two circles x 2 + y 2 = 4 and x 2 + (y 1) 2 = 1 is neither an x-simple nor a y-simple region. (iii). If S is a surface with boundary in space and if S has constant density, then the y coordinate of the center of mass of S is the average value of the function y on S. (iv). If T (u, v) = (x(u, v), y(u, v), z(u, v)) is a linear transformation mapping R 2 onto a plane in R 3 and if N = T u T v is the standard normal vector to T (R 2 ), then for any region D in the uv-plane that has an area, the area of T (D) is N area(d). (v). ( f) = 0 for all C 2 functions f(x, y, z) on R 3. 10
(vi). If f(x, y, z) is a C 2 function on R 3 then Div Grad(f) = 2 f x 2 + 2 f y 2 + 2 f z 2. (vii). If F = (P, Q) is a C 1 vector field on the domain W = R 2 (0, 0), [that is, W is R 2 with the origin removed], and if Q x = P y on domain W, then C P dx + Qdy = 0 for all simple closed curves C in W. (viii). If S is the hemisphere x 2 + y 2 + z 2 = 4, z 0 in R 3 with upward pointing unit normal n and if F = ( yz, y 2, z) then S F n ds < 0. (ix). On every smooth surface with or without boundary in R 3 there are two continuous unit normal vector fields. (x). If f(u) is any continuous function for u > 0, and if r = x 2 + y 2 + z 2, then the vector field f(r)(x, y, z) is conservative in the region R 3 (0, 0, 0). 11
10(a)[6]. For each of the regions W in R 3 described below, suppose F is a C 1 vector field with Curl F = 0 throughout W. In each case, decide if you can you be certain that F is a conservative vector field throughout W? Answer YES or NO. There are six regions. One point for each correct answer. (i). R 3 with the x axis removed. (ii). R 3 with the circle x 2 + y 2 = 1 in the xy-plane removed. (iii). R 3 with the origin removed. (iv). R 3 with the plane z = 0 removed. (v). The octant x > 0, y > 0, z > 0 in R 3. (vi). The region in R 3 described as all (x, y, z) with 4 < x 2 + y 2 + z 2 < 25. (b)[6]. Write T(true) or F(false) in the margin to the left of each of the six statement below. One point for each correct answer. (i). If F is a C 1 vector field on all of R 2 with Div F = 0 then the flux of F across any simple closed curve C in R 2 is always zero. (ii). There exists a vector field G on R 3 with Div( G) = z 2 y 2 x 2. (iii). If F is a C 1 vector field on all of R 3 and if F = 1 at all points of a solid region W in R 3 and if S is the boundary of W then the outward flux S F n ds always equals the volume of W. (iv). If F is a C 1 vector field on all of R 3 and if C C in R 3, then F = 0. F ds = 0 for all simple closed curves (v). If F is a C 2 vector field on R 3 (0, 0, 0), that is, everywhere except the origin, and if S is the sphere of radius 1 and center (0, 0, 0), then S Curl( F ) ds = 0. (vi). If F is a conservative vector field, C 1 on all of R 3, and if S is a closed oriented surface, that is, with no boundary, then S F ds = 0. 12
11. Consider the vector field on R 3 (0, 0, 0) given by F = 1 (x, y, z) where r = r3 x 2 + y 2 + z 2. Let S a be the sphere of radius a given by x 2 + y 2 + z 2 = a 2 with outward unit normal n. (a)[4]. Calculate the flux integral S a F n ds, WITHOUT using any parametrization. SHOW WORK. [You may assume the surface area of S a is 4πa 2. ] (b)[2]. Routine derivative calculations yield Div( F ) = 0 on the domain of F. Why doesn t Gauss s theorem give the result S a F n ds = 0? (c)[2]. Is it possible that F = Curl( G) for some vector field G on R 3 (0, 0, 0)? Give a reason why or why not. 13
12[10]. The graph on the next two pages plots a vector field F (x, y) = (P (x, y), Q(x, y)), oriented curves A, B, C, and points M, N, R, T. One assumes F is a C 1 vector field. Answer the following ten questions. There are two copies of the graph so you can tear off the last page while contemplating the questions. One point for each correct answer. (i). Is the circulation of F around curve A positive, negative, or 0? (ii). If n is the continuous unit normal to curve C pointing downward (except near N), is the flux across C in the n direction positive, negative, or 0? (iii). Is the quantity C F ds positive, negative, or 0? (iv). Is the value of p x + Q y at T most likely positive, negative, or 0? (v). Is the quantity Q x P y at M most likely positive, negative, or 0? (vi). Is the divergence of F at N most likely positive, negative, or 0? (vii). If D is the plane region bounded by curve A, is the quantity D positive, negative, or 0? Div( F ) dxdy (viii). If F is a fluid flow velocity field, would a small paddle at N most likely spin clockwise, counterclockwise, or not at all? (ix). If F is a force field, is the work done by F on a particle moving around directed curve B positive, negative, or 0? (x). Is it possible that F = f for some function f(x, y)? 14
y R T A B M N C x 15
y R T A B M N C x 16