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midterm2_b - MATH 251 SECOND HOUR EXAM Version B Professor...

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Unformatted text preview: MATH 251 - SECOND HOUR EXAM Version B Professor Weibel November 19, 2009 Section Section 7 (W2 10:20AM) Section 8 (W3 12 PM) Section 9 (W4 1:40PM) There are 100 points on the exam. The notation [n] at the start of a problem means that the problem is worth 77, points. Rules: No calculators. Remember—show your work to get partial credit. If you need more space, write “over” and use the back of the page. J acobians and Change of Variables. Suppose that under the 1—1 differentiable trans— formation :c = x(u, v) and y = y(u, 11) that the region R in the Jay—plane corresponds to the region S in the uv—plane. Then //[email protected]=//fwww where the J acobian ‘g—g’—Z; y) 8W: 1)) du dv 6—1: is det (37 23 6_y 8—”). For polar coordinates, d2: dy— — 1' d?" d0 611, 6—11 The 3—D analogue (between coordinates u,v,'w and say, z) amounts to the change-of— a($,y,z) 8(x’y’z) 6(u,’u,'w) 8(u,’u,w) the analogous 3 x 3 matrix of partial derivatives. For cylindrical coordinates, dw dy dz = 7" dr d6 dz. For spherical coordinates, dm dy dz = p2 sin(¢) dp d¢ d0. variables relation d3: dy dz = } is the determinant of du dv dw where 1 Criterion for the Existence of a Potential Function. If E 2 (F1, F2) is a vector field in a simply connected plane region, F is a gradient field if and only if 8F1 /6‘y ': 8F2/8x. —; If E 2 (F1, F2, F3) is a vector field in a simply connected region in 3—space, F is a gradient vector field if and only if the three cross partials agree: for all 2' and j, 8F, / 81:,- = BFj/ami. Line integrals. Let f"(t) parametrize a path 0 (to S t 3 t1). Then f0 f(3:, y,z)ds = t1 H0 f(f~(t))(ds/dt)dt. Iff‘ is a vector field, then [C E(m, y, z)-d§= tie, F(r (t)) (dr/dt)dt If F = VU, f0 F . d§ = U(f’(t1))— 0mm». Surface Area. If z : z(a:, y) defines a surface S in 3-space over a region R in the (m, 3;) plane, its surface area is //SdA=//Rllfilldwdy, Where||fi||=WIWj [7] Integrate f (m, y) = my over the region bounded by the line y = a: and the parabola y = 1:2 ~ 2:. (These curves meet at the origin and at 0417.) [5] Sketch the domain of integration, and express the iterated integral in the opposite order: 5 25 / f (iv, y) dy div III [10] A Wire bracelet in the shape of the curve y = sin(a:) (0 _<_ x g 7r/2) has density p(a:, y) = y. What is its mass? (Be careful With minus signs.) 3/2 [8] Find the surface area of the surface 8' given by z = 23: + By which lies over the unit square 0 3 33,3; _<_ 1 in the (ac 3/) plane. 953: 32”“ SA- JJW‘JW 25:3 :XS‘WAXA —jWAx O “‘ zik‘t ”(Ch H03 :Di‘l O 2 gammdiomJ [12] Set as = 311. + 2'0 and y = —u + '0. Let R denote the region in the (x,y)-plane corresponding to the unit square 0 S u, v S 1 in the (u, v)—plane. (a) Draw the picture of the region R in the (7:, y)-plane (b) Apply the change of variables formula to find ffR a: dx dy. ‘l (M) ,< E \ 21:) O \ .l 0 [8] Let U (x, y) 2 fig‘ and consider the vector field I? = VU. Determine f0 F - d§ Where C is the path f’(t) = (1 — t,t), 0 S t S 1. lite] : (l ' O) 5—3 L“ ‘3‘- (OI‘) $EA§3r U(O)\\——U(J‘O) G”: :14 :EJ [25] Consider the region between the (m, y)—p1ane and the cone m2 + 3/2 = 2'2 over the disk 0 S m2 + y2 S 4. Compute its volume in three ways: (a) Integration in Cartesian Coordinates: set this integral up, including all limits of inte— gration, but do not evaluate this integral. (b) Integration in Cylindrical Coordinates. a.“ '1 r ’«k P V: j szch—dg = afij‘ (ale (if 1 3m jargdr ‘— QT: (31:: “3-3;: 0 (0) Integration in Spherical Coordinates. Hint: 4'—=—p-ees(9§—_§2- F‘ r. f S'IALQ E: 1 R '1 ‘11.; \113 V ‘ fats/Ge AS Afifidg 2 a; 3 5% 4. egg ’8 CW :93 3: (JUL )5th ALL 3 1T 5“}? 3 if L’ _ H33 2 Q‘KAQ __ ““21 -\-%\E: “33} ’ 351C Lso —- 3 ban 115‘ 3 [10] Consider the average value of 1 / :3 inside the circle 7‘ = 2 cos(6) of radius 1 about the point (1,0). Show your work. ”Wk noose E: J. _L_ PJfiAG firm“ rcos‘B .i o a. i ’3 - : ,fi 5 :1 are - a "E a [15] Consider the vector field F = (yz, mz, a: + z). (a) Is there a potential function U such that F = VU? Explain your answer. No. 95-7 last EA. 21' 9* (b) Compute the work f0 F - ds’ done in moving from P = (0,0,0) to Q = (3, 1, 5) along the path f(t) 2 (3t, 75, 5t). ELFLLW ’-‘- <56) 58‘} $E> €m1<3;"5> \ l XOF‘A—S ; YG6€+ISS+W©Q 4% D l 3 5 (308+ woe) 3’: U l .3 1 lot +ROC lo \I ...
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