mat293_q2_2007_solutions - MAT293F VECTOR CALCULUS Quiz 2...

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Unformatted text preview: MAT293F VECTOR CALCULUS Quiz 2 15 October 2007 10:05 am — 10:55 am Closed Book, no aid sheets, no calculators Instructor: J. W. Davis Family Name: A . \C) . \bo-Lfl‘; Given Name: 80‘ i M , Student #: TA Name/Tutorial Section: FOR MARKER USE ONLY Note: The following integrals and formulae may be useful: [coszfidfl = i0 + lsin20 + C; fsin20d0 = 2 4 la — —1—sin20 + C 2 4 gfcpdm Qdy = [magi gde; Page 1 of 6 1) For the coordinate transformation x = rcosfi, y=rsin6for r€[0, 00] and 6€[0, 7272], by deriving expressions for both J acobians, verify that my) 2 [may 50,6) any) Hint: Recall that d/dx (arctan x) = 1/(1+x2). (10marks) In.- \ Q 1: Fang xr: c039 3C9 ram. \ ‘- 1 r a (23:. value? ‘Er" 5““0 :9 an MW _ °C9\ s v c.0119 “arse w lL‘HQi - ‘3‘: Io \ -".S 4L: ‘3 _ ., _ = _ ,La ,_t = 3.; 93 HG: )‘L 15 x341"- Afl _ it “1 .32., x Us L5 - _ __..— + __.— _____ A W“ at Q: a 1w M“ 373411 new - in: ._ i : .L Lxlki‘ Bah" 114*: 1 f" ___ ~\ QLMG) _ JL'KQ-gi - sign-Ll) éerey Page 2 of 6 2) Let R be the plane region of unit density that is bounded by the hyperbolas: xy=1, xy=3, xZ-yZ =1 and x2-y2=4. Use an appropriate coordinate transformation to find the polar moment of inertia 10 = [Roz + yodxdy of this region. Sketch the region in both the x-y plane and the u—v plane. :10 marks) 1.2.“;3 \ /C‘F LL .1303 15% is 3 L 5: u‘ U“: xlfll r —L7 'L 'L 1 x =4 U. 13.3.3 R 331::- x u» ‘ “3 “3 “L J :Z‘K, L7. '3' hid-3 JC“ v — L5 1 -: “2‘31_ 111' "-'- "1kx'11. “31) lkxkfi) 13L *1»; stun a .1; .. 'L WLR‘LS 'L To "' J (11*;313 = S R S Page 3 of 6 3) Find the second degree polynomial approximation to the function f (x, y) = ‘lxz + y3 near the point (1,2). (Smarks) Jaw u M We) ‘ 3 fix ‘ fi’? 9“ (NA =' if; x” “3 u ’5 A: L .. “L R .1 Rh) :5} J|1 2. SM v kit-N-Xb) + it‘ll» 11"“35) '7‘)!” _ xlfisvxl _ "33 _’ Q [\kL)= "f: [1%5311‘2 LIKES) ’7. M 59(1 Qx‘j = li‘li3‘ix1‘Ff) ' 53L _ -3 39;; = -3 1 kxk33)3f7_ $1.3 (KL) cl -711 -\‘_L L i 1 .5 ‘ 3:31 = 339%) +21% CAI“) 311 d‘ _ 3.3 kxkx'fi - E‘g‘ [w Samba?) = 33 W LIL-.0413) I 4 I, 7; vutL a, +1 MW e 3 4 away; + a m5 at H ) 3 Page 4 of 6 4) Verify GTeen’s theorem for the integral: (jide + sin ydy C where C is the triangle with vertices (1,1), (2,2), (2,0). (14 marks) 5 Q‘ L: =. 1—71 3) )Lr—t CS (épcfizl £31714: Q1 CL 1.11 =0 1:? C, 05-35:?— “5: " C5 fibx ==7 x124: ZZDLZ\ LX—Z—E I. ‘3 C(Ltw‘bfl M? ?é>¢_ + -‘ j :L— % B c. P— ‘3 at) C, '?-— 1-3 _ t [1%) Jung? Gu- stw‘g - 9m Vbt) 4%" bed: . 1'— \ BMLHDAX = ) JC [ 7:th + smbfi) (wit) 0‘ t z L -' _ i L- t: _ 1—4.) ‘) (1{_{1aslmL1-t))cit ’ L4 5 “UL :(l .L m .—. cenl “*1 : 4-93. .._.‘ _-l +3 'i' C l 3 C1: ?= dJL': G : 5:“...E :cL‘b 'L S Zt-o + Siva-i: At -— J 55* ‘L’C " [‘C‘fitl Page 5 of 6 C3 _ \s 1 i g 1 Z L ‘* CmKV-"EBla '— 3‘ (03‘ ‘" E + (m1 =\'§+Lu~ «pm :7 é}?éxj =- cm‘ “3- +\ —cur3?, fig— +gm2— (,o'b\ C. _ .._5_ F .3 G la.) Q a sums =3 é—i *0 A? , .13 :- XS ‘3’) ‘73 x Z L j if? - £5) :11 “ (~L) 0111 = ‘JJXJ 4‘3 3“ ¢ an I \z 3‘ ‘3 2 Pl 7. 2. )(_ .— E (Ix-b _ L 7 - xbc 1+1) “ —J 41 L51:- . = ., 1x. "a. ._ d _ h _. _. +[ __ a Dc - L “a, 1 L 3 3 5) Given a surface defined parametrically by r(u, v) = x(u, v)i + y(u, v)j + z(u, v)k, show that the surface area can be found from: LdS= majjdudv (8 marks) Page 6 of 6 ...
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This note was uploaded on 10/18/2010 for the course ENGINEERIN MAT293 taught by Professor J.davis during the Fall '08 term at University of Toronto- Toronto.

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mat293_q2_2007_solutions - MAT293F VECTOR CALCULUS Quiz 2...

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