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**Unformatted text preview: **(60 Math 2015 Applied Multivariate and Vector Calculus: Test5 Friday November 27, 2009
10:30am to 11:20am Name: SOLO”! to N g Student Number: Instructions: Complete all 4 of the following problems in the space provided. Notes and
calculators are not permitted. All cell phones and pagers are to be turned off. 1. (a) Show that the vector ﬁeld g WM“ lit; ~
F(x, y) = (3x2 — 3y2 — 2x)i + (—Gwy + 2y)j is conservative and then ﬁnd a potential function f (x, y) such that F = V f . (b) Use your result in part (a) to compute the line integral /CF($,y) -dr Where C is any curve connecting the points (0, 0) and (1,3). Le} m: ”$23 Jag-”Ill. N‘ = 4013 + 2}) 9M
—‘ 3 " (9 DE : “(a I ,\
3‘3 “A ’DBL ‘3 CD 2. Using spherical coordinates, evaluate the triple integral /// x2+y2+22dV
D Where D is the region that lies above the xy-plane, below the cone 4b = 7r/ 4, and between the spherespzaandp=bfor0<a<b. S me‘k)
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C Where F(m,y, z) = (y +1)i— (a: + 4)j + (a: + y + 7r)k. Do this when (a) C is the helical
curve a: = cost, y = sint7 z = t/Tl', for 0 g t 3 7r. Repeat the calculation of the integral
when G is the straight line between the points (1,0,0) and (—1,0, 1). Comparing your
results from parts (a) and (b), What can you conclude about F? A [00 Ht) : ng‘bﬁmt ,‘t/TL‘] Dét éTr ‘ lull” ‘5 ['Sln‘t, cost, UTE}
if. E“ AV 3 g . FlH/7 Al?“
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C‘s0,o3 mot Lv-l,o,1) W7: "50+ JCS U‘nM :1: k,\‘0,|3> (1,0,0) :6- 2,0,1) 5) PW): (Lo,o)+ {:LJAO”)
: (4‘—LbIOI+/7 bété‘ =3 ‘6‘ Hi): (4'2, 0‘ I > A "‘ A
Fm = FF'lth;
= \IJb—s, -\-Lt+ﬂ~[—2,o.u]c\t : \:Z +£1~£h+w>3dﬁ~ = (Ir-'1 *Z-mzk \.¢ 3 E'd‘? = 3‘ hpqdvcjcﬁ» —-® 0 9mm. (mull/‘5 do (:0 W [b) are, dgg‘QUWhQ CD 4. Consider a region D of the xy plane whose boundary C is a piecewise smooth simple
closed curve. Further, suppose the functions qS and d) have continuous ﬁrst and second
order partial derivatives on D. Use Green’s theorem to prove the following important identity g WWIH‘l% .
£¢V¢-dr=—j€¢v¢‘dr-
Fxrsl“ WscMr LHs- (4} M z 521,» A} = 4, D1? Tm 53 Ctrwn‘s “ll/124mm éc Zl’vxl'dﬂé‘ : :M'Bcﬁ'aj’1L4DR—1iz’9j23’ 2:1
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