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Unformatted text preview: University of Waterloo, Waterloo, Ontario Faculty of Mathematics
Department of Applied Mathematics AM 231
Calculus 4
Midterm test February 14th 2007 Time 1%Hrs. Instructor: C.G. Hewitt
NO AIDS
N O CALCULATORS There are 6 questions on this paper.
Please attempt them all.
Relevent justiﬁcation and sketches will help your grade. Question Mark H
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Family Name ________________
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u . Q1 Smarks In this question we consider the pArt the hyperbole :2 — y2 = 1 starting
at (1 0) and ending at.( (V l + 17., p) where p is apositive real number .You
do not have to evaluate any of the integrals 1n this question, but state
)1/
your ﬁnal solution as an integral with respect to y. 3’ 0,301.) :; (at) 3.) Write down an expression for the length of the curve. a m 3
(/11 f " 7/1 JMJul/Mummy
/:‘C I: iC/+%L)é M. (I, 3 (tiy)" "j b)If 0(33, y) denotes the mass per unit length of the curve then write
down an expression for the mass of the curve. a a
M: L/(Jr :L/rclrx’i": M169 0) If F = (P, Q) denotes a. force ﬁeld in the xy plane, then write down
an expression for the work done by the ﬁeld in moving an object along lemma 1? Q
Wafﬂe :— P5 my
C o (Hf/4 d) If J = (R, S) denotes a. mass ﬂux density (mass ﬂow per unit area
per unit time), then write down an expression for the rate at which ﬂuid 9
m :3fzrws— DPW 4y
7c mass ﬂows across the curve. (1 t7? ”27‘ . 7°"
“ 25)."; ' "in the 151311e, and prowde a’bnef physical mterpretaho
,, ‘ small paddle wheel. . 7 (Mu/lb". 1.1; FA,— 7 CD 4’“ hﬂQJ'A/Ibu cock t/W‘m ~ﬂ~ 4 A 0
v 1')" ”9" AM "MooAIM“ ,5; b) Prove that if F 15 constant then the circulation 1s zero. y F, (0,711.; At. '06? _2_P :0 7+ '33 “Gm57“ (/FJj QJJ (Q’ ;,)// :a' = mun/Ma @ c) Deﬁne the vorticity of: F and provide a brief physical interpretation
in terms of a small paddle wheel. .4 (625 r310 Aaé A" male l“! J91#.%ﬂ _
74. Jana/s: Jan” I.“ n/df' ”Ab/«[email protected] (1) Show that if F: vd) then f F dx= ¢(g(b))— ¢(g(a)) where C Joins
g(a) to g(b) and F and d) are leunctions. ‘4‘ Q L/E/i : 174141 = f/%53*%5’ 3$/# e) Show that the vector ﬁeld F = (221/ + y2,'2$y + 1:2) is conservative
and thus evaluate ch.dx along any path joining (0, 0) to (1, 1). @ iffy Z 2312% “(($113) :0 3 ¢) dyﬁ‘l’ ¢_y‘~ Q.
4P : V">*5"‘ ﬂ 7A. 1': $0”, ¢le 2’2_ f) Show that the vector ‘ﬁeld F = (3211+ y2,2zy+:1:2) is not conservative.
// I ‘ f. 4’ :7 .
QY’)) 25+27L67Lf2a): —>‘.50 0 I9 an} (”0,911 Q310n1arks _ v v 1 1 _ ‘ 1
a‘) Prove that if a particlemoves on the surface of a sphere centred at , l
(0, 0, 02 then its veloc1ty vector is orthogonal to its position vector H , O ,  I _ 0 , '
.Ja yon 4’»; as/ h‘tié nae ar/Jgsma/ (D b) A planet of mass m moves in the gravitational ﬁeld of a star and experiences a force
k r
F = —3
7. where r is the position of the planet relative to the star and 'r = r”,
and k is a constant. The motion of the planet is governed by Newton’s second law: d2r Show that the planet moves in a plane by showing that r and ‘31—: lie in a
ﬁxed plane. ‘ Q4 14 marks La) State Gteenstheorem d ' ' ,  y ,C
"1+ . 0,, 1.1,...wa amtw 4...“, IIWJM am‘5J an}: :41ka J . C9 ,4 m. w [email protected]~ 7x... Jc Fa’Hély :ﬂJ/(otPauﬂ , , Q b) Evaluate the following integral two ways, (i) as a line integral (ii) as a double integral.
1(22y—22)d2+(z+y2)dy ’— ,‘/C {7er Q4; where C is the closed curve of the region bounded by y— = 12 and :1:— — 3/2 957‘" (a , 4:(2]{"j £[[a)/J.
 ‘ 1 f fmqj." /J¢¢M¢I‘r __':‘,Q510mrks// é ‘, __ _‘ A, A
“ ,ﬁLetF— 5(ﬂw; 53—+1P% andconsnderthelmemtegralofFaioundvanous‘r j
simple piecewise Clclosed curves :in the :s— y plane. ' ' , ~ * ‘ I a.) Expldin why f0 F. dr 15 undeﬁned for any siniple piecewise Clclosecl
curve in the :c— y plane which goes through (0,0). A»; wan/2 lo a?“
@ (fluvii é A //‘IV /@~/ I I I,b)Prove that fc F.dr is zero for any simple piecewise Clclosed curve in
the :1:  1/ plane which does not contain (0,0) either on C or inside 0. @973 : #— [(0,9)[4271] 7" CWT)" ‘CZ’X‘JJ
1. L
/>¢+y) @
__ <7~‘*.>")‘U‘ 4 (#WJ‘) 4:" . O . (90+?) 1 —.3
not»;
c)Prove that fc F.dr is 27r for any simple piecewise Clclosed curve tra—
versed anticlockwise in the :1:  y plane which contains (0, 0). W {1. (b o In” 609.4! Juno«dd, 2
’4 6‘“; ﬁn.” [‘9 (an! 2':
[lg ~ 4 Mh/f.’~// (”a w M A“ ”A" 7
‘ ﬁQS 15311?st l f ‘ ‘ " ‘
‘ ‘ I In thisquéstidﬁ weirdensider the bene‘i ‘ “1 I ‘ ‘
8.) Express the surface area of this ‘cone as a double integral and then 0
evaluate the double integral.
x: ”49 y:r.u§9 2=V‘/ol 93.640.
2:5
94;:27’ 0‘2 54 :2 o__< hfach
 n J L
— in to 2}_ch9.ma, / ’a
I .
Q;,gag_ 77‘) If /__Y‘C€:§,i/_95I)./ ' I, l / t x '
,4: 0,»; WW :(n; "IL/5‘22 : mum b) The cone is coated with a thin layer of density p = poi—Eifﬁil. Express
the mass of this layer as a double integral and then evaluate «this integral. ‘1
, ...
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