# midsolns - University of Waterloo Waterloo Ontario Faculty...

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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 «:21 /8J Q2 k / 12 ”63 /1T[l Q4 ' / 14 Q5 my Q6 / 16 Total _ Family Name ________________ First Name __________________ Signature ____________________ I.D. # ______________________ , 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 (ti-y)" "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 x-y 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’“ h-ﬂQJ'A/Ibu cock t/W‘m ~ﬂ~ 4 A 0 v 1')" ”9" AM "Moo-AIM“ ,5; b) Prove that if F 15 constant then the circulation 1s zero. y F,- (0,711.; At. '06? _2_P :0 7+ '33 “Gm-57“ (/F-Jj 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 : 1741-41 = 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 23-12% “((\$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+27L-67L-f2a): -—>‘.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/(ot-Pauﬂ , , 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 “ ,ﬁ-LetF— 5(ﬂw; 53—+1P% andconsnderthelmemtegralofFai-oundvanous‘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?“ @ (flu-vii 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 anti-clockwise 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/_95-I)./ ' 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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