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Tue Nov 8 15:17:41 2011 UTC (18 months, 2 weeks ago) by aubreyja
File size: 2892 byte(s)
Rogawski problems contributed by publisher WHFreeman. These are a subset of the problems available to instructors who use the Rogawski textbook. The remainder can be obtained from the publisher.

    1 ## DBsubject('Calculus')
2 ## DBchapter('Fundamental Theorems of Vector Analysis')
3 ## DBsection('Greens Theorem')
4 ## KEYWORDS('calculus')
5 ## TitleText1('Calculus: Early Transcendentals')
6 ## EditionText1('2')
7 ## AuthorText1('Rogawski')
8 ## Section1('17.1')
9 ## Problem1('25')
10 ## Author('JustAsk - Kobi Fonarov')
11 ## Institution('W.H.Freeman')
12 ## UsesAuxiliaryFiles('image_17_1_27.png')
13
14 DOCUMENT();
15
22
23 TEXT(beginproblem());
24
25 $c2=random(2,9); 26$c3=random(2,9);
27 $curl=random(2,9); 28$c23=$c2+$c3;
29
30 $answer=Real(($c23+$curl*23)*$PI);
31
32 $curve="\mathcal{C}"; 33$domain="\mathcal{D}";
34 $FF="\mathbf{F}"; 35 36 TEXT('<SCRIPT>jsMath.Extension.Require("AMSmath");</SCRIPT>') 37 if$displayMode eq 'HTML_jsMath';
38
39 Context()->texStrings;
40
41 BEGIN_TEXT
42 \{ textbook_ref_exact("Rogawski ET 2e", "17.1","25") \}
43 $PAR 44 Referring to Figure 11, suppose that 45 $46 \oint_{{curve}_2} FF \cdot d\mathbf{s} = c2\pi,\qquad \oint_{{curve}_3} FF \cdot d\mathbf{s} = c3\pi 47$ 48 Use Green's Theorem to determine the circulation of $$FF$$ around $${curve}_1$$, 49 assuming that curl$$_z(FF)=curl$$ on the shaded region.$PAR
50 \{image("image_17_1_27.png", width=>146, height=>186)\}
51 $PAR 52 $$\int_{{curve}_1} FF \cdot d\mathbf{s} =$$ \{ans_rule()\} 53$PAR
54
55 END_TEXT
56
57 ANS($answer->cmp); 58 59 Context()->texStrings; 60 SOLUTION(EV3(<<'END_SOLUTION')); 61$PAR
62 $SOL 63 We must calculate $$\int_{{curve}_1} FF \cdot d\mathbf{s}$$.$BR We use  Green's Theorem for the region $$domain$$ between the three circles $${curve}_1$$, $${curve}_2$$, and $${curve}_3$$. $PAR 64 Because of orientation, the line integrals $$\int_{-{curve}_2} FF \cdot d\mathbf{s} =-\int_{{curve}_2} FF \cdot d\mathbf{s}$$ and $$\int_{-{curve}_3} FF \cdot d\mathbf{s} =-\int_{{curve}_3} FF \cdot d\mathbf{s}$$$BR must be used in applying Green's Theorem. That is,
65 $\int_{{curve}_1} FF \cdot d\mathbf{s} -\int_{{curve}_2} FF \cdot d\mathbf{s} -\int_{{curve}_3} FF \cdot d\mathbf{s} =\iint_{domain} \text{curl} (FF) \,dA$
66 We substitute the given information to obtain
67 $\int_{{curve}_1} FF \cdot d\mathbf{s} -c2\pi -c3\pi =\iint_{domain} curl \,dA=$$68 curl\iint_{domain} 1\cdot \,dA=curl \,\mathrm{Area} (domain) \quad \mathbf{(1)}$
69 The area of $$domain$$ is computed as the difference of areas of discs. That is,
70 $\mathrm{Area} (domain)=\pi \cdot 5^2-\pi \cdot 1^2-\pi \cdot 1^2=23\pi$
71 We substitute in $$\mathbf{(1)}$$ and compute the desired circulation:
72 $\int_{{curve}_1} FF \cdot d\mathbf{s} -c23\pi =curl\cdot 23\pi$
73 or
74 $\int_{{curve}_1} FF \cdot d\mathbf{s} =\{c23+curl*23\}\pi.$
75 END_SOLUTION
76
77 ENDDOCUMENT();