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| 1 : | aubreyja | 2584 | ## 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 : | |||
| 16 : | loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl"); | ||
| 17 : | loadMacros("Parser.pl"); | ||
| 18 : | loadMacros("freemanMacros.pl"); | ||
| 19 : | loadMacros("PGauxiliaryFunctions.pl"); | ||
| 20 : | loadMacros("PGgraphmacros.pl"); | ||
| 21 : | loadMacros("PGchoicemacros.pl"); | ||
| 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(); |
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