## DBsubject('Calculus')
## DBchapter('Fundamental Theorems of Vector Analysis')
## DBsection('Greens Theorem')
## KEYWORDS('calculus')
## TitleText1('Calculus: Early Transcendentals')
## EditionText1('2')
## AuthorText1('Rogawski')
## Section1('17.1')
## Problem1('25')
## Author('JustAsk - Kobi Fonarov')
## Institution('W.H.Freeman')
## UsesAuxiliaryFiles('image_17_1_27.png')

DOCUMENT();

loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
loadMacros("Parser.pl");
loadMacros("freemanMacros.pl");
loadMacros("PGauxiliaryFunctions.pl");
loadMacros("PGgraphmacros.pl");
loadMacros("PGchoicemacros.pl");

TEXT(beginproblem());

$c2=random(2,9);
$c3=random(2,9);
$curl=random(2,9);
$c23=$c2+$c3;

$answer=Real(($c23+$curl*23)*$PI);

$curve="\mathcal{C}";
$domain="\mathcal{D}";
$FF="\mathbf{F}";

TEXT('<SCRIPT>jsMath.Extension.Require("AMSmath");</SCRIPT>')
       if $displayMode eq 'HTML_jsMath';

Context()->texStrings;

BEGIN_TEXT
\{ textbook_ref_exact("Rogawski ET 2e", "17.1","25") \}
$PAR
Referring to Figure 11, suppose that
\[
\oint_{{$curve}_2} $FF \cdot d\mathbf{s} = $c2\pi,\qquad \oint_{{$curve}_3} $FF \cdot d\mathbf{s} = $c3\pi
\]
Use Green's Theorem to determine the circulation of \($FF\) around \({$curve}_1\),
assuming that curl\(_z($FF)=$curl\) on the shaded region. $PAR
\{image("image_17_1_27.png", width=>146, height=>186)\} 
$PAR
\(\int_{{$curve}_1} $FF \cdot d\mathbf{s} = \) \{ans_rule()\}
$PAR

END_TEXT 

ANS($answer->cmp);

Context()->texStrings;
SOLUTION(EV3(<<'END_SOLUTION'));
$PAR
$SOL
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
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,
\[ \int_{{$curve}_1} $FF \cdot d\mathbf{s} -\int_{{$curve}_2} $FF \cdot d\mathbf{s} -\int_{{$curve}_3} $FF \cdot d\mathbf{s} =\iint_{$domain} \text{curl}  ($FF) \,dA \]
We substitute the given information to obtain
\[\int_{{$curve}_1} $FF \cdot d\mathbf{s} -$c2\pi -$c3\pi =\iint_{$domain} $curl \,dA= \]\[ 
$curl\iint_{$domain} 1\cdot \,dA=$curl \,\mathrm{Area} ($domain) \quad \mathbf{(1)}\]
The area of \($domain\) is computed as the difference of areas of discs. That is,
\[ \mathrm{Area} ($domain)=\pi \cdot  5^2-\pi \cdot  1^2-\pi \cdot  1^2=23\pi  \]
We substitute in \(\mathbf{(1)}\) and compute the desired circulation:
\[ \int_{{$curve}_1} $FF \cdot d\mathbf{s} -$c23\pi =$curl\cdot 23\pi  \]
or
\[ \int_{{$curve}_1} $FF \cdot d\mathbf{s} =\{$c23+$curl*23\}\pi.  \]
END_SOLUTION

ENDDOCUMENT();