View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/17_Fundamental_Theorems_of_Vector_Analysis/17.1_Green's_Theorem/17.1.25.pg

    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 
   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();