View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/17_Fundamental_Theorems_of_Vector_Analysis/17.2_Stokes'_Theorem/17.2.13.pg

    1 ## DBsubject('Calculus')
    2 ## DBchapter('Fundamental Theorems of Vector Analysis')
    3 ## DBsection('Stokes Theorem')
    4 ## KEYWORDS('calculus')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('17.2')
    9 ## Problem1('13')
   10 ## Author('JustAsk - Kobi Fonarov')
   11 ## Institution('W.H.Freeman')
   12 ## UsesAuxiliaryFiles('image_17_2_13.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 $context = Context();
   26 $context->variables->add(y=>'Real');
   27 
   28 
   29 $a=random(2,9);
   30 $b=random(2,9);
   31 
   32 $fy=Formula("$b*x^($b-1)*e^(x^($b))")->reduce();
   33 $az=Formula("$a*e^y - e^(x^($b))")->reduce();
   34 
   35 
   36 $curve="\mathcal{C}";
   37 $domain="\mathcal{D}";
   38 $surf="\mathcal{S}";
   39 $FF="\mathbf{F}";
   40 $curl="\text{curl}";
   41 $ii="\mathbf{i}";
   42 $jj="\mathbf{j}";
   43 $kk="\mathbf{k}";
   44 $BB="\mathbf{B}";
   45 $AA="\mathbf{A}";
   46 $rr="\mathbf{r}";
   47 $cc="\mathbf{c}";
   48 $GG="\mathbf{G}";
   49 
   50 TEXT('<SCRIPT>jsMath.Extension.Require("AMSmath");</SCRIPT>')
   51        if $displayMode eq 'HTML_jsMath';
   52 
   53 Context()->texStrings;
   54 
   55 BEGIN_TEXT
   56 \{ textbook_ref_exact("Rogawski ET 2e", "17.2","13") \}
   57 $PAR
   58 Let \(I\) be the flux of \($GG = \left<$a e^y,$fy,0\right>\)
   59 through the upper hemisphere \($surf\) of the unit sphere. $PAR
   60 $BBOLD (a) $EBOLD Find a vector field \(\mathbf{A}\) such that \($curl(\mathbf{A})=$GG\).
   61 $BR
   62 $BBOLD (b) $EBOLD Calculate the circulation of \(\mathbf{A}\) around \(\partial $surf\).
   63 $BR
   64 $BBOLD (c) $EBOLD Compute the flux of \($GG\) through \($surf\)
   65 $PAR $PAR
   66 $BBOLD (a) $EBOLD \(\mathbf{A}=\) \{ans_rule()\} \(+ \left<C_1,C_2,C_3\right>\) $BR
   67 $BBOLD (b) $EBOLD \(\int_{$curve} \mathbf{A}\cdot d\mathbf{s}=\) \{ans_rule()\} $BR
   68 $BBOLD (c) $EBOLD \(I =\) \{ans_rule()\}
   69 $PAR
   70 
   71 END_TEXT
   72 
   73 Context("Vector");
   74 ANS(Vector(Real(0),Real(0),Formula($az))->cmp);
   75 Context("Numeric");
   76 ANS(Real(0)->cmp);
   77 ANS(Real(0)->cmp);
   78 
   79 
   80 Context()->texStrings;
   81 SOLUTION(EV3(<<'END_SOLUTION'));
   82 $PAR
   83 $SOL $PAR
   84 $BBOLD (a) $EBOLD We search for a vector field \(\mathbf{A}\) so that \($GG=$curl  (\mathbf{A})\). That is,
   85 \[\left< \frac{\partial {\mathbf{A}}_3}{\partial y}-\frac{\partial {\mathbf{A}}_2}{\partial z},\frac{\partial {\mathbf{A}}_1}{\partial z}
   86 -\frac{\partial {\mathbf{A}}_3}{\partial x},\frac{\partial {\mathbf{A}}_2}{\partial x}-\frac{\partial {\mathbf{A}}_1}{\partial y} \right> =
   87 \]\[
   88  \left<  $a e^y ,$fy,0 \right> \]
   89 We note that the third coordinate of this curl vector must be zero;
   90 this can be satisfied if \(\mathbf A_1 = 0\)
   91 and \(\mathbf A_2 = 0\). $BR
   92 With this in mind, we let \(\mathbf{A}= \left< 0,0, $a e^y - e^{x^{$b}} \right>\). $BR
   93 The vector field \(\mathbf{A}= \left< 0,0,$a e^y - e^{x^{$b}} \right>\) satisfies this equality. Indeed,
   94 \[ \frac{\partial {\mathbf{A}}_3}{\partial y}-\frac{\partial {\mathbf{A}}_2}{\partial z}=$a e^y ,\quad
   95    \frac{\partial {\mathbf{A}}_1}{\partial z}-\frac{\partial {\mathbf{A}}_3}{\partial x}=$fy,\]\[
   96    \frac{\partial {\mathbf{A}}_2}{\partial x}-\frac{\partial {\mathbf{A}}_1}{\partial y}=0\]
   97 
   98 $BBOLD (b) $EBOLD
   99 The boundary \($curve\) is the circle \( x^2+ y^2=1\), parametrized by
  100 \[\gamma (t)=(\cos t, \sin t,0),\quad 0\le t\le 2\pi \]
  101 \{image("image_17_2_13.png", width=>208, height=>186)\} $BR
  102 We compute the following values:
  103 \[\mathbf{A}\left(\gamma (t)\right)=
  104   \left< 0,0,$a e^y - e^{x^{$b}} \right> \bigg|_{x=\cos t\text{,}y=\sin t}=
  105   \left< 0,0,$a e^{\sin t}- e^{\cos^{$b}t} \right>\]
  106 \[{\gamma }'(t)=\left< - \sin t, \cos t,0 \right>\]
  107 \[\mathbf{A}\left(\gamma (t)\right)\cdot {\gamma }'(t) =\]\[
  108   \left< 0,0, $a e^{\sin t}- e^{{ \cos}^{$b}t} \right> \cdot  \left< - \sin t, \cos t,0 \right> =
  109   0\]
  110 Therefore,
  111 \[ \int_{$curve} \mathbf{A}\cdot d\mathbf{s} =\int_0^{2\pi } 0 \,dt=0 \quad \mathbf{(1)} \]
  112 $BBOLD (c) $EBOLD We found that \($GG=$curl  (\mathbf{A})\), where \(\mathbf{A}= \left< 0,0,$a e^y - e^{x^{$b}} \right>\). $PAR
  113 We compute the flux of \(\mathbf{G}\) through \($surf\). By Stokes' Theorem,
  114 \[
  115 \iint_{$surf} $GG\cdot d\mathbf{S} =
  116 \iint_{$surf} $curl  (\mathbf{A})\cdot d\mathbf{S} =
  117 \int_{$curve} \mathbf{A}\cdot d\mathbf{s}
  118 \]
  119 Combining with \(\mathbf{(1)}\) we get
  120 \[\iint_{$surf} $GG\cdot d\mathbf{S} =0\]
  121 
  122 
  123 
  124 
  125 END_SOLUTION
  126 
  127 ENDDOCUMENT();