View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/17_Fundamental_Theorems_of_Vector_Analysis/17.3_Divergence_Theorem/17.3.9.pg

    1 ## DBsubject('Calculus')
    2 ## DBchapter('Fundamental Theorems of Vector Analysis')
    3 ## DBsection('Divergence Theorem')
    4 ## KEYWORDS('calculus')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('17.3')
    9 ## Problem1('9')
   10 ## Author('JustAsk - Kobi Fonarov')
   11 ## Institution('W.H.Freeman')
   12 ## UsesAuxiliaryFiles('image_17_3_7_a.png','image_17_3_7_b.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 $context->variables->add(z=>'Real');
   28 
   29 $z0=random(2,9);
   30 
   31 $a=random(2,9);
   32 $b=random(2,9);
   33 $c=random(2,9);
   34 
   35 $fi=Formula("$a*x")->reduce();
   36 $fj=Formula("$b*z")->reduce();
   37 $fk=Formula("$c*y")->reduce();
   38 
   39 $sol=$a*$z0;
   40 $answer=Real($PI*$sol);
   41 
   42 $curve="\mathcal{C}";
   43 $domain="\mathcal{D}";
   44 $surf="\mathcal{S}";
   45 $FF="\mathbf{F}";
   46 $curl="\text{curl}";
   47 $ii="\mathbf{i}";
   48 $jj="\mathbf{j}";
   49 $kk="\mathbf{k}";
   50 $dive="\text{div}";
   51 $rec="\mathcal{R}";
   52 $nn="\mathbf{n}";
   53 $TT="\mathbf{T}";
   54 
   55 TEXT('<SCRIPT>jsMath.Extension.Require("AMSmath");</SCRIPT>')
   56        if $displayMode eq 'HTML_jsMath';
   57 
   58 Context()->texStrings;
   59 
   60 BEGIN_TEXT
   61 \{ textbook_ref_exact("Rogawski ET 2e", "17.3","9") \}
   62 $PAR
   63 Verify the Divergence Theorem for the vector field and region: $PAR
   64 \($FF = \left< $fi ,$fj ,$fk \right>\) and the region \(x^2+y^2\le 1\),
   65 \(0\le z \le $z0\) $PAR
   66 \(\iint_{$surf} $FF \cdot d\mathbf{S} =\) \{ans_rule()\} $PAR
   67 \(\iiint_{$rec} $dive ($FF) \,dV=\) \{ans_rule()\} $PAR
   68 
   69 END_TEXT
   70 
   71 ANS($answer->cmp);
   72 ANS($answer->cmp);
   73 
   74 Context()->texStrings;
   75 SOLUTION(EV3(<<'END_SOLUTION'));
   76 $PAR
   77 $SOL $BR
   78 \{image("image_17_3_7_a.png", width=>260, height=>230)\} With \(z_0=$z0\) $PAR
   79 Let \($surf\) be the surface of the cylinder and \($rec\) the region enclosed by \($surf\). $BR We compute the two sides of the Divergence Theorem:
   80 \[\iint_{$surf} $FF \cdot d\mathbf{S} =
   81 \iiint_{$rec} $dive ($FF) \,dV\]
   82 We first calculate the surface integral. $PAR
   83 $BBOLD Step 1. $EBOLD Integral over the side of the cylinder. $PAR
   84 The side of the cylinder is parametrized by
   85 \[\Phi (\theta ,z)=(\cos \theta , \sin \theta ,z),\quad 0\le \theta \le 2\pi ,\quad 0\le z\le $z0\]
   86 \[$nn = \left<  \cos \theta , \sin \theta ,0 \right>\]
   87 Then,
   88 \[$FF \left(\Phi (\theta ,z)\right)\cdot $nn  = \]\[
   89 \left< $a \cos \theta ,$b z,$c\sin \theta  \right> \cdot  \left<  \cos \theta , \sin \theta ,0 \right>= \]\[
   90 $a \cos^2\theta +$b z  \sin \theta
   91 \]
   92 We obtain the integral
   93 \[\iint_{\mathrm{side}} $FF \cdot d\mathbf{S}  =
   94 \int_0^{$z0} \int_0^{2\pi } \left($a \cos^2\theta +$b z  \sin \theta \right) \,d\theta  \,dz=\]\[
   95 \{$a*$z0\} \int_0^{2\pi } \cos^2\theta  \,d\theta +\left(\int_0^{$z0} $b z \,dz\right)\left(\int_0^{2\pi }  \sin \theta  \,d\theta \right)=\]\[
   96 \{$a*$z0\}\cdot \left(\frac{\theta }{2}+\frac{\sin 2\theta }{4}\bigg|_0^{2\pi }\right)+0=
   97 \{$a*$z0\}\pi\]
   98 $BBOLD Step 2. $EBOLD Integral over the top of the cylinder. $PAR
   99 The top of the cylinder is parametrized by
  100 \[ \Phi(x,y)=(x,y,$z0)\]
  101 with parameter domain \($domain=\lbrace (x,y):\, x^2+ y^2\le 1\rbrace\). The upward pointing normal is
  102 \[$nn ={$TT }_x \times {$TT }_y =
  103 \left< 1,0,0 \right>\times \left< 0,1,0 \right> =
  104 \]\[$ii \times $jj =$kk
  105 = \left< 0,0,1 \right>\]
  106 Also,
  107 \[$FF \left(\Phi (x,y)\right)\cdot $nn = \left< $a x,\{$b*$z0\},$c y \right>\cdot \left< 0,0,1 \right>=$c y\]
  108 Hence,
  109 \[\iint_{\mathrm{top}} $FF \cdot d\mathbf{S} =\iint_{$domain} $c y \,dA=0\]
  110 The last integral is zero due to symmetry. $PAR
  111 \{image("image_17_3_7_b.png", width=>240, height=>240)\} $PAR
  112 $BBOLD Step 3. $EBOLD Integral over the bottom of the cylinder. $PAR
  113 We parametrize the bottom by
  114 \[ \Phi (x,y)=(x,y,0),\quad (x,y)\in $domain \]
  115 The downward pointing normal is \($nn = \left< 0,0,-1 \right>\). Then
  116 \[$FF \left(\Phi (x,y)\right)\cdot $nn = \left< $a x,0,$c y \right>\cdot \left< 0,0,-1 \right> =-$c y \]
  117 We obtain the following integral, which is zero due to symmetry:
  118 \[\iint_{\mathrm{bottom}} $FF \cdot d\mathbf{S} =\iint_{$domain} -$c y \,dA=0\]
  119 Adding the integrals we get
  120 \[
  121 \iint_{$surf} $FF \cdot d\mathbf{S}  = \]\[
  122 \iint_{\mathrm{side}} $FF \cdot d\mathbf{S} +\iint_{\mathrm{top}} $FF \cdot d\mathbf{S} +\iint_{\mathrm{bottom}} $FF \cdot d\mathbf{S}
  123 =\]\[$sol\pi +0+0=$sol\pi \quad\mathbf{(1)}
  124 \]
  125 $BBOLD Step 4. $EBOLD Compare with integral of divergence. $PAR
  126 \[$dive  ($FF ) =
  127 $dive \left< $a x,$b z,$c y \right>=\]\[
  128 \frac{\partial }{\partial x}($a x)+\frac{\partial }{\partial y}($b z)+\frac{\partial }{\partial z}($c y)=$a \]
  129 \[\iiint_{$rec} $dive  \left($FF \right) \,dV =
  130 \iiint_{$rec} $a \,dV=\]\[$a\iiint_{$rec} \,dV=$a\, \mathrm{Vol} ($rec)
  131 = $a\cdot \pi \cdot $z0=$sol\pi\quad\mathbf{(2)}\]
  132 The equality of \(\mathbf{(1)}\) and \(\mathbf{(2)}\) verifies the Divergence Theorem.
  133 END_SOLUTION
  134 
  135 ENDDOCUMENT();