View of /branches/UGA/8.3.2.pg

    1 ## DESCRIPTION
    2 ## Multivariable Calculus
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS('calculus','line integrals')
    6 
    7 
    8 ## DBsubject('Calculus')
    9 ## DBchapter('Differential Forms and Integration on Manifolds')
   10 ## DBsection('Line Integrals')
   11 ## Date('December 30, 2009')
   12 ## Author('Ted Shifrin')
   13 ## Institution('UGA')
   14 ## TitleText1()
   15 
   16 
   17 DOCUMENT();
   18 loadMacros("PG.pl",
   19            "PGbasicmacros.pl",
   20            "PGchoicemacros.pl",
   21            "PGanswermacros.pl",
   22            "PGauxiliaryFunctions.pl",
   23            "PGmatrixmacros.pl",
   24            "Parser.pl",
   25            );
   26 
   27 
   28 $showPartialCorrectAnswers = 1;
   29 
   30 
   31 Context("Vector")->variables->are(x=>'Real',y=>'Real',z=>'Real',t=>'Real',
   32 u=>'Real',v=>'Real');
   33 
   34 $k = random(2,5);
   35 $l = random(1,4);
   36 $ll = 2*$l;
   37 
   38 
   39 @gs1 = ("t^{$k}","t","1-sin(t)^2");
   40 @gs2 = ("t^{$k}","t","cos(t)^2");
   41 @a = ("0","0","0");
   42 @b = ("1","1","\pi/2");
   43 
   44 @hs1 = ("t^2","1-cos($l t)^2","t");
   45 @hs2 = ("t","sin($l t)","sqrt(t)");
   46 @c = ("0","0","0");
   47 @d = ("1","\pi/$ll","1");
   48 
   49 @fs1 = ("y","$l y","y-$l x","y^2");
   50 @fs2 = ("$k x","x^2","$l y-x","$k x");
   51 @fansg = ("($k+1)/2","$l/2+1/3","0","$k/2+1/3");
   52 @fansh = ("($k+2)/3","2*$l/3+1/5","1/3","$k/3+1/2");
   53 
   54 
   55 @Fs1 = ("$l y","$l x","$k y-x","y^2");
   56 @Fs2 = ("$l x","$l y","$k x+y","2xy");
   57 @Fans = ("$l","$l","$k","1");
   58 
   59 @choice1=NchooseK($#gs1,2);
   60 @choice2=NchooseK($#fs1,1);
   61 
   62 @subg1=@gs1[@choice1]; @suba=@a[@choice1]; @subb=@b[@choice1];
   63 @subg2=@gs2[@choice1]; @subc=@c[@choice1]; @subd=@d[@choice1];
   64 @subh1=@hs1[@choice1]; @subh2=@hs2[@choice1];
   65 
   66 @subf1=@fs1[@choice2];@subf2=@fs2[@choice2];
   67 @subF1=@Fs1[@choice2];@subF2=@Fs2[@choice2];
   68 
   69 @subFans=@Fans[@choice2];
   70 @subfansg=@fansg[@choice2];
   71 @subfansh=@fansh[@choice2];
   72 
   73 $Fans = $subFans[0];
   74 $fansg = $subfansg[0];
   75 $fansh = $subfansh[0];
   76 
   77 $g[1] = Formula("$subg1[0]")->reduce; $g[2] = Formula("$subg2[0]")->reduce;
   78 $a = $suba[0]; $b = $subb[0];
   79 $h[1] = Formula("$subh1[0]")->reduce; $h[2] = Formula("$subh2[0]")->reduce;
   80 $c = $subc[0]; $d = $subd[0];
   81 $k[1] = Formula("$subg1[1]")->reduce; $k[2] = Formula("$subg2[1]")->reduce;
   82 $aa = $suba[1]; $bb = $subb[1];
   83 
   84 
   85 $f[1]=Formula("$subf1[0]")->reduce; $f[2]=Formula("$subf2[0]")->reduce;
   86 $F[1]=Formula("$subF1[0]")->reduce; $F[2]=Formula("$subF2[0]")->reduce;
   87 
   88 
   89 foreach $i (1..2){
   90 $Dg[$i] = $g[$i]->D('t')->reduce;
   91 }
   92 
   93 $g = ColumnVector("<$g[1],$g[2]>");
   94 $h = ColumnVector("<$h[1],$h[2]>");
   95 $gg= ColumnVector("<$k[1],$k[2]>");
   96 
   97 
   98 Context()->texStrings;
   99 BEGIN_TEXT
  100 Suppose \(C_1\) is parametrized by
  101 $PAR
  102 \[ \mathbf{g}(t) = $g \ , \quad $a\le t\le $b \ ,\]
  103 $PAR
  104 \(C_2\) is parametrized by
  105 \[ \mathbf{h}(t) = $h \ , \quad $c\le t\le $d \ ,\]
  106 $PAR
  107 and \(C_3\) is parametrized by
  108 \[ \mathbf{k}(t) = $gg \, \quad $aa\le t\le $bb \ .\]
  109 $PAR
  110 Let \(\omega = ($F[1])\,dx + ($F[2])\,dy\) and \(\phi = ($f[1])\,dx + ($f[2])\,dy\).
  111 
  112 $PAR
  113 Then
  114 \(\displaystyle\int_{C_1}\omega =\) \{ans_rule(10)\}
  115 $PAR
  116 \(\displaystyle\int_{C_2}\omega =\) \{ans_rule(10)\}
  117 $PAR
  118 \(\displaystyle\int_{C_1}\phi =\) \{ans_rule(10)\}
  119 $PAR
  120 \(\displaystyle\int_{C_2}\phi =\) \{ans_rule(10)\}
  121 $PAR
  122 \(\displaystyle\int_{C_3}\phi =\) \{ans_rule(10)\}
  123 $PAR
  124 What conclusions do you draw?
  125 
  126 END_TEXT
  127 
  128 ANS(num_cmp($Fans),num_cmp($Fans),num_cmp($fansg),num_cmp($fansh),num_cmp($fansg));
  129 
  130 ENDDOCUMENT();
  131 
  132 
  133 \{ mbox( '\( \mathbf{g}(t)= \)', display_matrix([['\( $g[1] \)'],['\( $g[2] \)'],['\( $g[3]\) ']]),'.') \}