## DESCRIPTION
## Multivariable Calculus
## ENDDESCRIPTION

## KEYWORDS('calculus','differential forms','pullback')


## DBsubject('Calculus')
## DBchapter('Differential Forms and Integration on Manifolds')
## DBsection('Differential Forms')
## Date('December 30, 2009')
## Author('Ted Shifrin')
## Institution('UGA')
## TitleText1()


DOCUMENT();
loadMacros("PG.pl",
           "PGbasicmacros.pl",
           "PGchoicemacros.pl",
           "PGanswermacros.pl",
           "PGauxiliaryFunctions.pl",
           "PGmatrixmacros.pl",
           "Parser.pl",
           );
 

$showPartialCorrectAnswers = 1;


Context("Vector")->variables->are(x=>'Real',y=>'Real',z=>'Real',t=>'Real',
u=>'Real',v=>'Real');


$a = non_zero_random(-3,3,1);
$b = random(1,5);
$c = random(1,5);

@gs1 = ("e^($a t)","$a cos($b t)","$a t");
@gs2 = ("cos($b t)","$a sin($b t)","$b t^2");
@gs3 = ("sin($b t)","$c t","$c t^3");

@hs1 = ("cos($b u)","u cos($b v)","sin(u) cos(v)", "u cos(v)");
@hs2 = ("sin($b u)","u sin($b v)","sin(u) sin(v)", "u sin(v)");
@hs3 = ("$c v","$c v","cos(u)","$b u");

@fs1 = ("y","-y","y^2");
@fs2 = ("z","x","xz");
@fs3 = ("x","z","yz");

@Fs1 = ("z","x","y^2 z");
@Fs2 = ("x","y","x z^2");
@Fs3 = ("y","z","x^2 y");

@choice1=NchooseK($#gs1,1);
@choice2=NchooseK($#hs1,2);

@subg1=@gs1[@choice1]; @subf1=@fs1[@choice1];
@subg2=@gs2[@choice1]; @subf2=@fs2[@choice1];
@subg3=@gs3[@choice1]; @subf3=@fs3[@choice1];

@subh1=@hs1[@choice2]; @subF1=@Fs1[@choice2];
@subh2=@hs2[@choice2]; @subF2=@Fs2[@choice2];
@subh3=@hs3[@choice2]; @subF3=@Fs3[@choice2];

$g[1] = Formula("$subg1[0]")->reduce; $f[1]=Formula("$subf1[0]");
$g[2] = Formula("$subg2[0]")->reduce; $f[2]=Formula("$subf2[0]");
$g[3] = Formula("$subg3[0]")->reduce; $f[3]=Formula("$subf3[0]");

$h[1] = Formula("$subh1[0]")->reduce; $F[1]=Formula("$subf1[0]");
$h[2] = Formula("$subh2[0]")->reduce; $F[2]=Formula("$subf2[0]");
$h[3] = Formula("$subh3[0]")->reduce; $F[3]=Formula("$subf3[0]");

foreach $i (1..3){
$Dg[$i] = $g[$i]->D('t')->reduce;
$Dhu[$i] = $h[$i]->D('u')->reduce;
$Dhv[$i] = $h[$i]->D('v')->reduce;}

$g = ColumnVector("<$g[1],$g[2],$g[3]>");
$h = ColumnVector("<$h[1],$h[2],$h[3]>");

$pb1 = Formula("$Dg[1]*$f[1]+$Dg[2]*$f[2]+$Dg[3]*$f[3]") -> 
   substitute(x=>$g[1],y=>$g[2],z=>$g[3]) -> reduce;
$pb2 = Formula("$Dhu[1]*$f[1]+$Dhu[2]*$f[2]+$Dhu[3]*$f[3]") -> 
   substitute(x=>$h[1],y=>$h[2],z=>$h[3]) -> reduce;
$pb3 = Formula("$Dhv[1]*$f[1]+$Dhv[2]*$f[2]+$Dhv[3]*$f[3]") -> 
   substitute(x=>$h[1],y=>$h[2],z=>$h[3]) -> reduce;
$pb4 = Formula("$F[1]*($Dhu[2]*$Dhv[3]-$Dhu[3]*$Dhv[2])+$F[2]*($Dhu[3]*$Dhv[1]-$Dhu[1]*$Dhv[3])+$F[3]*($Dhu[1]*$Dhv[2]-$Dhu[2]*$Dhv[1])") -> 
   substitute(x=>$h[1],y=>$h[2],z=>$h[3]) -> reduce;



Context()->texStrings;
BEGIN_TEXT
Suppose 
\( \mathbf{g}(t) = $g \).

$PAR
Let \(\omega = $f[1]\,dx + $f[2]\,dy+ $f[3]\,dz\).

$PAR
Then \(\mathbf{g}^*\omega =\) \{ans_rule(20)\} \(dt\) .

$PAR
Suppose
\( \mathbf{h}\left(\begin{array}{c} u\\v \end{array}\right) = $h \).
$PAR
With the same \(1\)-form \(\omega\),
$PAR
\(\mathbf{h}^*\omega = \)\{ans_rule(20)\} \(du\) + \{ans_rule(20)\} \(dv\) .

$PAR
Now consider the \(2\)-form \(\phi = $F[1]\,dy\wedge dz + $F[2]\,dz\wedge dx + $F[3]\,dx\wedge dy\). Then 
$PAR
\(\mathbf{h}^*\phi =\) \{ans_rule(30)\} \(du\wedge dv\) .

END_TEXT

ANS($pb1->cmp,$pb2->cmp,$pb3->cmp,$pb4->cmp);

ENDDOCUMENT();


\{ mbox( '\( \mathbf{g}(t)= \)', display_matrix([['\( $g[1] \)'],['\( $g[2] \)'],['\( $g[3]\) ']]),'.') \}