## 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([['$$g[1]$$'],['$$g[2]$$'],['$$g[3]$$ ']]),'.') \}