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# View of /branches/UGA/8.3.2.pg

Sat Jul 24 17:11:33 2010 UTC (2 years, 10 months ago) by ted shifrin
File size: 3308 byte(s)
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    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();
19            "PGbasicmacros.pl",
20            "PGchoicemacros.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([['$$g[1]$$'],['$$g[2]$$'],['$$g[3]$$ ']]),'.') \}